viscosity solutions assumptions viscosity solutions to pde on … · 2011-11-02 · level set...

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Classical surface . . .

Development of . . .

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Viscosity solutions to PDE on Riemannian manifolds

October 19, 2007

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1. Eikonal equations

Let Ω be an open bounded subset of Rn. Assume that∂Ω is C1. Then the function v(x) = dist(x, ∂Ω) satisfiesthe equation |∇v| = 1 on a neighborhood of ∂Ω inRn, as well as the boundary condition v = 0 on ∂Ω.However, there is no classical solution to the stationaryeikonal problem

|∇v(x)| = 1, x ∈ Ω, (1.1)

v = 0 on ∂Ω.

There are lots of almost everywhere solutions whichare everywhere differentiable, as shown by Deville andMatheron [12] (on Rn) and Deville and Jaramillo [11](on manifolds)However, the only natural (at least from a geometricpoint of view) solution of this equation should be theLipschitz function dist(·, ∂Ω), which is not everywheredifferentiable.

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On the other hand, the function u(t, x) = t − v(x) =t− dist(x, ∂Ω) satisfies the evolution eikonal equation

ut − |∇u(x)| = 0 (1.2)

on a neighborhood of 0 × ∂Ω in R × Rn, but againthere is no classical global solution. This equation canbe rewritten as

ut

|∇u|= 1, (1.3)

which means that the level sets Γt := x : u(t, x) = 0evolve in time with normal velocity equal to 1, and Γ0 =∂Ω. Again, there is no global solution to this equationand the only natural solutions should be those whichsatisfy the condition u(t, ·)−1(0) = x : dist(x, ∂Ω) =t (and hence cannot be everywhere differentiable).

Equation 1.3 is a very special example of a more generalclass of surface evolution equations.

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2. Classical evolution of hypersurfaces. Examples

In the last 30 years there has been a lot of interest inthe evolution of hypersurfaces of Rn by functions of theircurvatures. In this kind of problem one is asked to finda one parameter family of orientable, compact hyper-surfaces Γt which are boundaries of open sets Ut andsatisfy

V = −G(ν, Dν) for t > 0, x ∈ Γt, and (2.1)

Γt|t=0 = Γ0

for some initial set Γ0 = ∂U0, where V is the normalvelocity of Γt, ν = ν(t, ·) is a normal field to Γt at eachx, and G is a given (nonlinear) function.

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Two of the most studied examples are the evolutions by

• mean curvature (when one takes V = H := div ν,the sum of all principal curvatures in the directionof ν); and by

• Gaussian curvature (when V = K is the Gaussiancurvature of Γt, that is the product of all principalcurvatures in the direction of ν).

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In both cases (mean and Gaussian curvature), shorttime existence of classical solutions has been estab-lished.For strictly convex initial data U0, it has been shownthat Ut shrinks to a point in finite time, and moreover,Γt looks more and more spherical at the end of the con-traction.This has been done by B. Andrews, M. Gage, M.A.Grayson, R. Hamilton, G. Huisken, K. Tso and others.See [2, 18, 23, 24, 26, 27, 39].

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In the case n = 2, it has moreover been proved (byGrayson) that (not necessarily convex) embedded planecurves Γt moving by their mean curvatures become con-vex in finite time, and afterwards they shrink to a point(with round limiting shape).This is not true in the case of surfaces (n ≥ 3).

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The classical motion of hypersurfaces by their mean cur-vatures has also been studied in the setting of Rieman-nian manifolds M of dimension n.When n = 2, the mean curvature flow Γt is sometimescalled curve shortening, because the flow lines in thethe space of closed curves are tangent to the gradientfor the length functional; this means that the curve isshrinking as fast as it can using only local information.

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Curve shortening processes can be used to find embed-ded geodesics on surfaces, especially spheres. For in-stance, Grayson [24] proved that if Γ0 : S1 → M isa smooth curve embedded in M , then Γt : S1 → Mexists for t ∈ [0, t∞) satisfying

∂C

∂t= kν,

where k is the curvature of C and ν is its unit normalvector. If t∞ is finite then C(t) converges to a point. Ift∞ is infinite, then the curvature of C converges to 0 inthe C∞ norm (and one expects to find a geodesic in thelimit).

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Grayson used this result to give a rigorous proof of theLusternik-Schnirelmann theorem: a 2-sphere with asmooth Riemannian metric has at least three sim-ple closed geodesics.

