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Regularity Theory

Lihe Wang

Contents

1 Schauder Estimates 51.1 The Maximum Principle Approach . . . . . . . . . . . . . . . 71.2 Energy Method . . . . . . . . . . . . . . . . . . . . . . . . . . 131.3 Compactness Method . . . . . . . . . . . . . . . . . . . . . . 171.4 Boundary Estimates on Flat Boundaries . . . . . . . . . . . . 231.5 Estimates Near the Boundary . . . . . . . . . . . . . . . . . . 25

1.5.1 Barrier Functions . . . . . . . . . . . . . . . . . . . . . 251.5.2 Hopf Lemma and Slope Estimates . . . . . . . . . . . 261.5.3 Cα Estimates on Lipschitz Domains . . . . . . . . . . 28

1.6 Boundary Estimates on Curved Domains . . . . . . . . . . . 301.7 Patching of Interior Estimates with Boundary Estimates . . 321.8 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2 Lp Estimates 372.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.2 The geometry of functions and sets . . . . . . . . . . . . . . . 38

2.2.1 Geometry of Holder spaces . . . . . . . . . . . . . . . 382.2.2 Lp spaces . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.3 W 2,p Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . 432.4 W 1,p estimates . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3 Harmonic Maps 493.1 Introduction and Statement of Results . . . . . . . . . . . . . 493.2 Weakly Harmonic Maps . . . . . . . . . . . . . . . . . . . . . 503.3 Stationary Harmonic Maps . . . . . . . . . . . . . . . . . . . 533.4 C1,γ Regularity . . . . . . . . . . . . . . . . . . . . . . . . . 56

3

4 CONTENTS

Chapter 1

Schauder Estimates

The main topic of this chapter is to show that, due to Schauder, if

−∆u = f in B1 (1.1)

and if f is Holder continuous, then u is C2,α

loc, i.e. for any 1 ≤ i, j ≤ n,uij is Holder continuous with the same exponent. This is the so-called theSchauder estimates.

We will give three closely related proofs. The most elementary andgeometric one is the maximum principle approach, followed by the energymethod and the most important, flexible compactness method.

Let us examine Schauder’s theorem more closely. Clearly, it is a perfecttheorem: if u was C2,α, its second order derivatives and hence its Laplacianare Cα.

Next let us examine the estimates with respect to scaling:

−∆u(rx)r2

= f(rx)

Hence it is expected that

∆u(rx)− u(0)− r∇u(0) · x

r2≈ f(0).

We see that u(rx)−u(0)−r∇u(0)·xr2 so does the second order derivative satisfies an

equation with right hand side close to a constant and the closeness is gettingbetter as how small the r is. All the proofs are based on this observation.

This leads us to look at the questions how we can say a function is C2,α.Let us recall the definition of continuity: a function is continuous if it is

5

6 CHAPTER 1. SCHAUDER ESTIMATES

more and more close to a constant in a smaller and smaller neighborhood,where the closeness is characterized by the fact that the error, which is thepopular o(1), is smaller than any positive constant. Similarly, the definitionof differentiability goes like that a function is differentiable if it is more andmore close to a linear function in smaller and smaller neighborhoods, wherethe closeness to a linear function is characterized by saying that the error,which is the popular o(|x|), between the function and the linear function issmaller than any linear function.

Now let us recall the definition of Cα continuity. A function u is Cα at0 if

|u(x)− u(0)| ≤ C|x|α.

Here C|x|α is of course smaller than any constants. It is little bit more thancontinuity. It gives a qualitative control of the speed of how the functionapproaches the constant.

It seems puzzling about why we use Cα and C2,α. The reason is that likemost of mathematics, it is hard to say non qualitative things. Continuityis a very unstable process. However Cα is a very stable measurement: anyuniform limit of functions with the same Holder continuity is still enjoyingthe same Holder continuity.

Similarly we can define C1,α, C2,α. A function is C2,α at 0, for example,if there is a second order polynomial P (x) such that

|u(x)− P (x)| ≤ C|x|2+α.

It says that u can be approximated by a second order polynomial and theapproximation is getting better and and better as x goes to 0. P is also theTaylor polynomial at 0 and the main thing is that the qualitative control oferror of the approximation as C|x|2+α.

Now let us examine the difficulty. The main difficulty is that it is hard toget any information about the second order derivatives of the solutions. Theequation says and only says that the sum of some second order derivatives isunder control. This information certainly is not sufficient for us to determinethe second order derivatives. It can be seen later that the second orderderivatives actually depend, given the equation, on the boundary values.However, our problem is not to find them. We just want to show that thesolution has a second order polynomial approximation and to show the orderof approximation is like |x|2+α.

The trivial and psychologically suggestive case is when f = 0. In thiscase u is harmonic and we can take P as the Taylor polynomial of u. We

1.1. THE MAXIMUM PRINCIPLE APPROACH 7

have seen from the previous chapter that any order derivatives of a harmonicfunction are under control.

Now consider the case when f is small. First we need a good guess aboutthe second order polynomial. This leads us to the conclusion if f is small, weshould able to say the solution is almost harmonic. That is the solution isclose to a harmonic function. Then we have the first approximation, whichwe simply take as the second order polynomial of the harmonic function.

Here we has not talk about in what sense that f is small and in whatsense the solution is close to harmonic. Actually it turns out that no mat-ter how we measure the smallness or closeness, we always obtain the sameconclusion at the end. This equivalence is exactly the content of the Morreyor Campanato embedding theorem. All theorems are equivalent.

We will use L∞ norm to measure the smallness when we use the maxi-mum principle approach, H1 for energy approach and L2 for the compact-ness.

1.1 The Maximum Principle Approach

In this section we give the most elementary proof for Schauder estimates.One of the drawback of this approach is that we have to assume solvability ofDirichlet problem and assume that the solution is C2. However, we actuallyshow that if u is C2 than u is C2,α and moreover that its C2,α norm has abound which is independent of of its C2 norm. In other words, we obtain aC2,α aprori estimate.

We can overcome this shortcoming, if we use the existence theory ofviscosity solutions and the corresponding maximum principle. However, indoing that we have to develop a lot of machinery, which we will postponeto later chapters.

Lemma 1.1 Suppose u is a C2 solution of −∆u = f(x) in B1. Then

u(x) ≤ sup∂B1

u+1− |x|2

2nsupB1

(f) for x ∈ B1. (1.2)

Proof. Apply the maximum principle to the subharmonic function

u(x)− sup∂B1

u− 1− |x|2

2nsupB1

(f).


Corollary 1.2 Suppose u is a C2 solution of (4.1) in B1. Then

‖u‖L∞(B1) ≤ ‖u‖L∞(∂B1) +1

2n‖f‖L∞(B1) . (1.3)

Proof. Apply Lemma 1.1 to −u, we obtain,

u(x) ≥ inf∂B1

u+1− |x|2

2ninfB1

(f). (1.4)

Clearly (4.3) follows from (4.2) and (4.4).This Corollary says that the solution is in L∞ norm is controlled by a

multiple of the L∞ norm of its Laplacian and the L∞ of its boundary data.Now we start to prove the C2,α estimates. We will show the discrete

version first.

Lemma 1.3 For any 0 < α < 1, there exist constants C0, 0 < λ < 1,ε0 > 0 such that for any function f and solution of ∆u = f in B1 with|u| ≤ 1 and ‖f‖L∞(B1) ≤ ε0, there is a second order harmonic polynomial

p(x) =12xTAx+Bx+ C

such that|u(x)− p(x)| ≤ λ2+α for |x| ≤ λ

and|A|+ |B|+ |C| ≤ C0,

where C0 is a universal constant.

Let us first explain why we need the above statement. The essense of thetheorem about C2,α regularity of the solution is that the solution is almost asecond order polynomial. We want to show that the solution, after subtracta second order polynomial, decays as C|x|2+α. These are clearly infinitelymany inequalities, for all x ∈ B1.

Now it comes one of the most important techniques in modern analysis.Instead of showing these uncountablely many inequalities, we show theseinequality in countable rings or balls with sizes as a geometric series. Wefirst show that the solution is close to a second polynomial such that theerror, measured only in some smaller ball Bλ, decays as C|x|2+α. The beautylies in that we show this only up to a fixed size of balls, not all the way upto the origin 0. In this way, we reduce the burden of the proof a lot.


Let us explain it further. The following is clearly equivalent:(1) For all x ∈ B1,

|u(x)− P (x)| ≤ C|x|2+α.

(2) For k = 0, 1, . . .,

|u(x)− P (x)| ≤ Cλk(2+α) for |x| ≤ λk.

The key is to show (2) for k = 1 first and reduce the general k to k = 1 byscaling. Mathematically (1) means that the second order Taylor polynomialof u is exactly P at 0. It is very rigid. It is very hard to find such P forgeneral u. The advantage of (2) is that it behaviors well under small errorsin P , where as (1) is rigid. In practice, we will always to prove (2) in thefollowing form,

(3) For k = 0, 1, . . .,

|u(x)− Pk(x)| ≤ Cλk(2+α) for |x| ≤ λk,

provided that we could provide the convergence of Pk, which in general isthe case since the fast convergence of Pk to u.

Pk is ususally born to be convergent or as

|Pk(x)− Pk−1(x)| ≤ 2Cλk(2+α) for |x| ≤ λk.

We will find Pk is an approximation of u in Bλk where f is very muchapproximated by a constant. Once f is also like a constant, the solutionwill close to a harmonic function and (3) is proved by taking Pk as theTaylor polynomial of the harmonic function. This approximation of Pk freesus from the rigidity P and gives us enough freedom so we can find Pk.Then the estimates for Bλk(0) gave better and better approximation. Theseapproximations are improved iteratively, which is much easier to an one stepshot of (1).

This key lemma gives approximation in Bλ. Later we will repeat thesame procedure to estimate Bλ2 , Bλ3 and so on. This requires countablymany steps.

Then there is another beautiful feature of PDEs. They behave nicelyunder scaling. Scaling, we mean to consider the equation in a blow-up thecoordinates

v(x) = u(λkx).

In general we can find a similar equation for v as that for u. The estimatesfrom Bλk to Bλk+1 is exactly the same as those from B1 to Bλ. This scaling


consideration reduces the burden of the proof further from countably manysteps to one step.

Proof of Lemma 1.3 Let h be the harmonic function such that h = u on∂B1. From Corollary 1.2,

|u(x)− h(x)| ≤ 12n

supB1

|f(x)| .

Clearly |h| ≤ 1 in B1(0) by the maximum principle and |u| ≤ 1. Hence itssecond order Taylor polynomial at 0

p(x) =12xTAx+B · x+ C

has universal bounded coefficients and more over it is harmonic.By the mean value theorem, we have for x ∈ B 1

2,

|h(x)− p(x)| ≤ 16|x|3 sup

B 12

(x)|∇3h|.

By Theorem 3.18, we have supB 1

2(x)|∇3h(x)| ≤ C for some universal constant

C. Finally, we get

|u(x)− p(x)| ≤ |h(x)− p(x)|+ 12n

supB1(0)

|f(x)|

≤ C

6|x|3 +

12n

sup |f | .

Now, taking λ small enough, such that the first term is less or equal to

12λ2+α for |x| ≤ λ

and then, taking ε0 such that

12n

supB1(0)

|f | ≤ 12nε0 ≤

λ2+α

2,

the lemma follows.


Theorem 1.4 Suppose u is a C2 solution of (4.1) and suppose f is Holdercontinuous at 0, i.e.,

[f ]Cα(0) = sup|x|≤1

|f(x)− f(0)||x|α

< +∞ .

