de giorgi estimates and elliptic equations
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De Giorgi Estimates and Elliptic Equations
Esteban Adam Navas
Department of MathematicsUniversity of California, Riverside
May 18, 2012 / Graduate Student Seminar
Esteban Adam Navas De Giorgi Estimates and Elliptic Equations
Introduction to Elliptic Equations
Elliptic Partial Differential Equations (PDEs) aregeneralizations of Laplace’s Equation, −∆u = 0 andPoisson’s equation, −∆u = f
where ∆ = ∂2
∂x21
+ ∂2
∂x22
+ · · ·+ ∂2
∂x2n
u : U ⊂ Rn → R is a scalar function of n variablesu(x) := u(x1, . . . , xn)
U is an open, bounded subset of Rn
Esteban Adam Navas De Giorgi Estimates and Elliptic Equations
Introduction to Elliptic EquationsGeneral Elliptic PDEs
General Elliptic PDEs look like:
Lu = −n∑
i,j=1
Dj(aij(x)Diu) +n∑
i=1
bi(x)Diu + c(x)u = f (x)
or
Lu = −div(A(x) · Du) + b · Du + c(x)u = f (x)
Notation:
Diu =∂u∂xi
Du = (D1u, · · · ,Dnu)
A(x) = (aij (x)) b(x) = (b1(x), · · · ,bn(x))
Esteban Adam Navas De Giorgi Estimates and Elliptic Equations
Introduction to Elliptic EquationsLu = −Dj(aij(x)Diu) + bi(x)Diu + c(x)u = f (x)
A(x) = (aij(x)) is a symmetric, positive definite coefficientmatrix.This imposes the uniform ellipticity condition onthe operator L, that there exists λ > 0 such that:
n∑i,j=1
aij(x)ξiξj ≥ λ|ξ|2
for all ξ ∈ Rn
That is, A(x) has smallest eigenvalue ≥ λ.
Esteban Adam Navas De Giorgi Estimates and Elliptic Equations
Introduction to Elliptic EquationsLu = −Dj(aij(x)Diu) + bi(x)Diu + c(x)u = f (x)
Our goal is to prove Holder continuity of weak solutions toa special case of the general elliptic equation:b ≡ 0,c ≡ 0,f ≡ 0 and aij ∈ L∞.
This special case is the equation −n∑
i,j=1
Dj(aij(x)Diu) = 0
Esteban Adam Navas De Giorgi Estimates and Elliptic Equations
Introduction to Elliptic EquationsLu = −Dj(aij(x)Diu) + bi(x)Diu + c(x)u = f (x)
Example:
Take A(x) = (δij(x)) = In, b ≡ 0, c ≡ 0.
Esteban Adam Navas De Giorgi Estimates and Elliptic Equations
Weak Solutions to Lu = f
From here on out we deal with the equation
−Dj(aij(x)Diu) + c(x)u = f (x)
b ≡ 0summation convention over repeated indices is used
(e.g. cix i =n∑
i=1
cix i )
Unless aij are differentiable (C1) functions, this equationmakes no sense in the classical sense.If we want to consider more general coefficients, we willneed a more general derivative.
Esteban Adam Navas De Giorgi Estimates and Elliptic Equations
Weak Solutions to Lu = f
From here on out we deal with the equation
−Dj(aij(x)Diu) + c(x)u = f (x)
b ≡ 0summation convention over repeated indices is used
(e.g. cix i =n∑
i=1
cix i )
Unless aij are differentiable (C1) functions, this equationmakes no sense in the classical sense.
If we want to consider more general coefficients, we willneed a more general derivative.
Esteban Adam Navas De Giorgi Estimates and Elliptic Equations
Weak Solutions to Lu = f
From here on out we deal with the equation
−Dj(aij(x)Diu) + c(x)u = f (x)
b ≡ 0summation convention over repeated indices is used
(e.g. cix i =n∑
i=1
cix i )
Unless aij are differentiable (C1) functions, this equationmakes no sense in the classical sense.If we want to consider more general coefficients, we willneed a more general derivative.
Esteban Adam Navas De Giorgi Estimates and Elliptic Equations
Weak Solutions to Lu = fThe Weak Derivative
Suppose u,v ∈ L1loc(U).
v is a weak partial derivative of u if for any testfunction φ ∈ C∞
c (U),∫U
uDiφdx = −∫
Uvφdx
We say v = Diu exists weakly if such v exists.
That is, Diu exists in the weak sense if we can integrate itby parts against any such test function and "move thederivative" to u.