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3. Development of singularities. Why not allow non-smooth initial data?

For dimension n ≥ 3 it has been shown [22] that a hy-persurface evolution Γt may develop singularities beforeit disappears.Grayson’s example consists of two disjoint spherical sur-faces connected by a sufficiently thin neck. The inwardcurvature of the neck is so large that it will force theneck to pinch before the two spherical ends can shrinkappreciably.

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Other examples: a thin torus shrinks to a ring. A fattorus becomes singular before it can shrink to a point.

Hence it is natural to try to develop weak notions ofsolutions to (2.1) which allow to deal with singularitiesof the evolutions, and even with nonsmooth initial dataΓ0.

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There are two mainstream approaches concerning weaksolutions of (2.1): the first one uses geometric measuretheory to construct (generally nonunique) varifold solu-tions [6, 29], while the second one adapts the theory ofsecond order viscosity solutions developed in the 1980’s[9] to show existence and uniqueness of level-set weaksolutions to (2.1).In this talk we will focus on this second approach, whichwas initiated in 1991 by L.C. Evans and Spruck [15] inthe case of the mean curvature evolution, and indepen-dently by Y.G. Chen, Y. Giga and S. Goto [7] for moregeneral equations (but not including the Gaussian cur-vature evolution).

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4. Level set method. Equations

A smooth function u : [0, T ] × Rn → R with Du :=Dxu 6= 0 has the property that all its level sets evolveby (2.1) if and only if u is a solution of

ut + F (Du, D2u) = 0, (4.1)

where F is related to G in (2.1) through of the followingformula:

F (p, A) = |p|G(

p

|p|,

1

|p|

(I − p⊗ p

|p|2

)A

). (4.2)

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Example 1. Mean curvature evolution. If u is afunction on [0, T ]×Rn such that Du(t, x) 6= 0 for all t, xwith u(t, x) = c, then each level set Γt = u(t, ·) = cevolves according to its mean curvature if and only if usatisfies

ut

|Du|= div

(Du

|Du|

),

which in turn is equivalent to ut + F (Du, D2u) = 0,where

F (ζ, A) = −trace

((I − ζ ⊗ ζ

|ζ|2

)A

). (4.3)

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Example 2. Gaussian curvature evolution.Now, if u is a function on [0, T ] × Rn such thatDu(t, x) 6= 0 for all t, x with u(t, x) = c, then alllevel sets Γt = u(t, ·) = c evolve according to theirGaussian curvature if and only if u satisfies

ut

|Du|= det

(DT

(∇u

|∇u|

)),

where DT stands for the orthogonal projection onto TΓt

of the derivative in Rn. This equation is equivalent tout + H(Du, D2u) = 0, where

H(ζ, A) = −|ζ|det

((I − ζ ⊗ ζ

|ζ|2

)A +

ζ ⊗ ζ

|ζ|2

).

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The level set method consists of the following steps

• For a given initial hypersurface Γ0 which is theboundary of a bounded open set Ut we take anauxiliary function u0 which is at least continuousand bounded and such that Γ0 = u−1

0 (0), U0 =u−1

0 (0,∞).

• We solve (in some weak sense) the initial value prob-lem

ut + F (Du, D2u) = 0, u(0, x) = u0(x)

• We then set Γt = u(t, ·)−1(0), Γt = u(t, ·)−1(0,∞),and hope that Γt is a generalized solution in somesense. In particular Γt should equal the classicalmotion whenever the latter exists, and should al-ways be independent of the function u0 chosen torepresent Γ0.

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Because equation (4.1) is very singular (meaning that Fis undefined at Du = 0, admits no continuous extensionto Rn × Rn2

, and in general F (p, A) is not bounded asp → 0), classical PDE theory and Sobolev spaces arenot useful to solve the problem.

It turns out that an adequate notion of solution of (4.1)is that of viscosity solution.

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5. Viscosity solutions to second order PDE

Second order subdifferentials, parabolic semi-jets.Let f : (0, T )× Rn → (−∞, +∞] be a lower semicon-tinuous (LSC) function. We define the parabolic secondorder subjet of f at a point z0 = (t0, x0) ∈ (0, T )×Rn

by

P2,−f (z0) = (Dtϕ(z0), Dxϕ(z0), D2xϕ(z0)) :

ϕ ∈ C2, f − ϕ attains a local minimum at z0.