Then u(x) is C2,α at 0, i.e. there is a second order polynomial P (x) =12x

TAx+B · x+ C such that

|u(x)− P (x)| ≤ D|x|2+α

for |x| ≤ 1, with

|D| ≤ C0

([f ]Cα(0) + |f(0)|+ ‖u‖L∞(B1)

)and

|A|+ |B|+ |C| ≤ C0

([f ]Cα(0) + |f(0)|+ ‖u‖L∞(B1)

),

where C0 is a universal constant depending only on the dimension.

Proof. We first discuss the normalization of the estimates. First, we mayassume f(0) = 0 otherwise, let v(x) = u(x)− f(0)

2n |x|n. Then

∆v = f(x)− f(0).

Clear the estimates of v will translates to that for u. Second, we can assume‖u‖L∞(B1) ≤ 1 and [f ]Cα(0) ≤ ε0, where ε0 is the constant in Lemma 1.3.There are two ways of realizing this normalization.

The linear method: consider

v(x) = ε0u

‖u‖L∞(B1) + [f ]Cα(0).

Clearly we have ‖v‖L∞(B1) ≤ 1 and [∆v]Cα(0) ≤ ε0. Again the estimate of vtranslates to that for u.

The nonlinear method: consider

v(x) =u(rx)

‖u‖L∞(B1) + 1.

Actually, the division of ‖u‖L∞(B1)+1 is also depends on the linear structure.We say this is a nonlinear method because of the scaling u(rx). We will seelater that many nonlinear equations behaves well under this scaling.

1.2. ENERGY METHOD 13

Clearly we proved the (k + 1)-th step by letting

pk+1(x) = pk(x) + λ(2+α)kp0(x

λk) .

Finally, it is elementary to see that Ak, Bk, Ck converge to A∞, B∞, C∞respectively and the limiting polynomial

p∞(x) =12xTA∞x+B∞x+ C∞

satisfies|pk(x)− p∞(x)| ≤ Cλk(2+α)

for any |x| ≤ λk. Hence for λk+1 ≤ |x| ≤ λk,

|u(x)− p∞(x)| ≤ |u(x)− pk(x)|+ |pk(x)− p∞(x)|≤ λk(2+α) + C|x|2+α

=1

λ2+αλ(k+1)(2+α) + C|x|2+α

≤ 1λ2+α

|x|2+α + C|x|2+α

= (1

λ2+α+ C) |x|2+α .

1.2 Energy Method

This section is parallel to §4.1. Since we will use energy method, our methodsin this section apply to weak solutions. Consequently, this yields a regularitytheory for weak solutions, which is stronger than a-priori estimates.

However, it is parallel to the previous section. The only difference be-tween these two sections are the ways we measure the approximations: wemeasure it by L∞ for u and f in the maximum principle approach while weuse H1 semi-norm for u and L2 norm for f in this section.

We see that, by Theorem 1.12 and 1.13, the Holder continuity does notdepend on the way we measure the approximation of u from a second orderpolynomial once we prove the estimates in a domain.

Lemma 1.5 Suppose u is a H1 weak solution of−∆u = f in B1(0)u = 0 on ∂B1(0) .


Then ∫B1

|∇u|2 ≤ C∫B1

f2 .

Lemma 1.6 For any 0 < α < 1, there exist C0, 0 < λ < 1 and ε0 > 0 such

that for∫\B1|∇u|2 ≤ 1 and u is a solution of (4.1), there is a second order

harmonic polynomial

p(x) =12xTAx+B · x+ C

such thatλ2∫\Bλ(0)|∇(u− p)|2 ≤ λ2(2+α)

and |A|+ |B| ≤ C0 a universal constant, provided∫\B1f2 ≤ ε2

0 .

Proof. Let v be the solution of∆v = fv = 0 on ∂B1(0) .

By the previous lemma, we have∫B1

|∇v|2 ≤ C∫B1

|f |2.

Let h = u− v. Then h is harmonic. Clearly, we have∫B1

|∇h|2 ≤ 2∫B1

|∇u|2 + 2∫B1

|∇v|2 ≤ 2|B1|,

and consequently,

|∇h(x)|2 + |∇3h(x)|2 ≤ C∫B1

|∇h|2 ≤ Cfor |x| ≤ 12

and

|∇2p|+ |∇p|(0) ≤ C0 for some universal constant C0. Now, let p(x) be thesecond order Taylor polynomial of h at 0. We have

|∇(h− p)(x)| ≤ C0|x| for |x| ≤ 12.

1.2. ENERGY METHOD 15

Therefore, we obtain,

λ2∫\Bλ|∇(u− p(x))|2dx ≤ 2λ2

∫\Bλ|∇(u− h)|2dx+ 2λ2

∫Bλ

|∇(h− p)|2dx

≤ 2λ2−nC20

∫B1

f2dx+ 2λ2Cλ4|Bλ| .

Now, taking λ small such that the second term is less than or equal to

12λ2(2+α)|Bλ| ,

and then taking ε0 such that the first term is less than or equal to

12λ2(2+α)|Bλ| ,

the lemma follows.

Theorem 1.7 Suppose u is a weak solution of (4.1) and suppose f is Holdercontinuous at 0 in the L2 sense, i.e.

[f ]L2;α(0) =: sup0<r≤1

1rα

√∫\Br|f(x)− f(0)|2 < +∞ ,

where f(0) is defined as f(0) =: limr→0∫\Brf . Then u(x) is C2,α at 0 in H1

semi-norm, i.e.,

[u]H1;2+α

0(0) = inf

P2

sup0<r≤1

r−1−α

√∫\Br|∇(u(x)− P2(x))|2 < +∞,

where P2 is taken over the set of second order polynomials. Moreover the

coefficients of P2 and [u]H1;2+α

0(0) are bounded by [f ]L2;α(0) and

∫B1

|∇u|2dxas

[u]H1;2+α

0(0) ≤ C([f ]L2;α(0) + (

∫B1

|∇u|2dx)12 ).

Proof. It is exactly the translation of the proof of Theorem 1.4, while wehave to use Lemma 1.6 repeatedly. See also the proof of Theorem 1.11.

We can do the normalization as before so that [f ]L2;α(0) is small and∫\B1|∇u|2dx ≤ 1. Now, we prove the following inductively: there are har-

monic polynomials,



such thatλ2k

∫\Bλk|∇(u− pk)|2 ≤ λ2k(2+α)

and


|Bk −Bk+1| ≤ C λ(α+1)k.

k = 0 is the condition on u .

k = 1 is exactly the previous lemma .

Let us assume it is true for k. Let


λ(2+α)k.

Then

∆w =f(λkx)λαk

and∫\B1|∇w|2dx ≤ 1

Now, we apply the previous lemma on w. There is a harmonic polynomialp with bounded derivatives, such that

λ2∫\Bλ|∇(w − p)|2 ≤ λ2(2+α)

provided ∫\B1

|f(λkx)|2

λ2αk≤ ([f ]L2;α(0))2 ≤ ε2

0 .

Now, we scale back,

λ2(k+1)∫\Bλk+1|∇(u(x)− pk(x)− λ(2+α)kp0

( xλk

))|2 ≤ λ2(k+1)(2+α) .


pk+1(x) = pk(x) + λ(2+α)kp(x

λk) .

Finally, it is elementary to see that Ak, Bk converge to A∞, B∞ respectivelyand the limiting polynomial

p∞(x) =12xTA∞x+B∞x

1.3. COMPACTNESS METHOD 17

satisfiesλ2k)

∫\Bλk|∇(Pk − p∞)|2 ≤ Cλ2k(2+α).

Hence for λk+1 ≤ r ≤ λk,

r2∫\Br|∇(u− p)|2 ≤ 2r2

∫\Br|∇(u− Pk)|2 + |∇(Pk − p)|2

≤ 2∫\Bλk|∇(u− Pk)|2 + |∇(Pk − p)|2

≤ (1 + C)λ2k(2+α)

=1 + C

λ2(2+α)λ(k+1)(2+α)

≤ 1 + C

λ2(2+α)r2(2+α) .

1.3 Compactness Method

In this section, we discuss the most important method in nonlinear analysis— the compactness method. It has wide applications. It is an extremelypowerful tool for the regularity theory. This method was discovered by De-Giorgi in the 60’s when he studied Plateau problem. The method presentedhere applies to weak solutions and hence gives a regularity theory, which issomewhat better than a priori estimates.

The big advantage of compactness method is its potential applicationto nonlinear equations. The maximum principle and energy methods havetheir limitations, namely we have to use the solvability of Dirichlet problemsand more seriously, the equation of the difference of the solution and itsapproximation. We will come back to this point when we deal with nonlinearequations.

Lemma 1.8 Suppose −∆u = f in B1. Then for any smooth function ηwith compact support in B1, we have∫

B1

η2|∇u|2dx ≤∫B1

η2f2dx+∫B1

(4|∇η|2 + η2)u2dx .

Proof. This is Caccoippoli type inequality. Multiplying the equation byη2u, we have ∫

η2u(−∆u) =∫η2ufdx .


As before, integrating by parts, we have∫η2|∇u|2 =

∫η2uf − 2

∫η∇η · u∇u

≤ 12

∫η2f2 +

12

∫η2u2 +

12

∫η2|∇u|2 + 2

∫|∇η|2u2 .

Cancelling 12

∫η2|∇u|2, we obtain the inequality immediately.

The following theorem uses the so-called compactness argument. It isalso called the indirect method or blow-up method. The major advantage ofthis method compared with the maximum principle method and the energymethod is that it requires no sovabilty of the Dirichlet problem.

Theorem 1.9 For any ε > 0, there exists a δ = δ(ε) > 0 such that for any

weak solution of −∆u = f in B1 with∫\B1

u2dx ≤ 1 and∫\B1

f2 ≤ δ2, there

exists a harmonic function h such that∫B 1

2

|u− h|2 ≤ ε2 .

Remark Later it will be clear that the argument of this lemma does notdepends on the linear structure of the equation. It can be generalized tosolutions of different equations if the two equation are close.

Proof. We prove it by contradiction. Suppose there exist ε0 > 0, un andfn with

−∆un = fn∫\B1

u2n ≤ 1 ,

∫\B1|fn|2 ≤

1n,

so that for any harmonic function h in B 12,

∫B 1

2

|un − h|2dx ≥ ε20 .

By the previous lemma, ∫B 1

2

|∇un|2dx ≤ C .


Hence un has a subsequence, which we still denoted as un, such that

un → u weakly in H1(B 12) and

un → u strongly in L2(B 12) .

We will show that u itself is harmonic in B 34, which is a contradiction.

Actually, for any test function ϕ ∈ C∞0 (B 34),∫

∇ϕ∇un =∫ϕfndx .

Letting n go to infinity, we have∫∇ϕ∇u = 0 .

Hence u is harmonic!The rest of the proof for C2,α is almost the same as that of the previous

section. We state it as the following.

Lemma 1.10 For any 0 < α < 1, there exist constants C0, 0 < λ <1, δ0 > 0 such that for any function f and solution of ∆u = f in B1

with∫\B1

u2dx ≤ 1 and∫\B1

f2dx ≤ δ20, there is a second order harmonic

polynomial

p(x) =12xTAx+Bx+ C

such that ∫\Bλ|u− p(x)|2 ≤ λ2(2+α)

and|A|+ |B|+ |C| ≤ C0,

where C0 is a universal constant.