Esteban Adam Navas De Giorgi Estimates and Elliptic Equations
Weak Solutions to Lu = fThe Weak Derivative
Suppose u,v ∈ L1loc(U).
v is a weak partial derivative of u if for any testfunction φ ∈ C∞
c (U),∫U
uDiφdx = −∫
Uvφdx
We say v = Diu exists weakly if such v exists.
That is, Diu exists in the weak sense if we can integrate itby parts against any such test function and "move thederivative" to u.
Esteban Adam Navas De Giorgi Estimates and Elliptic Equations
Weak Solutions to Lu = fThe Weak Derivative
Suppose u,v ∈ L1loc(U).
v is a weak partial derivative of u if for any testfunction φ ∈ C∞
c (U),∫U
uDiφdx = −∫
Uvφdx
We say v = Diu exists weakly if such v exists.
That is, Diu exists in the weak sense if we can integrate itby parts against any such test function and "move thederivative" to u.
Esteban Adam Navas De Giorgi Estimates and Elliptic Equations
Weak Solutions to Lu = fThe Weak Derivative
Example:
Let n = 1 and U = (0,2). Then the weak derivative of
u(x) =
{x , 0 < x ≤ 11, 1 ≤ x < 2
is
v(x) =
{1, 0 < x ≤ 10, 1 < x < 2
Esteban Adam Navas De Giorgi Estimates and Elliptic Equations
Weak Solutions to Lu = fThe Weak Derivative
Example:
Let n = 1 and U = (0,2). Then the function
u(x) =
{x , 0 < x ≤ 12, 1 ≤ x < 2
has no weak derivative.
Esteban Adam Navas De Giorgi Estimates and Elliptic Equations
Weak Solutions to Lu = fThe Weak Derivative
Fact:
Weak partial derivatives are unique.
Esteban Adam Navas De Giorgi Estimates and Elliptic Equations
Weak Solutions to Lu = fThe Sobolev Space H1(U)
Suppose u : U ⊂ Rn → R and u ∈ L1loc(U).
The Sovolev Space H1(U) = W 1,2(U) is the space of allu such that Diu exists weakly and u, Diu ∈ L2(U) for each1 ≤ i ≤ n.
H1(U) = {u ∈ L1loc(U) ∩ L2(U) :
Diu exists weakly and ||Diu||L2(U) <∞}
Esteban Adam Navas De Giorgi Estimates and Elliptic Equations
Weak Solutions to Lu = fThe Sobolev Space H1(U)
Suppose u : U ⊂ Rn → R and u ∈ L1loc(U).
The Sovolev Space H1(U) = W 1,2(U) is the space of allu such that Diu exists weakly and u, Diu ∈ L2(U) for each1 ≤ i ≤ n.
H1(U) = {u ∈ L1loc(U) ∩ L2(U) :
Diu exists weakly and ||Diu||L2(U) <∞}
Esteban Adam Navas De Giorgi Estimates and Elliptic Equations
Weak Solutions to Lu = fDefinition
A weak solution to Lu = −Dj(aij(x)Diu) + c(x)u = f (x)is a function u ∈ H1(U) such that, for all test functionsφ ∈ H1
0 (U),
−∫
UDj(aij(x)Diu)φdx +
∫U
c(x)φdx =
∫U
f (x) dx
⇒∫
Uaij(x)DiuDjφ+ c(x)φdx =
∫U
f (x) dx
Notice that we no longer need differentiability of aij for thisPDE to make sense.
Esteban Adam Navas De Giorgi Estimates and Elliptic Equations
Weak Solutions to Lu = fDefinition
A weak solution to Lu = −Dj(aij(x)Diu) + c(x)u = f (x)is a function u ∈ H1(U) such that, for all test functionsφ ∈ H1
0 (U),
−∫
UDj(aij(x)Diu)φdx +
∫U
c(x)φdx =
∫U
f (x) dx
⇒∫
Uaij(x)DiuDjφ+ c(x)φdx =
∫U
f (x) dx
Notice that we no longer need differentiability of aij for thisPDE to make sense.
Esteban Adam Navas De Giorgi Estimates and Elliptic Equations
Results and RegularityLu = −Dj(aij(x)Diu) + c(x)u = f (x)
Unique weak solutions u ∈ H1(U) to Lu = f exist forgeneral elliptic PDEs.
If aij ∈ C∞(U) then the weak solution is C∞(U) smooth.If aij ∈ L∞(U) then the best regularity we can hope for isHolder continuous with exponent α ∈ (0,1).
that is, supx,y∈B1/2
|u(x)− u(y)||x − y |α
≤ C.