Similarly, for an upper semicontinuous (USC) functionf : (0, T )× Rn → [−∞, +∞), we define the parabolicsecond order superjet of f at (z0) by

P2,+f (z0) = (Dtϕ(z0), Dxϕ(z0), D2xϕ(z0)) :

ϕ ∈ C2, f − ϕ attains a local maximum at z0.

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Standard notion of viscosity solution. Intro-duced by M. Crandall and P.-L. Lions in the 1980’s.

Let F : (0, T ) × Rn × R × Rn × Rn2

be a continuousfunction. Consider the equation

ut + F (t, x, u,Du, D2u) = 0, (5.1)

where u is a function of (t, x). We say that an uppersemicontinuous function u : (0, T ) × Rn → R is a vis-cosity subsolution of (5.1) provided that

a + F (t, x, u(t, x), ζ, A) ≤ 0

for all (t, x) ∈ (0, T )× Rn and (a, ζ, A) ∈ P2,+u(t, x).

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Similarly, a viscosity supersolution of (5.1) is a lowersemicontinuous function u : (0, T )×Rn → R such that

a + F (t, x, u(t, x), ζ, A) ≥ 0

for every (t, x) ∈ (0, T ) × Rn and (a, ζ, A) ∈P2,−u(t, x). If u is both a viscosity subsolution and aviscosity supersolution of ut + F (t, x, u,Du,D2u) = 0,we say that u is a viscosity solution.

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Here is some motivation for this definition. Assume thatF is elliptic (that is decreasing in the variable D2u).

If u is a classical solution then we have ut(z) +F (Du(z), D2u(z)) = 0 for all z = (t, x). Then, if ϕ issuch that u−ϕ attains a minimum at z, we have ϕt(z) =ut(z), Dϕ(z) = Du(z), and D2u(z) ≥ D2ϕ(z). SinceF is elliptic we get

ϕt(z) + F (z, u(z), Dϕ(z), D2ϕ(z)) ≥ut(z) + F (z, u(z), Du(z), D2u(z)) = 0,

that is, u is a supersolution at z. A similar argumentshows that u is a subsolution.

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Therefore every classical solution of ut+F (Du, D2u) =0 is a viscosity solution.

This is no longer true if F is not elliptic, as shown bythe example: u(t, x) = t + x2 − 2, which is a classicalsolution of ut +u+uxx−x2−1 = 0, but not a viscositysolution.

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Refined notion of viscosity solution. Introducedby H. Ishii and P. Souganidis in 1995 [32].

Due to the singularity of F in (4.1), the standard notionof viscosity solution does not make sense for functions ϕat points z0 = (t0, x0) such that Dϕ(z0) = 0 and u−ϕattains a maximum or a minimum at z0.In order to deal with this difficulty, Ishii and Souganidissuggested to make the class A(F ) of test functions ϕsmaller, in a clever, rather technical way which we willnot detail here, so that, at a point z0 = (t0, x0) whereDϕ(z0) = 0, one can show that

limz→z0

F (Dϕ(z), D2ϕ(z)) = 0,

and then to demand that a subsolution u of (4.1) shouldsatisfy that if u−ϕ has a maximum at z0 and Dϕ(z0) =0 then

ϕt(z0) ≤ 0.

A similar condition is required of u to be a supersolu-tion.The class A(F ) of test functions is a proper subset ofC2, but remains dense in C0.

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6. Assumptions on F

The function F is assumed to be of the form

F (ζ, A) = |ζ|G(

ζ

|ζ|,

1

|ζ|

(I − ζ ⊗ ζ

|ζ|2

)A

), (6.1)

where G is any (nonlinear) function such that:

(A) F is continuous off ζ = 0.(B) F is elliptic, that is,

P ≤ Q =⇒ F (ζ, Q) ≤ F (ζ, P ).

Besides, because(I − ζ⊗ζ

|ζ|2

)(ζ ⊗ ζ) = 0, any function

F of the form (6.1) also satisfies

(C) F is geometric, that is,

F (λζ, λA + µζ ⊗ ζ) = λF (ζ, A)

for every λ > 0, µ ∈ R.