Proof. The proof is the same as that of Lemma 1.6. Let h be the harmonicfunction of the previous lemma with some ε < 1 to be determined. Hencewe have ∫

B 12

|h|2 ≤ 2∫B1

|u|2 + 2∫B 1

2

|u− h|2 ≤ 2 + 2ε2 ≤ 4.

By the estimates of harmonic functions, we have

|∇3h(x)|2 ≤ C∫\B1|u|2 ≤ C for |x| ≤ 1

4.


Now, let p(x) be the second order Taylor polynomial of h at 0. We have

|(h− p)(x)| ≤ C0|x|3 for |x| ≤ 14.

Therefore for each 0 < λ < 14 , we have,∫

\Bλ|u− p|2dx ≤ 2

∫\Bλ|u− h|2dx+ 2

∫\Bλ |h− p|

2dx

≤ 2ε2

|Bλ|+ 2C2

0λ6 .

Now, taking λ small such that the second term is less than or equal to

12λ2(2+α),

and then taking δ = δ0 so that ε so small that the first term is less than orequal to

12λ2(2+α) ,

and the lemma follows.

Theorem 1.11 For each 0 < α < 1 and the dimension n, there is a con-stant C0 so that for all weak solution u of −∆u = f in B1 for [f ]L2;α(0) <∞then there exists a second order polynomial P (x) = 1

2xTAx + Bx + C such

that ∫\Br|u− P (x)|2 ≤ C1r

2(2+α)

where the constant C1 depends on∫B1u2dx and [f ]L2;α(0). Moreover

−∆P = f(0)

and

C21 + |A|2 + |B|2 + |C|2 ≤ C0

([f ]L2;α(0)2 + |f(0)|2 +

∫B1

u2dx),

where f(0) is defined as before.

Proof. We can do the normalization as before so that f(0) = 0 and [f ]L2;α(0) ≤δ0 and

∫\B1

u2dx ≤ 1. Now, we prove the following inductively: there are

harmonic polynomials,



such that ∫\Bλk|u− pk|2 ≤ λ2k(2+α),

and


|Bk −Bk+1| ≤ C λ(α+1)k

|Ck − Ck+1| ≤ C λ(α+2)k .

k = 0 is the condition on u .

k = 1 is exactly the previous lemma .

Let us assume it is true for k. Let


λ(2+α)k.

Then

−∆w =f(λkx)λαk

and∫\B1

w2dx ≤ 1

Now, we apply the previous lemma. There is a harmonic polynomial p withbounded coefficients, such that∫

\Bλ|u− p|2 ≤ λ2(2+α)

provided ∫\B1

|f(λkx)|2

λ2αk≤ [f ]L2;α(0)2 ≤ δ2

0 .

Now, we scale back,∫\Bλk+1|u(x)− pk(x)− λ(2+α)kp

( xλk

)|2 ≤ λ(k+1)(2+α) .


pk+1(x) = pk(x) + λ(2+α)kp0(x

λk) .

Finally, it is elementary to see that Ak, Bk, Ck converge to A∞, B∞, C∞respectively and the limiting polynomial

p∞(x) =12xTA∞x+B∞x+ C∞


satisfies|pk(x)− p∞(x)| ≤ Cλk(2+α)

for any |x| ≤ λk. Hence for λk+1 ≤ r ≤ λk,∫\Br|u(x)− p∞(x)|2 ≤ 2

∫\Br|u(x)− pk(x)|2 + |Pk(x)− p∞(x)|2

≤ 2∫\Bλk|u(x)− Pk(x)|2 + |Pk(x)− p(x)|2

≤ (1 + C)λ2k(2+α)

≤ 1 + C

λ2(2+α)r2(2+α) .

We remark that sometimes the following definition is also convenient.

Definition 1

[f ]L2;α(0) = sup0<r≤1

1rα

(∫\Br|f − fBr |

2) 1

2 ,

wherefBr =

∫\Brf.

Now we have showed C2,α estimates by three methods all of which arethe second order polynomial approximation.

We can use the same method to prove Cα, C1,α and any Ck,α estimatesfor the solution if we replace the second order polynomial approximation toconstant, linear and k−th order polynomial approximations.

Definition 2

[f ]L2;−1+α(0) = sup0<r≤1

1r−1+α

(∫\Br|f |2

) 12 .

We remark that we don’t have to normalized by averages or f(0) in ourdefintion. Such average doesn’t converge when [f ]L2;−1+α(0) < ∞. We alsoremark that from the Holder inequality we have that if f is in Lp for p > n,then

[f ]L2;−1+α(0) ≤ ‖f‖Lp(B1),

for α = 1− np .

Let me state two of the theorem that our methods apply.

1.4. BOUNDARY ESTIMATES ON FLAT BOUNDARIES 23

Theorem 1.12 Suppose u is a weak solution of −∆u = f in B1 and[f ]L2,−1+α(0) < ∞, then there exists a linear function L(x) = Bx + C suchthat ∫

\Br|u− L(x)|2 ≤ C0r

2(1+α)

where the constant C0 depends on∫B1u2dx and [f ]L2,−1+α(0). Moreover

C0 + |B|2 + |C|2 ≤ C([f ]2

C−1+α

L2+∫B1

u2dx).

Definition 3

[f ]L2;−2+α(0) = sup0<r≤1

1r−2+α

(∫\Br|f |2

) 12 .

We remark that if f is in Lp for n > p > n2 , then

[f ]L2;−2+α(0) ≤ ‖f‖Lp(B1),

for α = 2− np .

Theorem 1.13 Suppose u is a weak solution of −∆u = f in B1 then thereexists a constant A such that∫

\Br|u−A|2 ≤ C2

0r2α

where the constant C0 depends on∫B1u2dx and [f ]L2;−2+α(0). Moreover

C20 + |A|2 ≤ C

([f ]2L2;−2+α +

∫B1

u2dx).

We also remark that we can prove similar theorems by using [f ]Lp;−2+α(0)for any 1 ≤ p ≤ ∞. In order to carry out the argument the only estimatesthat we need is an estimate of u by the Lp norm of ∆u.

1.4 Boundary Estimates on Flat Boundaries

Let us start with an example of boundary estimates when the boundary isflat. We will use the L∞ approach since it has the best geometric intuitions.


We will denote x = (x′, xn) and Rn+ = x : xn ≥ 0,

Tr = x′ ∈ Rn−1: |x′| < rTr(x′0) = Tr + x′0 for x′0 ∈ Rn−1

B+r = B1(0) ∩Rn

+

B+r (x0) = Br + (x0, 0).

Lemma 1.14 Suppose u in weak solution of ∆u = 0 in B+1 and u = 0 on

T1 and |u(x)| ≤ 1 in C+1 . Then u is smooth in B+

12

. Moreover

|Dαu(x)| ≤ Cα

for x ∈ B+12

.

Proof. The conclusion is trivially proved by the odd extension, which isalso weakly and therefore classically harmonic in B1.

Lemma 1.15 Suppose∆u = 0 in B+

1

|u|L∞(B+1 ) ≤ 1

u(x′, 0) = ϕ(x′) on T1 .

(1.5)

Then there exist constants a and a universal constant C such that

−C|x|2 + infT1

ϕ ≤ u(x)− axn ≤ C|x|2 + supT1

ϕ .

Proof. Let v be the harmonic function (provided by the Poisson integralformula)

−∆v = 0v |T1 = sup

T1

ϕ

v |∂+(B+1 ) = u

.

Clearlyv ≥ u

andu ≥ v − sup

T1

ϕ+ infT1

ϕ .

1.5. ESTIMATES NEAR THE BOUNDARY 25

The lemma follows by applying the previous lemma to the function v −supT1

ϕ and the fact that

∇v(0) = (0, · · · , 0, ∂nv(0)) =: (0, · · · , 0, a).

Corollary 1.16 For any 0 < α < 1 there are universal positive constantsε0 and λ < 1 so that for any solution of

∆u = 0in B+1

|u|L∞(B+1 ) ≤ 1

u(x′, 0) = ϕ(x′) on T1 ,

there are constants a and b such that

|u(x)− axn − b|L∞(B+λ

) ≤ λ1+α

provided oscT1ϕ ≤ ε0.

An iteration of this corollary proves the following theorem.

Theorem 1.17 Suppose −∆u = 0 in B+1 (0) and ϕ(x′) = u(x′, 0) is C1,α at

0. Then u(x) is C1,α at 0. Moreover

[u]C1,α(0) ≤ C([ϕ]C1,α(0) + ‖u‖L∞(B1)

).

We can similarly prove Ck,α, for k > 1, estimates at the boundary.

1.5 Estimates Near the Boundary

In this section, we introduce the basic tools for boundary estimates. Wewill also prove Cα boundary estimates on Lipschitz domain, which is a veryimportant class of domain since they are invarant under scaling.

1.5.1 Barrier Functions

Let us start out this section with some discussion on the most importantfunction for elliptic equations: the Newtonian potential,

p(x) =

1

|x|n−2 x 6= 0 (n ≥ 3)

−ln|x| x 6= 0 n = 2.


As seen in previous chapter,

∆p(x) = −Cnδ0(x) ,

where δ0(x) is the delta function with pole at 0.We will use the following function repeatedly: 0 < r,R <∞ .

hr,R(x) =p(x)− p(R)p(r)− p(R)

.

Clearly, hr,R is harmonic in between BR and Br and

hr,R = 0 on ∂BR and (1.6)hr,R = 1 on ∂Br . (1.7)

It is also clear that, if r < R, hr,R is convex in the radial direction and

∂hr,R∂n

≤ Cr,R < 0 . (1.8)

As showed in the previous section that harmonic functions takes bound-ary data as regularly as the boundary data.

1.5.2 Hopf Lemma and Slope Estimates

The following important theorem (according to tradition we call it Lemma)says that the normal derivative of a harmonic function depends on the val-ues of u far away in a qualitative way. This also tells us the informationbetween the values of a harmonic function is infinitely fast and the first or-der derivative of a harmonic function is lifted if we lift the function a littlebit somewhere else.

Lemma 1.18 (Hopf boundary point lemma) Suppose ∆u = 0 in B1,u ≥ 0 in B1. If u(x0) = 0 for x0 ∈ ∂B1 then u(x) ≥ C|x−x0|u(0) along theradial direction of x0 for some C depending only on the dimension.

Proof. From Harnack inequality, we have

u(x) ≥ cu(0) for |x| ≤ 12.

Hence, we haveu(x) ≥ cu(0)h 1

2,1(x)

by applying the maximum principle to the annulus 12 ≤ |x| ≤ 1. The

lemma follows immediately from the property of h 12,1(x).


Lemma 1.19 Suppose Ω is a domain such that B1 ∩ Ω = ∅ and x0 ∈∂B1 ∩ ∂Ω. Let ∆u = 0 in Ω with u ≤ 1 in Ω with u ≤ 0 on ∂Ω ∩B2. Thenu(x) ≤ C|x − x0| for |x| ≤ 1, where C is a constant depending only on thedimension.

Proof. Applying the maximum principle to the function

u(x)− h2,1(x)

on the domain Ω ∩B2, we have

u(x) ≤ h2,1(x) in Ω ∩B2 .

The lemma follows immediately since h2,1(x) is a smooth function.Lemma 1.18 and 1.19 the control of the bebavior of the harmonic func-

tion near the boundary. One immediate consequence of Lemma 1.19 is thefollowing gradient estimates.

Lemma 1.20 Suppose Ω is a domain such that B1 ∩ Ω = ∅ and x0 ∈∂B1 ∩ ∂Ω. Let ∆u = 0 in Ω with |u| ≤ 1 in Ω with u = 0 on ∂Ω∩B2. Thenif u is differentiable at x0, |∇u(x0)| ≤ C.