Esteban Adam Navas De Giorgi Estimates and Elliptic Equations
Results and RegularityLu = −Dj(aij(x)Diu) + c(x)u = f (x)
Unique weak solutions u ∈ H1(U) to Lu = f exist forgeneral elliptic PDEs.If aij ∈ C∞(U) then the weak solution is C∞(U) smooth.
If aij ∈ L∞(U) then the best regularity we can hope for isHolder continuous with exponent α ∈ (0,1).
that is, supx,y∈B1/2
|u(x)− u(y)||x − y |α
≤ C.
Esteban Adam Navas De Giorgi Estimates and Elliptic Equations
Results and RegularityLu = −Dj(aij(x)Diu) + c(x)u = f (x)
Unique weak solutions u ∈ H1(U) to Lu = f exist forgeneral elliptic PDEs.If aij ∈ C∞(U) then the weak solution is C∞(U) smooth.If aij ∈ L∞(U) then the best regularity we can hope for isHolder continuous with exponent α ∈ (0,1).
that is, supx,y∈B1/2
|u(x)− u(y)||x − y |α
≤ C.
Esteban Adam Navas De Giorgi Estimates and Elliptic Equations
Results and RegularityLu = −Dj(aij(x)Diu) + c(x)u = f (x)
Unique weak solutions u ∈ H1(U) to Lu = f exist forgeneral elliptic PDEs.If aij ∈ C∞(U) then the weak solution is C∞(U) smooth.If aij ∈ L∞(U) then the best regularity we can hope for isHolder continuous with exponent α ∈ (0,1).
that is, supx,y∈B1/2
|u(x)− u(y)||x − y |α
≤ C.
Esteban Adam Navas De Giorgi Estimates and Elliptic Equations
Theorem (De Giorgi)Special Case c ≡ 0, f ≡ 0, U = B1
Suppose −Dj(aij(x)Diu) = 0 weakly in B1, whereaij ∈ L∞(B1) and ||aij ||∞L (B1) ≤ Λ. Then
supB1/2
|u(x)|+ supx ,y∈B1/2
|u(x)− u(y)||x − y |α
≤ C||u||L2(B1)
Esteban Adam Navas De Giorgi Estimates and Elliptic Equations
Proof:
Step 1: Let u+ = max{u,0}. Then u+ is locally bounded:
supBr/2
u+ ≤ C||u+||L2(Br )
for any r .
Step 2: (Density Theorem) Suppose u is a positivesolution in B2 with |{x ∈ B1; u ≥ 1}| ≥ ε|B1|. Then there isa constant C such that infB1 u ≥ C.Step 3: (Oscillation Theorem) DefineoscBr u = supBr
u − infBr u. Suppose that u is a boundedsolution of Lu = 0 in B2. Then there exists a γ ∈ (0,1)independent of r such that
oscB1/2u ≤ γ oscB1u
Esteban Adam Navas De Giorgi Estimates and Elliptic Equations
Proof:
Step 1: Let u+ = max{u,0}. Then u+ is locally bounded:
supBr/2
u+ ≤ C||u+||L2(Br )
for any r .Step 2: (Density Theorem) Suppose u is a positivesolution in B2 with |{x ∈ B1; u ≥ 1}| ≥ ε|B1|. Then there isa constant C such that infB1 u ≥ C.
Step 3: (Oscillation Theorem) DefineoscBr u = supBr
u − infBr u. Suppose that u is a boundedsolution of Lu = 0 in B2. Then there exists a γ ∈ (0,1)independent of r such that
oscB1/2u ≤ γ oscB1u
Esteban Adam Navas De Giorgi Estimates and Elliptic Equations
Proof:
Step 1: Let u+ = max{u,0}. Then u+ is locally bounded:
supBr/2
u+ ≤ C||u+||L2(Br )
for any r .Step 2: (Density Theorem) Suppose u is a positivesolution in B2 with |{x ∈ B1; u ≥ 1}| ≥ ε|B1|. Then there isa constant C such that infB1 u ≥ C.Step 3: (Oscillation Theorem) DefineoscBr u = supBr
u − infBr u. Suppose that u is a boundedsolution of Lu = 0 in B2. Then there exists a γ ∈ (0,1)independent of r such that
oscB1/2u ≤ γ oscB1u
Esteban Adam Navas De Giorgi Estimates and Elliptic Equations
Proof:(Continued)
Step 4: Holder continuity proven from the OscillationTheorem.
supx ,y∈B1/2
|u(x)− u(y)||x − y |α
≤ C
Esteban Adam Navas De Giorgi Estimates and Elliptic Equations
Thank you for your attention!References:
Evans, Partial Differential EquationsLin, Han, Elliptic Partial Differential Equations
Esteban Adam Navas De Giorgi Estimates and Elliptic Equations
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