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Examples of F ’s satisfying (A-C).1. Mean curvature evolution equation forlevel sets.2. Positive Gaussian curvature evolutionequation for level sets. The Gaussian curvatureevolution equation is given by ut + H(Du, D2u) = 0,where

H(ζ, A) = −|ζ|det

((I − ζ ⊗ ζ

|ζ|2

)A +

ζ ⊗ ζ

|ζ|2

).

However, the function H is not elliptic, so this problemcannot be treated, in its most general form, with thetheory of viscosity solutions. Nevertheless, if our initialdata u(0, x) = g(x) satisfies that D2g(x) ≥ 0 (that is,if the initial hypersurface Γ0 = x ∈ M : g(x) = chas nonnegative Gaussian curvature) then it is reason-able, and consistent with the classical motion of convexsurfaces by their Gaussian curvature, to assume thatD2u(t, x) ≥ 0 for all (t, x) with u(t, x) = c (that is, Γt

will have nonnegative Gaussian curvature as long as itexists).

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In this case our equation becomes

ut + F (Du, D2u) = 0,

with

F (ζ, A) = −|ζ|det+

(1

|ζ|

(I − ζ ⊗ ζ

|ζ|2

)A +

ζ ⊗ ζ

|ζ|2

),

and where

det+(A) =

n∏j=1

maxλj, 0

if λ1, ..., λn are the eigenvalues of A.This F is elliptic and satisfies (A-C). This equationmodels the wearing process of a (nonconvex) stone atthe seashore [31].

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7. Existence and uniqueness

Under assumptions (A-C) it is possible to show thatthere is a unique level set evolution of Γ0, that is aunique family of sets Γt, Dt, of the form Γt = x :u(t, x) = 0, Dt = x : u(t, x) > 0, where u is aviscosity solution of

ut + F (Du, D2u) = 0

defined on [0,∞)× Rn.

The proof is done it two steps: first one establishes acomparison principle (this is the hardest part of the the-ory, the proof involves deep results of real analysis andconvex functions). Then one may construct a solutionby combining the so-called Perron’s method, compar-ison, and stability of the equation (that is, limits ofsubsolutions are subsolutions, etc), as follows.

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Comparison:Let u, v be a viscosity subsolution and supersolution,respectively, of ut+F (Du, D2u) = 0, defined on [0, T )×Ω. Assume that u ≤ v on the parabolic boundary0 × Ω ∪ [0, T )× ∂Ω. Then u ≤ v on [0, T )× Ω.

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Stability

Lemma 1. Assume that u.s.c. (respectively l.s.c.)functions uk are subsolutions (supersolutions, re-spectively). Assume also that uk converges locallyuniformly to a function u. Then u is subsolution(supersolution, respectively).

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Perron’s method:Assume comparison holds for the equation

ut + F (Du, D2u) = 0u(0, x) = g(x).

(7.1)

Let u and u be a subsolution and a superso-lution of (7.1), respectively, satisfying u∗(0, x) =u∗(0, x) = g(x). Then w = supv : u ≤ v ≤u, v is a subsolution is a solution of (7.1).

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Now we can prove existence of solutions to the surface evolution problem asfollows. Assume that comparison holds for the equation

ut + F (Du, D2u) = 0u(0, x) = g(x).

(7.2)

Take a continuous function g such that g−1(0) = Γ0. Since Γ0 is compact we canassume that g equals some positive constant outside a closed ball B0 containingΓ0.

Let us first produce a solution of (7.2) for g ∈ A(F ).Let us define u(t, x) = −Kt + g(x), and u(t, x) = Kt + g(x), where K :=

supx∈B |F (Dg(x), D2g(x))| (which is finite because g ∈ A(F ) and B is compact).It is immediately seen that u is a subsolution and u is a supersolution of (7.2),and obviously u∗(0, x) = u∗(0, x) = g(x). According to Perron’s method andcomparison, there exists a unique solution u of (7.2).

Now take g a continuous function. It is easy to check that the class A(F )of admissible test functions satisfies the hypotheses of the Stone-Weierstrasstheorem, hence it is dense in the space of continuous functions. Therefore wecan find a sequence gk of functions in A(F ) such that gk → g uniformly onbounded sets. Let uk be the unique solution of (7.2) with initial datum gk. Bycomparison, for any ball B ⊃ B0, (uk) is a Cauchy sequence in C([0,∞) × B),hence it converges to some u ∈ C([0,∞) × Rn) uniformly on bounded sets. Bystability, it follows that u is a solution with initial datum u(0, x) = g(x).