Proof. We apply Lemma 1.19 to u and −u to obtain

|u(x)| ≤ C|x− x0| for x ∈ B2 ∩ Ω.

The estimates follows.

A nice application of Lemma1.20 is that it can easier adapted to sit-uations that a domain is a small perturbation from a hyperplane and theboundary is also approximately constatant.

Lemma 1.21 Suppose u ≤ 1 and subharmonic in B3 ∩Ω. If the domain isflat as

B3 ∩ xn ≥ ε ⊂ B3 ∩ Ω ⊂ B+3 ,

then there is a universal constant C so that

u(x) ≤ supB3∩∂Ω

u+ C(√ε+ xn).


Proof. Without lose of generality, we may assume that supB3∩∂Ω = 0. Foreach y ∈ T1 = B1 ∩ xn = 0, we see that B1(y,−1) is outside Ω. Nowlet t be the maximum of 0 ≤ t ≤ 1 so that the balls B1(y, t) is outside Ω.From the flatness condition we see that 0 ≤ t ≤ ε and ¯B1(y, t)∩∂Ω containsat least one point, say y0 with |yo − y| ≤ C

√ε. Now we apply Lemma1.20

(with B1 replaced by B1(y, t)) to obtain

u(x) ≤ C|x− y0|.

Therefore u(0, xn) ≤ C(|xn − (y0)n|+ |y′0|) ≤ C(xn +√ε).

Exercise Consider the domain

Cα = (x, y): angle((x, y), (1, 0)) <π

α .

Suppose ∆u(x) = 0 in Cα ∩B1

u(x) = 0 on (∂Cα) ∩B1 .

Prove that1. u is Lipschitz at 0 if and only if α ≥ 1.2. u is Ck(0) if and only if α is an integer or α ≥ k.

1.5.3 Cα Estimates on Lipschitz Domains

Theorem 1.22 Suppose Ω is a Lipschitz domain. Let 0 ∈ ∂Ω. Supposeϕ(x) is Cα at 0:

[ϕ]Cα(0) = sup0<|x|≤1

x∈∂Ω

|ϕ(x)− ϕ(0)||x|α

< +∞ .

Then any solution of ∆u = 0 in Ωu = ϕ(x) on (∂Ω) ∩B1

is Holder continuous at 0 with exponent β = β0 ∧ α for some β0 dependingon the Lipschitz character of the domain.

The only thing we need to show for Cα regularity is that the oscillationof the solution is smaller if we restrict the solution to a smaller domain.

Proof. Without loss of generality, we may suppose ∂Ω is a Lipschitz graphin the xn direction and Ω is in the x+

n direction. Suppose its Lipschitz


constant is K. From trigonometry,

Br0(0,−1/3) ⊂ Ωc for r0 =1

3√K2 + 1

.

Next, as usual, we may suppose, u(0) = 0, [ϕ]Cα(∂Ω)(0) ≤ ε0 and |u(x)| ≤ 1in B1(0). Applying the maximum principle to the function

u(x)− sup(∂Ω)∩B1

u− hr0,2/3(x− (0,−1/3))

on the domainΩ ∩B2/3((0,−1/3)) ,

we obtainu(x) ≤ sup

∂Ω∩B1

u+ hr0, 23(x− (0,−1

3)) .

By a symmetric argument, we can obtain

u(x) ≥ inf∂Ω∩B1

u− hr0, 23 (x, (0− 13

)) .

Hence|u(x)| ≤ Osc[ϕ] + γ for |x| ≤ 1

6 ,

whereγ = sup

|x|≤ 56

hr0, 23(x) < 1 .

If Osc[ϕ] ≤ 1−γ2 , then

|u(x)| ≤ 1 + γ

2=: γ1 < 1

Now, letting γ2 = max( 16α , γ1) we will prove

supB 1

6k(0)

|u| ≤ γk2

and the theorem follows.We will prove it inductively as

supB 1

6k(0)|u| ≤ osc

B 16k−1

(0)

[ϕ] + γ supB 1

6k−1(0)|u| (1.9)

≤ [ϕ]Cα(0)1

6α(k−1)+ γγk−1

2 (1.10)

≤ ([ϕ]Cα(0) + γ)γk−12 . (1.11)

It follows immediately by if [ϕ]Cα(0) is small enough.


1.6 Boundary Estimates on Curved Domains

A combination of the methods in §1.1 to §1.6 provides different proof forthe following theorems.

Theorem 1.23 Suppose Ω is a Lipschitz domain, 0 ∈ ∂Ω. ϕ is C1,α at 0and ∂Ω is C1,α at 0. Let u be a solution of

∆u = f(x) in B1 ∩ Ωu = ϕ on (∂Ω) ∩B1 .

Then u is C1,α at 0 if [f ]C−1,α(0)<∞. Moreover

[u]C1,α(0) ≤ C([f ]C−1,α(0) + [ϕ]C1,α(0) + ‖u‖L∞(B1∩Ω)

),

for some C depending only on [∂Ω]C1,α and the dimension.

Theorem 1.24 Suppose Ω is a Lipschitz domain, 0 ∈ ∂Ω. ϕ is C2,α at 0and ∂Ω is C2,α at 0. Let u be a solution of

∆u = f(x) in B1 ∩ Ωu = ϕ on (∂Ω) ∩B1 .

Then u is C2,α at 0 if [f ]Cα(0) <∞. Moreover

[u]C2,α(0) ≤ C([f ]Cα(0) + [ϕ]C2,α(0) + [u]L∞(B1∩Ω)

),

for some C depends only on [∂Ω]C2,α and the dimension.

Here we will outline the proof using the compactness method.

Lemma 1.25 If u is a weak solution of∆u = f(x) in B1 ∩ Ωu = ϕ on (∂Ω) ∩B1 .

Then for any η ∈ C∞0 (B1) we have∫B1∩Ω

η2|∇u|2 ≤∫B1∩Ω

η2f2 + C

∫B1∩Ω

(|∇η|2 + η2)|u|2 + C‖ϕ‖2H

12 (∂Ω∩B1)

.

1.6. BOUNDARY ESTIMATES ON CURVED DOMAINS 31

Lemma 1.26 For any ε > 0 there is a δ > 0 so that if u is a solution of∆u = f(x) in B1 ∩ Ωu = ϕ on (∂Ω) ∩B1 .

then there is a harmonic function h in B+12

with h(x′, 0) = 0 so that

∫B 1

2∩Ω|u− h|2 ≤ ε2,

provided the following set of smallness condition∫B1∩Ω

f2 ≤ δ2, ‖ϕ‖2H

12 (∂Ω∩B1)

≤ δ,

and a flatness condition on the boundary as

B1 ∩ Ω ⊂ B+1 and x ∈ B1 : xn ≥ δ ⊂ Ω.

The proof of this is an application of the compactness argument. C2,α andC1,α estimates will follow by approximations as before.

One can also use the maximum principle approach by the estimates inLemma1.21. We prove the key approximation.

Lemma 1.27 Suppose u is a solution of−∆u = f(x) in B3 ∩ Ωu = ϕ on (∂Ω) ∩B3 .

with |u| ≤ 1 and ‖f‖ ≤ δ and the domain is flat as

B3 ∩ Ω ⊂ B+3 and x ∈ B3 : xn ≥ ε ⊂ Ω.

then there is a harmonic function h in B+1 with h(x′, 0) = 0 so that

|u− h| ≤ C(√ε+ δ) in B1 ∩ Ω.

Proof. From Lemma1.21 applied to u± δ2n |x|

2, we see that

|u| ≤ C(xn +√ε+ δ). (1.12)


(We notice that this inequality will be enough for Cα estimates). In order toimprove the estimates, we consider the harmonic h with boundary conditionsas

−∆h = 0in B1

h(x) = C(√ε+ δ) for x ∈ ∂B+

1 \Ωh(x) = u for x ∈ ∂B1 ∩ Ω ,

and by the maximum principle and with estimate 1.12, we have

h ≥ −C1h1, 12(x− (y′,

12

))− C(√ε+ δ) in B1(y′,−1

2) ∩B+

1 ,

for some constant C1 and each |y′| ≤ 1.Therefore h ≥ −C

√ε for xn = ε. We apply the maximum principle to

u− h in the domain B1 ∩ xn ≥ ε to yield the estimates

u ≥ h− 2C(√ε+ δ),

for xn ≥ ε.

1.7 Patching of Interior Estimates with BoundaryEstimates

We only discuss estimates in L∞-norm. For L2 and H1 norms, there arecorresponding patching theorems.

Theorem 1.28 Let S be a set of functions defined on a closed bounded setΩ. Suppose the following hold:

(1) (Boundary Estimates) For any y ∈ ∂Ω, there exists a polynomial Pyof degree k such that

|u(z)− Py(z)| ≤ E|z − y|k+α, z ∈ Ω .

with ∑|σ|≤k

|DσP (z)| ≤ E .

(2) (Interior Estimates) For any x ∈ Ω0, letting dx = d(x, ∂Ω), thenthere exists a polynomial P of degree k such that

|Dσp(x)| ≤ Cd−|σ|x ‖u‖L∞(B dx2

) +Bdk−|σ|−2+αx (1.13)

|(u− px)(z)| ≤(A‖u‖L∞(b dx

2)

dk+αx

+D)|z − x|k+α for |z − x| ≤ dx

2.(1.14)

1.8. EXAMPLES 33

(3) (Invariance) For any u ∈ S, Px in (1), v = u − P also satisfy theestimates of (2).

Then S ⊂ Ck,α(Ω). In fact S ⊂ BM ⊂ Ck+α(Ω) for some M dependingon A, B, C, D and E.

Proof. For any x ∈ Ω0, there exists an y ∈ ∂Ω such that

d(x, y) = dx .

Use (2) for y, we have

|u(z)− Py(z)| ≤ E|z − y|k+α

for z ∈ Ω.Applying (1) to u− Py(z), we have

|u(z)− Py(z)− Px(z)| ≤ (AE +D)|z − x|k+α

for |z − x| < dx2 . Clearly, for |z − x| ≥ dx

2 ,

|u(z)− Py(z)− Px(z)| ≤ E|y − z|k+α + |Px(z)| (1.15)≤ C1(dx)k+α (1.16)≤ C1|z − x|k+α . (1.17)

1.8 Examples

Let us review our technique in this section.

Example 1 Let u be a solution of∆u = f(x) in Ωu |∂Ω= ϕ(x) on ∂Ω .

1. If ϕ ∈ Cα(∂Ω), ∂Ω ∈ C1, ‖f‖L∞ < +∞, then u ∈ Cα(Ω).2. If ϕ ∈ C1,α(∂Ω), ∂Ω ∈ C1,α, f ∈ L∞(Ω), then u ∈ C1,α(Ω).


3. If ϕ ∈ C2,α(∂Ω), ∂Ω ∈ C2,α, f ∈ C2,α(Ω), then u ∈ C2,α(Ω).

Proof. These are immediate consequences of scaled interior estimates, bound-ary estimates and the technique of patching.

Example 2 Let u be a solution of∆u = f(u)u |∂Ω= ϕ(x)

if ‖u‖L∞(Ω) ≤ C and ϕ, f ∈ C∞ then u ∈ C∞(Ω).

Proof. Homework.

Exercise Suppose ∆u = f(x) ∈ L∞(B1)

‖u‖L∞(B1) < +∞ .

Show‖∇u‖L∞(Br) ≤ C

( 11− r

‖u‖L∞(B1) + r‖f‖L∞)

for any 0 ≤ r ≤ 1.

Exercise (Hu and Wang) Let Ω ⊂ B1 be a C1,α domain χΩ be the charac-teristic function of Ω. Suppose

−∆u = χΩ in B1

‖u‖L∞(B1) < +∞ .

Then u ∈ C2,α(Ω) andu ∈ C2,α

loc(B1 − Ω) .

Hint Using‖u‖L1(B1) ≤ C‖∆u‖L1(B1)

if u |∂B1= 0.

Most of the techniques for the Dirichlet problem can be extended toNeumann boundary value problems. However, the following problem is a

1.8. EXAMPLES 35

negative result for mixed value problems.

Exercise Find a function u in B+1 such that

∆u = 0 in B+1

u = 0 on T1 ∩ xn−1 ≤ 0∂u

∂n= 0 on T1 ∩ xn−1 > 0

and u ∈ C12 (B+

12

) but not in C12

+ε(B+12

) for any ε > 0.

Chapter 2

Lp Estimates

In this chapter we will show a proof of the classical Calderon–Zygmundestimates established in [2] 1952. The estimates are among the fundamentalestimates for elliptic equations.

2.1 Introduction

In this paper, we will mainly discuss the theorem,

Theorem 2.1 (Calderon–Zygmund) If u is a solution of

4u = f in B2 (2.1)

then∫B1

∣∣∣D2u∣∣∣p ≤ C (∫

B2

|f |p +∫B2

|u|p)

for any 1 < p < +∞. (2.2)

The analytical tools in this chapter are the maximal function, energyestimates and Vitali Covering Lemma.

We recall the basic notations that we use. Br = x ∈ Rn : |x| < r, Qr =x = (x1, . . . , xn) ∈ Rn : −r < xi < r and Br(x) = Br+x, Qr(x) = Qr+x.For any measurable set A, |A| is its measure. For any integrable function u,we denote the average of u as

uA =∫Au =

1|A|

∫Au.

The classical proof of Calderon–Zygmund estimates, uses the singularintegrals

∂2u

∂xi∂xj(x) =

∫Rn

wij (y) f (x− y) dy (2.3)

37

38 CHAPTER 2. LP ESTIMATES

where wij is a homogeneous function of degree −n with cancellation con-ditions. The approach involves an L2–L2 estimate and an L1 to weak-L1

estimate. See details in the book of Stein [?].Our approach is more elementary. It gives an unified proof for elliptic,

parabolic and subelliptic operators. Our proof is built upon geometricalintuitions. Our basic tools in this approach are the standard estimates forthe equations, the Vitali covering lemma and Hardy–Littlewood maximalfunction.

Our approach is very much influenced by [4] and the early works in[3] and [29], in which the Calderon–Zygmund decompositions were used.Here we will use the Vitali covering lemma. Analytically the differencebetween the Calderon–Zygmund decomposition and Vitali covering lemmais not quite essential but subtle. One is on cubes and the latter is on balls.However we hope that Vitali covering lemma can easily adapted to morecomplicated situations where balls can be easily defined.

2.2 The geometry of functions and sets

2.2.1 Geometry of Holder spaces

We should start out with a geometric description of Holder space which isthe key to visualize the estimates.

First of all, the geometry of ‖u‖L∞(B1) ≤ 1 is that the graph of u is inthe box B1 × [−1, 1]. This gives a very mild control of u.

The Holder norm of u is actually very geometrical. Let us recall that uis Holder with [u]Cα(B1) ≤ 1 if

|u (x)− u (y)| ≤ |x− y|α for all x, y inB1. (2.4)

Geometrically, the graph of u is not a box anymore rather than a surfacewhich is away from spikes: |x|α. That is, if (x0, y0) is on the graph of u,then all the points (x, y) with y − y0 > |x − x0|α or y − y0 < |x − x0|α arenot on the graph.

This actually leads us to say that if u is Cα at x0 if

|u (x)− u (x0)| ≤ C |x− x0|α .

2.2.2 Lp spaces

The information carried by the Lp norm of a function is not as local as bythe Holder norms. In contrast to Holder spaces, there is no way to say afunction is in Lp function at a point.

2.2. THE GEOMETRY OF FUNCTIONS AND SETS 39

In order to examine the information carried by Lp norm of a function,let us recall the formula,∫

Ω|u|p dx = p

∫ ∞0

λp−1 |x ∈ Ω : |u| > λ| dλ. (2.5)

If∫

Ω|u|p dx = 1, one will have that

|x ∈ Ω : |u| > λ| ≤ 1λp, (2.6)

i.e., the measure|x :∈ Ω : |u| > λ|

decays for λ large. This tells us that if we randomly choose a point x, thenthe probability for |u(x)| > λ is small for λ large. The identity (2.5) showsthe decay of |x :∈ Ω : |u| > λ| in a precise way and this decay is the onlyinformation carried by the Lp norm. We also observe that the faster thisprobability decays the bigger the p is.

Now let us discuss how we can show a function is in Lp.First we see that one has to prove the decay of | |u| > λ |. As in the

Holder estimates, we should prove this decay inductively. A reasonableexpectation of this sort is to prove:

||u| > λ0| ≤ ε ||u| > 1| . (2.7)

The smaller ε or λ0 − 1 is, the faster the decay is. Here one should realizethat this estimate should be scaled to

||u| > λ0λ| ≤ ε ||u| > λ| (2.8)

with proper conditions on the data. As in the Holder space case, one shouldexpect that an inductive argument proves the decay.

The W 2,p theory of (2.1) says that D2u is in Lp if ∆u is. Hence areasonable expectation of an inductive estimate could be∣∣∣|D2u| > λ0

∣∣∣ ≤ ε (∣∣∣|D2u| > 1∣∣∣+ ||f | > δ0|

). (2.9)

Here we can scale (2.9) to∣∣∣|D2u| > λ0λ∣∣∣ ≤ ε (∣∣∣|D2u| > λ

∣∣∣+ ||f | ≥ δ0λ|)

(2.10)


which is the so-called good-λ inequality. It is elementary to see that (2.9)implies the estimate. This expectation (2.9), however, is not true. Onereason for the failure of (2.9) is that the condition

|D2u(x0)| ≤ 1 (2.11)

is unstable in the setting of W 2,p theory.Although (2.9) is not true, its modification (2.15) is true.The key modification is provided by one of the treasures in analysis: the

Hardy–Littlewood maximal function.For a locally integrable function v defined in Rn, its maximal function

is defined asMv (x) = sup

r>0

∫Br(x)

|v| dLn. (2.12)

We also useMΩv (x) =M(vχΩ) (x) ,

if v is not defined outside Ω or equivalently we replace or extend v by 0outside Ω. We will drop the index Ω if Ω is understood clearly in thecontext. We can also define the maximal function by taking the supremumin cubes.

Mv (x) = supQr(x)

∫Qr(x)

|v| dLn. (2.13)

It is clear that,Mv ≤ CMv ≤ CMv.

We will use the maximal function Mv defined in (2.12) on balls in thispaper.

The basic theorem for Hardy–Littlewood maximal function is the follow-ing:

Theorem 2.2

‖M (v) (x)‖Lp(Ω) ≤ C ‖v‖Lp(Ω) for any 1 < p ≤ +∞.

|x ∈ Ω :Mv(x) ≥ λ| ≤ C

λ‖v‖L1(Ω).

The first inequality is call strong p−p estimates and the second is call weak1 − 1 estimates. This theorem says that the measures of |v(x)| > λ andMv(x) > λ decay roughly in the same way. HoweverMu(x) ≤ 1 is muchmore stable and geometrical than |u(x)| ≤ 1 if u is merely an Lp function.The reason is that Mu is invariant with respect to scaling. Another aspect

2.2. THE GEOMETRY OF FUNCTIONS AND SETS 41

of the maximal function is that Mu ≥ λ and |u| ≥ λ have roughly thesame measure.

Likewise we will replace (2.11) by(M|D2u|2

)(x) ≤ 1. (2.14)

If M|D2u|2 (x0) ≤ 1, one would see that D2u (x) is really ≤ 1 at x0 in allscales in the sense of L2.

In fact we will show that∣∣∣B1 ∩M|D2u|2 > λ2

0

∣∣∣ ≤ ε (∣∣∣B1 ∩M|D2u|2 > 1

∣∣∣ +|B1 ∩ M(f2) > δ20|)

(2.15)where δ0 can be taken as small as possible since it is about the data.

Now comes the question, how can we prove (2.15)?Let’s first examine a limiting case that

M|D2u|2 > 1

has big mea-

sure, for example |B1 ∩M|D2u|2 > 1

| = |B1|, i.e.,

|M|D2u|2| ≤ 1

has

measure 0 or is a point, say.This further reduces the question to the following.Under what condition on the data f , and M|D2u|2 ≤ 1 at some point,

can we conclude thatM|D2u|2 > λ2

0

has small measure?

Let us again look at a limiting case of this question in which f = 0.However the answer is clear in this case since now u is harmonic. Thecondition that M|D2u|2 ≤ 1 at some point, provides a bound, say N0 onD2u and hence

M|D2u|2 > N2

0

= ∅ which has measure 0.

If f is small, thenM|D2u|2 > N2

0 + 1

will have small measure by thestandard energy estimates. Namely that

(N20 + 1)|x :M|D2u|2 > N2

0 + 1| ≤ C∫|D2u|2 ≤ C

∫|f |2 << 1,

which is exactly what we want.Let us go back to the general case of (2.15).First of all one should understand Lemma 2.7 by its scaling invariant

form: which says that if the density of the setM|D2u|2 > λ0

in a ball B:

|M|D2u|2 > λ0

∩B|

|B|≥ ε,

then B ⊂M|D2u|2 > 1

.

One immediately observes that if one can cover the set by balls (or bycubes as in the Calderon-Zygmund decomposition) so that the density

|M|D2u|2 > λ0

∩B|

|B|= ε


and at the same time these balls are disjoint, then clearly B is 1ε times as

big asM|D2u|2 > λ0

∩ B in this situation. We conclude that (2.15) is

true by taking sum over these balls.Of course the above expectation, that is the set is covered by disjoint

balls with the exact density in each of these balls, is not true. However, it isalmost true. There are two ways of arranging these coverings. One is calledCalderon Zygmund decomposition and the other is called Vitali coveringlemma. The Calderon Zygmund decomposition arranges cubes in an almostoptimal way so that in each cube the density is almost ε and the cubes aredisjoint. In Vitali covering lemma, one can arrange the balls so they aredisjoint and the balls cover a major portion of the set.

Lemma 2.3 (Vitali) Let C be a class of balls in Rn with bounded radius.Then there is a finite or countable sequence Bi ∈ C of disjoint balls such that

∪B∈CB ⊂ ∪i5Bi,

where 5Bi is the ball with the same center as Bi and radius five times big.

We will use the following in this paper.

Theorem 2.4 (Modified Vitali) Let 0 < ε < 1 and let C ⊂ D ⊂ B1 betwo measurable sets with |C| < ε|B1| and satisfying the following property:for every x ∈ B1 with |C ∩ Br(x)| ≥ ε|Br|, Br(x) ∩ B1 ⊂ D. Then |D| ≥

120nε |C|.

Proof. Since |C| < ε|B1|, we see that for almost every x ∈ C, thereis an rx < 2 so that |C ∩ Brx(x)| = ε|Brx | and |C ∩ Br(x)| < ε|Br| forall 2 > r > rx. By Vitali’ covering lemma, there are x1, x2, · · · , so thatBrx1

(x1), Brx2(x2), · · · are disjoint and ∪kB5rxk

(xk) ∩B1 ⊃ C.From the choice of Brxk , we have

|C ∩B5rxk(xk)| < ε|B5rxk

(xk)| = 5nε|Brxk (xk)| = 5n|C ∩Brxk (xk)|.

We also notice that

|Brxk (xk)| ≤ 4n|Brxk (xk) ∩B1|

since xk ∈ B1 and rxk ≤ 2.

2.3. W 2,P ESTIMATES 43

Putting everything together,

|C| = | ∪k B5rxk(xk) ∩ C|

≤∑k |B5rxk

(xk) ∩ C|≤ 5n

∑k ε|Brxk (xk)|

≤ 20n∑k ε|Brxk (xk) ∩B1|

= 20nε| ∪Brxk (xk) ∩B1|≤ 20nε|D|.

This finishes the proof.The reader can compare the above theorem with the following so called

Calderon–Zygmund Decomposition.

Lemma 2.5 (Calderon–Zygmund Decomposition) Let C ⊂ D ⊂ Q1

be measurable such that

1. |C| < ε,

2. if |Q ∩ C| ≥ ε |Q| for a cube then 3Q ⊂ D.

Then |C| ≤ ε |D|.

Notice that this lemma is from a scaling invariant micro-condition to|A| ≤ ε |B|, a global estimate.

The details are provided in the next section.We remark that one can prove Theorem 2.2 using either Theorem 2.4 or

Lemma 2.5.

2.3 W 2,p Estimates

Now we prove Theorem 2.1. We only need to prove it for p > 2 since thestatement for p < 2 follows from the standard duality argument.

The starting point of the estimates is the following classical estimates.See [9], page 317.

Lemma 2.6 If 4u = f in B1,u = 0 on ∂B1,

then ∫B1

∣∣∣D2u∣∣∣2 ≤ C ∫

B1

|f |2 .


Lemma 2.7 There is a constant N1 so that for any ε > 0, ∃ δ = δ(ε) > 0and if u is a solution of (2.1) in a domain Ω ⊃ B4, with

M(|f |2

)≤ δ2

∩M∣∣∣D2u

∣∣∣2 ≤ 1∩B1 6= ∅ (2.16)

then ∣∣∣∣M ∣∣∣D2u∣∣∣2 > N2

1

∩B1

∣∣∣∣ < ε |B1| . (2.17)

Proof. From condition (2.16), we see that there is a point x0 ∈ B1 so that∫Br(x0)

∣∣∣D2u∣∣∣2 ≤ 1 and

∫Br(x0)

|f |2 ≤ δ2, (2.18)

for all Br(x0) ⊂ Ω and consequently we have∫B4

∣∣∣D2u∣∣∣2 ≤ 2n and

∫B4

|f |2 ≤ 2nδ2.

Then ∫B4

∣∣∣∇u−∇uB4

∣∣∣2 ≤ C1.

Let v be the solution of the following equation4v = 0v = u−

(∇u)B4

· x− uB4 on ∂B4.

Then by the minimality of harmonic function with respect to energy in B4,∫B4

|∇v|2 ≤∫B4

∣∣∣∇u−∇uB4

∣∣∣2 ≤ C1.

Now we can use the local C1,1 estimates that there is a constant N0 so that∥∥∥D2v∥∥∥2

L∞(B3)≤ N2

0 . (2.19)

At the same time we have,∫B3

∣∣∣D2 (u− v)∣∣∣2 ≤ C ∫

B4

f2 ≤ Cδ2.

From the weak 1− 1 estimate,

λ|x ∈ B3 :MB3 |D2 (u− v) |2(x) > λ| ≤ C∫B3

∣∣D2 (u− v)∣∣2

≤ C∫B4f2

≤ Cδ2.


Consequently,

|x ∈ B1 :MB3 |D2 (u− v) |2(x) > N20 | ≤ Cδ2.

Now we claim that

x ∈ B1 :M|D2u|2 > N21 ⊂ x ∈ B1 :MB3 |D2(u− v)|2 > N2

0 ,

where N21 = max(4N2

0 , 2n).

Actually if y ∈ B3, then∣∣D2u(y)∣∣2 =

∣∣D2u(y)∣∣2 − 2

∣∣D2v(y)∣∣2 + 2

∣∣D2v(y)∣∣2

≤ 2∣∣D2u(y)−D2v(y)

∣∣2 + 2N20 .

Let x be a point inx ∈ B1 :MB3 |D2(u− v)|2(x) ≤ N2

0

.

If r ≤ 2 we have Br(x) ⊂ B3 and

supr≤2

∫Br(x)

|D2u|2 ≤ 2MB3(|D2(u− v)|2)(x) + 2N20 ≤ 4N2

0 .

Now for r > 2, since x0 ∈ Br(x) ⊂ B2r(x0), we have∫Br(x)

|D2u|2 ≤ 1|Br|

∫B2r(x0)

|D2u|2 ≤ 2n,

where we have used (2.18). This says that M(|D2u|2

)(x) ≤ N2

1 .This establishes the claim.Finally, we have

|x ∈ B1 :M(|D2u|2) > N21 | ≤ |x ∈ B1 :MB3(|D2(u− v)|2 > N2

0 |

≤ CN2

0

∫f2 ≤ Cδ2

N20< Cδ2 = ε |B1| ,

by taking δ satisfying the last identity above. This completes the proof.An immediate consequence of the above lemma is the following corollary.

Corollary 2.8 Assume u is a solution in a domain Ω and a ball B so that4B ⊂ Ω. If |

x :M

(∣∣D2u∣∣2) > N2

1

∩B| ≥ ε |B|, then

B ⊂x :M

(∣∣∣D2u∣∣∣2) (x) > 1

∪ Mf2 > δ2.


The moral of Corollary 2.8 is that the setx :M

(∣∣D2u∣∣2) > 1

is much

bigger than the setx :M

(∣∣D2u∣∣2) > N2

1

modulo M(f2) > δ2 if∣∣∣∣x :M

(∣∣∣D2u∣∣∣2) > N2

1

∩B

∣∣∣∣ = ε |B|

. As said in the previous section, we will cover a good portion of the setx :M

(∣∣D2u∣∣2) > N2

1

by disjoint balls so that in each of the balls the

density of the set is ε. As an application of Corollary 2.8, we will show thedecay of the measure of the set

x :M

(∣∣D2u∣∣2) > N2

1

.

The covering is a careful choice of balls as in Vitali covering lemma.

Corollary 2.9 Assume that u is a solution in a domain Ω ⊃ B4, with thecondition that

∣∣∣x ∈ B1 :M(∣∣D2u

∣∣2) > N21

∣∣∣ ≤ ε|B1|. Then for ε1 = 20nε,

1.∣∣∣B1| ∩

M(∣∣D2u

∣∣2) > N21

∣∣∣ ≤ ε1

(∣∣∣B1 ∩M(D2u

)2 (x) > 1∣∣∣

+∣∣∣B1 ∩

M(|f |2

)> δ2

∣∣∣) .2.∣∣∣B1 ∩

M(∣∣D2u

∣∣2) > N21λ

2∣∣∣ ≤ ε1

(∣∣∣B1 ∩x ∈ B1 :M

(D2u

)2> λ2

∣∣∣+∣∣∣B1 ∩

M(|f |2

)> δ2λ2

∣∣∣) .3.∣∣∣B1 ∩

M(∣∣D2u

∣∣2) > (N21

)k∣∣∣ ≤k∑i=1

εi1

∣∣∣∣B1 ∩M(|f |2

)> δ2

(N2

1

)k−i∣∣∣∣+εk1

∣∣∣B1 ∩M(∣∣D2f

∣∣2) > 1∣∣∣ .

Proof. (1) is a direct consequence of Corollary 2.8 and Theorem 2.4 on

C = B1 ∩M(∣∣D2u

∣∣2) > N21

,

D = B1 ∩M(∣∣D2u

∣∣2) > 1∪M(|f |2

)> δ2

.

(2) is obtained by applying (1) to the equation 4(λ−1u

)= λ−1f .

(3) is an iteration of (2) by λ = N1, (N1)2 , . . ..

Theorem 2.10 If 4u = f in B4 then∫B1

∣∣∣D2u∣∣∣p ≤ C ∫

B6

|f |p + |u|p .

Proof. Without lose of generality, we may assume that ‖f‖p is smalland the measure |x ∈ B1 : M|D2u|2 > N2

1 | ≤ ε|B1| by multiplying the


function by a small constant. We will show that M(∣∣D2u

∣∣2) ∈ Lp2 (B1)

from which it follows that D2u ∈ Lp(B1). Since f ∈ Lp, we have thatM(|f |2

)∈ L

p2 with small norm. Suppose

‖f‖Lp(B4) = δ.

Then+∞∑i=1

(N1)ip∣∣∣M (

|f |2)> δ2 (N1)2i

∣∣∣ ≤ pNp

δp(N − 1)‖f‖pLp(B1) ≤ C‖f‖

pLp(B2) ≤ C.

Hence∫B1

∣∣∣D2u∣∣∣p ≤ ∫

B1

(M(∣∣∣D2u

∣∣∣2)) p2

dx

= p

∫ ∞0

λp−1∣∣∣x ∈ B1 :M|D2u|2 ≥ λ2

∣∣∣ dλ≤ p

(|B1|+

∑+∞k=1 (N1)kp

∣∣∣x ∈ B1 :M∣∣D2u

∣∣2 > (N1)2k∣∣∣)

≤ p(|B1|+

+∞∑i=1

Nkp1

k∑i=1

εi1

∣∣∣x ∈ B1 :M|f |2 ≥ δ2N2(k−i)1

∣∣∣+∑∞k=1N

kp1 εk

∣∣∣x :M(∣∣D2u

∣∣2) ≥ 1∣∣∣)

≤ p

|B1|++∞∑i=1

N ip1 ε

i1

∑k≥i

N(k−i)p1

∣∣∣B1 ∩M|f |2 ≥ δ2N

2(k−i)1

∣∣∣+∞∑k=1

Nkp1 εk1

∣∣∣∣B1 ∩M∣∣∣D2u

∣∣∣2 ≥ 1∣∣∣∣)

≤ C,

if we take ε1 so that Np1 ε1 < 1 and the theorem follows.

We remark that our methods can be adapted to prove the same resultby using the Caldron Zygmund decomposition.

The advantage of Vitali covering lemma is that it holds on any man-ifolds whereas the Caldron Zygmund decomposition requires cubes whichgive clean cut in Euclidean spaces which are rare to find on manifolds.

2.4 W 1,p estimates

One can easily adapt the methods in the preceding section to obtain W 1,p

estimates of the following type:


Theorem 2.11 If

4u = divf =n∑i=1

∂ifi in B1

then ∫B 1

2

|∇u|p ≤ Cp∫B1

|f |p + |u|p for 1 < p < +∞.

Theorem 2.11 is proved by the following elementary energy estimateslemma and the steps as in the previous section.

Lemma 2.12 If 4u = divf in B1,u = 0 on ∂B1,

then ∫B1

|∇u|2 ≤∫B1

|f |2 .

For a proof of this elementary lemma, see [9], page 297.

Chapter 3

Harmonic Maps

3.1 Introduction and Statement of Results

In a recent work we have made some progress in understanding the regular-ity theory of biharmonic maps. Since the techniques that we use are basedon a simplified treatment of the regularity theory for harmonic map, wepresent here the argument for regularity of harmonic maps as an introduc-tion to our work for the more complicated situation of biharmonic maps. Toorient the reader, we briefly recall the basic references to the subject. Forharmonic maps of surfaces, Morrey ([21]), Schoen ([25]), and Helein ([13],[14]) provided the classic regularity results. In higher dimensions, the cor-responding regularity results are due to Hildebrandt-Kaul-Widman ([16]),Schoen-Uhlenbeck ([24]), Evans ([10]) and Bethuel([1]).

First we will present an elementary proof of Helein’s [13] regularity the-ory for weakly harmonic maps from compact surface to spheres. Our proofis more elementary because it does not rely on the structure theory of theHardy spaces ([8]). We will first derive the continuity for harmonic maps ofsurfaces, we will then indicate how our method can be adapted to study sta-tionary harmonic maps when the dimension of the manifold of the domainis greater than two, this again simplifies an earlier result of Evans [10]. Atthe end of the paper we will also give an easy proof of the C1,α regularityof harmonic maps once the solution is continuous.

We would like to remark that, regularity results of Helein [13] and Evans[10] have also been extended to arbitrary target manifolds (in [14] and [1], cfalso the excellent book of Helein [15] for a complete treatment of the theory).At this moment, it is not clear how to extend our method to treat the case ofgeneral target manifolds, we hope to extend this type of argument to cover

49

50 CHAPTER 3. HARMONIC MAPS

the general case.

3.2 Weakly Harmonic Maps

Suppose Mm, Nn are Riemannian manifolds we assume Nn to be isometri-cally embedded in Euclidean space Rk and let u:M → N be a smooth map.Denote its differential by du:TM → TN . In local coordinates xi on Mand ambient Euclidean coordinates uα, the Riemannian metric g on M isrepresented by g = gijdx

idxj ; and we denote du by the matrix(∂uα

∂xi

).

Definition 4 The energy density of u at x ∈M is defined by

e(u) =12|du|2 =

12∂uα

∂xi∂uα

∂xjgij

where gij is the inverse matrix of gij, i.e. gijgjk = δik.

Definition 5 The energy of u is defined by

E(u) =∫e(u) dVol

wheredVol =

√det(gij)dx1 . . . dxm .

Definition 6 A map u:M → N a.e. and du ∈ L2(M,Rk) is harmonic ifit is a critical point of the energy in the sense of calculus of variations, i.e.,if for each smooth deformation of ut such that u0 = u, we have

d

dtE(ut)

∣∣∣t=0

= 0 .

Theorem 3.1 (Helein [13], [14]) Any harmonic map from a surface isHolder continuous.

In the following we shall give a simple proof of above theorem in the casethe target manifold is the standard sphere.

Let us recall the following standard estimates from the linear theory.Denote by B1 the unit ball in Rn. Denote by

∫\B1

f the average integral off over B1.

3.2. WEAKLY HARMONIC MAPS 51

Lemma 3.2 Suppose u is a scalar weak solution of

div(A(x)du) = div(F ) =∑mi=1

∂F i

∂xion B

u = 0 on ∂B1

with λI ≤ A(x) ≤ ΛI and that A(x) is Holder continuous in B1, then forany 1 < q < ∞ there is a constant C depending only on q, the dimensionm, two elliptic constants λ,Λ and ‖A‖Cα(B1) such that

‖du‖Lq(B1) ≤ C‖F‖Lq(B1).

Theorem 3.3 Any harmonic map from a two dimensional disk to sphereSn is Holder continuous.

Proof. We first fix some 1 < q < 2 and denote p = 2q2−q . We want to show

that if E1(u) = 12

∫B1|∇u|2 is small enough, then for some fixed s < 1/2 we

have(∫\Bs |u−A1|p)

1p ≤ 1

2(∫\B1|u−A0|p)

1p , (3.1)

for some constant vectors A0, A1 satisfying

|A0 −A1|p ≤ C∫\B1|u−A0|p. (3.2)

Then a rescaling yields,

(∫\B

sk|u−Ak|p)

1p ≤ 1

2(∫\B

sk−1|u−Ak−1|p)

1p , (3.3)

with|Ak −Ak−1|p ≤ C

∫\B

sk−1|u−Ak−1|p. (3.4)

It then follows from (1.3) and (1.4) that

|Ak −Ak−1| ≤ Cp(12

)k(∫\B1|u−A0|p)1/p ≤ Cp(

12

)kE1[u] (3.5)

for some constant Cp depending only on C and p. Thus the sequence Akconverges exponentially to a vector A, with

|Ak −A| ≤ Cp(12

)kE1[u]. (3.6)

We have also from (1.3)

(∫\B

sk|u−Ak|p)

1p ≤ (

12

)k(∫\B1|u−A0|p)

1p . (3.7)


The Holder continuity of u follows from the decay estimates (3.6) and (3.7).Let us recall the equation of harmonic map to spheres u = (u1, . . . , un)

as−∆uα = uα|du|2. (3.8)

An important observation of Helein is to rewrite this equation when thetarget space is Sn:

−∆uα = uα|du|2 =n∑β=1

m∑k=1

(uαuβk − uβuαk )uβk . (3.9)

Let us assume∫B1|∇u|2 ≤ ε for some ε small. Let 1

2 ≤ r ≤ 1 be such that∫∂Br|u − A0|p ≤ 8

∫B1|u − A0|p for some constant vector A0; it turns out

the proof below works for any constant vector A0. Let h be the harmonicfunction such that u = h on ∂Br. We have

|dh(x)|p ≤ C∫Br|h−A0|p ≤ C(

∫B1

|u−A0|p) (3.10)

for |x| ≤ r4 . From (3.9) we have,

−∆(uα − hα) =n∑β=1

m∑k=1

∂k[(uαuβk − u

βuαk )(uβ −Aβ0 )]. (3.11)

denote E = 12

∫B1|Du|2 as before, then for any fixed 1 < q < 2, p = 2q

2−q wehave from Lemma 3.2 and the Holder inequality that∫

Br|d(u− h)|q ≤ C

∫Br

∑α,β,k[(u

αuβk − uβuαk )(uβ −Aβ0 )]q

≤ C∑α,β,k(

∫Br

[(uαuβk − uβuαk )]2)q2 (∫Br|uβ −Aβ0 |p)

2−q2

≤ CEq/2r (

∫Br

∑β |uβ −A

β0 |p)

2−q2

≤ CEq/2r (

∫B1|u−A0|p)

qp . (3.12)

Now for any s < 14 < r we have, via Sobolev inequality,

1s2

∫Bs|u− h(0)|pdx ≤ 2p−1

s2

∫Bs|u− h|pdx+ 2p−1

s2

∫Bs|h− h(0)|pdx

≤ 2p−1

s2(∫Br|u− h|p) + 2p−1

s2

∫Bs|h− h(0)|pdx

≤ 2p−1rp+2

s2(∫Br|d(u− h)|qdx)

pq + 2p−1Csp supB 1

4

|dh|p

≤ CEq/2rs2

(∫B1|u−A0|p) + Csp(

∫B1|u−A0|p). (3.13)

(3.14)

3.3. STATIONARY HARMONIC MAPS 53

Now, taking s small such that the second term is less than

12p+2

∫\B1|u−A0|p

and then taking E = E1 small such that the first term is less than

12p+2

∫\B1|u−A0|p

so we have by letting A1 = h(0)∫\Bs |u−A1|pdx ≤

12p

∫\B1|u−A0|p.

This proves (3.1). We then observe that for A1 = h(0) =∫\∂Brh =

∫\∂Bru,

|A1 −A0| = |h(0)−A0| = |∫\∂Br(u−A0)|

≤ (∫\∂Br |u−A0|p)

1p

≤ C(∫\B1|u−A0|p)

1p .

Thus (3.2) is satisfied for any constant vector A0. From (3.1) and (3.2), weconclude that u is Holder continuous by the arguments in (3.3) to (3.7).Remark The main point in the above argument is that the right hand sideof (3.11) is the divergence of a quadratic oscillation of u while the left handside of (3.11) is the divergence of the linear oscillation of u. The indices inLp norms in (3.13) are from different sources. The index p on the left is fromthe Sobolev embedding and the index p on the right hand side is from theHolder inequality (3.12). These two match only in dimension 2. In our laterwork on biharmonic maps we again observe a similar matching of indices indimension 4.

We will use BMO semi-norm to study the case when these indices donot match in Section 3.3, where we will have mq

m−q on the left hand side and2q

2−q on the right of the inequality (3.13).

3.3 Stationary Harmonic Maps

In this section we modify the argument in section 3.2 to give an alternativeproof of Evans’ theorem that the singular set of a stationary harmonic mapfrom an m-manifold to a sphere has m− 2 Hausdorff measure zero.


Definition 7 (Stationary Harmonic Maps) Let u be a harmonic map froma manifold M (possible with boundary) to another compact manifold N . Wesay that u is stationary if

d

dtE(u(ϕ(t))) = 0 at t = 0,

where ϕ(t) : M → M is a smooth one parameter family of diffeomorphismsuch that ϕ(0) = identity.

Definition 8 A function f defined on a smooth domain Ω ⊂ Rm is inBMO, i.e. function of bounded mean oscillation, if

||u||BMO(Ω) = supB∫\B|u− uB|dx < +∞ , (3.15)

where uB =∫\Bu, the supremum is taken over all balls with B ⊂ Ω.

We will use the following classical result of John and Nirenberg([18], cf alsoChapter IV of Stein [?]).

Theorem 3.4 For any 1 < p < ∞ there exists an Cp (which depends onlyon p and dimension m) such that if u ∈ BMO(Ω), then

1Cp||u||BMO(Ω) ≤ supB(

∫\B|u− uB|p)

1p ≤ Cp||u||BMO(Ω)

where the supremum is taken over all balls with B ⊂ Ω.

The following monotonicity formula is proved by Schoen-Uhlenbeck ([24]) inthe case of minimizing harmonic maps, and by P. Price ([23]) for stationaryharmonic maps.

Theorem 3.5 A stationary harmonic map from the m-dimensional Eu-clidean disk satisfies the following monotonicity formula: The scaling in-variant energy

E(r) = r2∫\Br|du|2 =

∫\B1|du(rx)|2 .

is monotonically increasing in r for all concentric balls Br = B(x, r) ⊂ B1.

The following regularity result is due to Evans ([10]):

Theorem 3.6 A stationary harmonic map from the m-dimensional Eu-clidean disk to sphere Sn is Holder continuous except a set of m-2 dimen-sional Hausdorff measure zero.

3.3. STATIONARY HARMONIC MAPS 55

Our proof below is patterned after the two dimensional argument. Inhigher dimensions the exponents resulting from the two inequalities (3.12)and (3.13) do not match so we show instead that the BMO norm of the mapdecays. In fact we have to show the decay of the map in every scale. Themonotonicity formula makes the control in every scale possible.

We will show that when E1(u) is small enough, we can choose s small,so that

||u||BMO(Bs) ≤12||u||BMO(B1), (3.16)

Then an iteration of (3.16) yields,

||u||BMO(Bsk

) ≤12k||u||BMO(B1). (3.17)

We start with the equation of a harmonic map u = (u1, . . . , un) withtarget Sn as in (3.8). We also assume

∫B1|du|2 ≤ ε for some ε small

here B1 is the unit ball in Rm. The monotonicity formula asserts Er =r2−m ∫

Br(0) |du|2 ≤ 2ε for all 0 < r < 1. A little reflection will show that forany Br(x) ⊂ B 1

2(0) we also have for some constant Cm:

r2∫\Br(x)|du|

2 ≤ Cmε. (3.18)

Now fix r and x0 with Br(x0) ⊂ B 12(0); we abbreviate Br for Br(x0); and

choose r/2 ≤ r1 ≤ r so that∫∂Br1

|u−A0| ≤ 8∫Br1

|u−A0| (3.19)

here A0 =∫\Br1u.

Let h be the harmonic function such that u = h on ∂Br1 . We have from(3.19):

|dh(x)|p ≤ Cpr−p1

∫\Br1 |u−A0|p

for |x| ≤ r14 . Then from equation (3.11) we have with Er = r2

∫\Br |du|

2, thatexactly as (3.12) that

∫\Br1 |d(u− h)|q ≤ CEq/2r1

(∫\B1|u−A0|

2q2−q

) 2−q2

r1−q. (3.20)


Now taking p = mqm−q , for any s < r1/4, we have, via Sobolev inequality and

the John-Nirenberg’s inequality as in Theorem 3.4.

1sm

∫Bs

|u− h(x0)|pdx ≤ 2p−1

sm∫Br1|u− h|pdx+ 2p−1

sm∫Bs|h− h(x0)|pdx

≤ C 2p−1rp+m1sm

(∫\Br1 |d(u− h)|q

) pq + 2p−1

sm∫Bs|h− h(x0)|pdx

≤ C 2p−1rp+m1sm

(∫\Br1 |d(u− h)|qdx

) pq + C2p−1sp supBr1/4 |dh|

p

≤ C 2p−1rm1sm E

p/2r1

(∫\Br1 |u−A0|

2q2−q

) p(2−q)2q

+ C sp

rp1

(∫\Br1 |u−A0|p

)≤ C rm1

smEp/2r1 (||u||BMO(Br))

p + C sp

rp1(||u||BMO(Br))

p . (3.21)

Now, taking s/r1 ≤ r0 small such that the second term is less than

122p+1

(‖u‖BMO(Br))p

and then taking E small depending only on r such that the first term is lessthan

122p+1

(‖u‖BMO(Br))p

so we have, by taking the supremum over balls B ⊂ Br(x0), for all s ≤ s0 =r1r0 and r sufficiently small that∫

\Bs |u− h(xo)|p ≤1

22p‖u‖pBMO(Br)

. (3.22)

Since ∫\Bs |u−

∫\Bsu|

p ≤ 2p∫\Bs |u− h(x0)|p,

we have ∫\Bs |u−

∫\Bsu|

p ≤ 12p‖u‖pBMO(Br)

. (3.23)

We then vary r and x0 with Br = Br(x0) ⊂ B 12(0). As Br runs over all

balls in B 12(0), Bs runs over all balls in Bs0(0), From this we conclude from

(3.23) that (3.5) holds and this finishes the proof of the theorem.

3.4 C1,γ Regularity

We will prove the following regularity result in this section.

3.4. C1,γ REGULARITY 57

Theorem 3.7 If u is a weakly harmonic map from Mm to Nn for m ≥ 2and u is continuous in an open set in Mm, then u is locally smooth there.

We use compactness argument to establish C1,γ regularity. Then thehigher order regularity of the map follows from the harmonic equation andstandard elliptic theory.

Suppose the u is a weak solution of

∆uα = fα(x,∇u), (3.24)

We would assume also that

|fα(x, P )| ≤ A(1 + |P |2), (3.25)

for some constant A.Our iteration scheme depends on a finer structure of the right hand side

of (3.25) defined as|fα(x, P )| ≤ A(1 + µ|P |2), (3.26)

for some constant A sufficiently small and with some constant µ suitablysmall.Remarks 1. We introduce (3.26) in order to trace the different decay rateof the constant term and the quradratic term in fα.2. Once we know u is continuous and satisfies a system of equations (3.24)and (3.25), then we can define

u1(x) =u(rx)− u(0)

c(r), (3.27)

where c(r) = r + supB1|u(rx) − u(0)|. Then u1 satisfies equation of same

type as that of (3.24) with

fα1 (x, P ) =r2

c(r)fα(rx,

c(r)rP ).

Thus f1 satifies equation of type (3.26) with

|f1(x, P )| ≤ A1(1 + µ|P |2)

with A1 = c(r)12A and µ = c(r)

12 , both can be made arbitrarily small when

r is sufficiently small. Also, the C1,γ estimates of u follows from that of u1.Thus we may assume without loss of generality that the harmonic map inour proof of Theorem 3.7 below satisfies both (3.24) and (3.26). This will


be the only place in our proof where we will use the assumption that u iscontinuous.3. We also remark that the C1,γ regularity theory holds if we replace the∆uα by any elliptic sytems, in particular it covers the case when ∆ is theLaplacian operator on a manifold.

Theorem 3.8 Suppose w is a solution of (3.24) satisfying (3.26) in B1 withµ|w| ≤ C1, and AC1 < 1 then∫

B 12

|∇w|2 ≤ C∫B1

(w2 + 1).

where C = C(C1, A) .

Proof. This is a Caccioppoli type inequality. Choose a smooth cut-off func-tion η with η(x) = η(|x|) of compact support in B1 and η = 1 on B 1

2.

Multiplying the equation (3.24) by η2w and integrate by parts, we have∫η2w(−∆w) =

∫η2wfdx .

As before, we have∫η2|∇w|2 =

∫η2wf − 2

∫η∇η · w∇w

≤∫A|w|η2 +

∫AC1η

2|∇w|2 + ε∫η2|∇w|2 + 1

ε |∇η|2w2

≤∫Aη2(|w|2 + 1) + 1

ε |∇η|2w2 + (C1A+ ε)

∫|η|2|∇w|2 .

Now taking ε small so that C1A + ε < 1, we obtain the inequality immedi-ately.

Lemma 3.9 For any given ε > 0, there is a δ > 0 such that if A ≤ δ then forany solution w of (3.24) with (3.26) satisfying µ|w| ≤ C1 and

∫\B1|w|2 ≤ 1,

there exists some harmonic function h defined in B 12

which approximates win the sense that: ∫

B 12

|w − h|2 ≤ ε2.

Proof. We prove the result by contradiction. Suppose there exist someε > 0, and sequences of wn and fn satisfying

−∆wn = fn(x,∇wn)∫\B1

w2n ≤ 1 ,

|fn| ≤ 1n(1 + µ|∇wn|2).


But for any harmonic function v in B 12, we have∫

|wn − v|2dx ≥ ε .

We then have by (3.4) in the previous lemma,∫B 3

4

|∇wn|2dx ≤ C .

Hence wn has a convergent subsequence, which we still denoted as wn,such that

wn → w weakly in H1 and wn → w strongly in L2(B 34) .

We will show that w itself is harmonic in B 34, which is a contradiction.

Actually, for any test function ϕ ∈ C∞0 (B 34),∫

∇ϕ∇wn =∫ϕfndx .

Thus if we let n go to infinity, we have∫∇ϕ∇w = 0 .

Hence w is actually harmonic. We have thus proved the assertion (3.9).

Corollary 3.10 For any 0 < γ < 1 and C1, there exist some ε > 0 and0 < λ < 1

2 such if A ≤ ε and w is a solution of (3.24) and (3.26) withµ|w| ≤ C1 and

∫B1|w|2 ≤ 1, then there is a linear function l(x) = Bx + C

such that ∫\Bλ|w − l|2 ≤ λ2(1+γ)

and |B|+ |C| ≤ C0 a universal constant.

Proof. Let h be the harmonic function such that∫B 1

2

|w − h|2 ≤ ε2,

as in the statement of Lemma 3.9. By the triangle inequality, we have∫B 1

2

|h|2 ≤ 2∫B1

(|w|2 + 1) ≤ C0 .


Since h is harmonic, we have

|∇2h(x)|2 ≤ C∫B 1

2

|h|2 ≤ CC0 for |x| ≤ 14,

for some constant C=C(m). If we now denote l(x) be the first order Taylorpolynomial of h at 0, we have for λ ≤ 1

4 ,∫\Bλ|w − l|2dx ≤ 2

∫\Bλ|w − h|2dx+ 2

∫\Bλ |h− l|

2dx

≤ Cλ−mε2 + Cλ4.

by (3.4) and (3.4), where C = C(m).Thus we can take λ small so that the second term in (3.4) is less than

12λ2(1+γ) ,

and then taking ε sufficiently small so that the first term in (3.4) is less than

12λ2(1+γ),

we obtain (3.10) in the corollary.We now prove Theorem 3.7.

Proof. (of Theorem 3.7) We first assert that by Remark(2) at the be-ginning of this section, we may assume that the continuous harmonic mapwe have also satisfy that |u(x)| ≤ 1 for x ∈ B1 and system (3.24) withconditions (3.26).

We will prove by induction the following statement (*):There exist some constants C0, 0 < λ < 1

2 and ε > 0 such that for|u| ≤ 1 and u is a solution of (3.24) with (3.26) with A ≤ ε, there are linearfunctions lk(x) = Bk · x+ Ck such that∫

\Bλk|u− lk|2 ≤ λ2(1+γ)k (3.28)

and the constants Bk and Ck satisfy

λk|Bk −Bk+1|+ |Ck − Ck+1| ≤ C0λ(γ+1)k, (3.29)

with C0 a universal constant.Assuming statement (*), we can then argue as in the proof of Theorem

3.3 of section 3.2 that both Bk and Ck converge in an exponentially decay


rate to B and C respectively, and u can be approximated by a linear functionl(x) = B · x+ C satisfying∫

\Bλk|u− l|2 ≤ 2λ2(1+γ)k

for each k. Thus u is in C1.γ by the usual Morrey estimates.To prove the statement (*), we notice that the statement for k = 0

follows from our assumption on u and the statement k = 1 follows fromCorollary 3.10. Thus we need only to establish the inductive step.

We assume statement (*) for up to k. We first observe that from (3.29)that |Bk| = |∇lk| ≤ C0

1−λγ ≤C0

1−2−γ , similarly |Ck| ≤ C01−2−γ .

Define w(x) = (u−lk)(λkx)

λ(1+γ)k . Then w satisfies,

∆wα = fα1 (x,∇w),

wherefα1 (x, P ) = λ(1−γ)kfα(λkx, λγkP +∇lk).

By our assumption that u satisfies (3.25), we have

|f1(x, P )| ≤ Aλ(1−γ)k(1 + 2|∇lk|2) + 2λ(1+γ)kA|P |2.

We verify that w satisfes the conditions of Corollary 3.26. To see this, wehave w satisfy (3.24) and (3.4), with

2|w|λ(1+γ)k = 2(u− lk)(λkx) ≤ 2(1 + |∇lk|+ |Ck|) ≤ 4(1 +C0

1− 2−γ)

and ∫\B1|w|2 = λ−2(1+γ)k

∫\Bλk|u− lk|2 ≤ 1.

Thus we may apply Corollary 3.10 to conclude that there exist some linearfunction l(x) = Bx+ C with |B|+ |C| ≤ C0 some universal bound, and∫

\Bλ|w − l|2 ≤ λ2(1+γ). (3.30)

We now define lk+1 = lk+λ(1+γ)kl( xλk

) and check that inequalities (3.28)and (3.29) in the statement (*) for k+ 1 follow directly from (3.30) and thebounds on l(x). We have thus finished the proof of the inductive step instatement (*) and hence the proof of Theorem 3.7.

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Address of the author:Lihe Wang, Department of Mathematics, University of Iowa, Iowa city, Iowa52240.

[email protected]

Prepared for the Korea Winter School, 2013

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