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8. Geometric consistency & Generalized evolution

Condition (C) (geometricity) and stability imply:

Theorem 2. Let θ : R → R be a continuous func-tion, and let u be a bounded continuous solution.Then v = θ u is also a solution.Moreover, if θ is nondecreasing and u is a subsolu-tion (resp. supersolution) then v = θ u is a subso-lution (resp. supersolution) as well.

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Let g be a continuous function on M with Γ0 = x ∈M : g(x) = 0, and assume that Γ0 is compact. Wemay also assume that g is constant outside a boundedneighborhood of Γ0, and in particular bounded. Let ube the unique solution of (4.1) with u(0, ·) = g. Wedefine

Γt = x ∈ M : u(t, x) = 0.Theorem 3. Assume that comparison and existencehold for (4.1). Let g : M → R be a continuous func-tion satisfying Γ0 = x ∈ M : g(x) = 0 and suchthat g is constant outside a bounded neighborhood ofΓ0. Let u be the unique continuous solution of (4.1)with initial condition g. Then

Γt = x ∈ M : u(t, x) = 0.

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Corollary 4. The definition of Γt = x ∈ M :u(t, x) = 0 does not depend upon the particularchoice of the function g satisfying Γ0 = x ∈ M :g(x) = 0.It can also be checked that the evolution Γ0 7→K(t)Γ0 := Γt thus defined has the semigroup property

K(t + s) = K(t)K(s).

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9. Consistency with the classical motion

Assume Γt is a classical evolution of Γ0. More precisely, suppose (Γt)t∈[0,T ] is afamily of smooth, compact, orientable hypersurfaces in Rn evolving according toa classical geometric motion, locally depending only on its normal vector fieldsand second fundamental forms. In particular, we assume that Γt is the boundaryof a bounded open set Ut ⊂ Rn and that there exists a family of diffeomorphismsof manifolds with boundary

φt : U0 → Ut, t ∈ [0, T ] ,

such that:

(i) φ0 =Id, and,

(ii) for every x ∈ Γ0 the following holds:

d

dtφt (x) = G

(ν

(t, φt (x)

),∇Γν

(t, φt (x)

)), (9.1)

where ν (t, ·) is a unit normal vector field to Γt, and the linear map

∇Γν (t, x) : (TΓt)x 3 ξ 7→ ∇Tξ ν (t, x) ∈ (TΓt)x

and ∇T stands for the orthogonal projection onto (TΓt)x of the derivativein Rn.

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Classical motion by mean curvature corresponds to taking G(ν,∇Γν

)=

tr(−∇Γν

)ν, whereas classical motion by Gaussian curvature is defined by G

(ν,∇Γν

)=

det(−∇Γν

)ν. The level set evolution equation induced by (9.1) is of the form

(4.1) where F is related to G through formula (6.1). As before, we assume thatF is continuous, elliptic, translation invariant and geometric.

Define d : [0, T ]×M → R, the signed distance function from Γt, as:

d (t, x) :=

dist (x,Γt) if x ∈ Ut

−dist (x,Γt) if x ∈ M \ Ut.

Theorem 5. Let u be the unique viscosity solution to the level set equation (4.1)with initial datum u|t=0 = d|t=0. Then, for every t ∈ [0, T ], the zero level set ofu (t, ·) coincides with Γt:

Γt = x ∈ M : u (t, x) = 0 .

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10. Why not make it on manifolds?

In the case of the F corresponding to the mean curva-ture evolution, this theory has been extended to Rie-mannian manifolds by T. Ilmanen [28].

Recently, in a joint work with M. Jimenez-Sevilla and F.Macia [5], we have extended this theory to Riemannianmanifolds of nonnegative curvature for all F ’s such thatF is continuous off Du = 0, elliptic, geometric, andlocally invariant by parallel translation. This includesthe case of the (positive) Gaussian curvature evolution.

When M is not of nonnegative curvature, the same re-sults hold if one additionally requires that F is uni-formly continuous with respect to D2u (but this ex-cludes the motion by positive Gaussian curvature).

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Open problem. Is there a (well defined, unique,consistent) generalized evolution of level sets by theirGaussian curvatures in Riemannian manifolds of nega-tive curvature?

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11. Counterexamples and conjectures in the settingof Riemannian manifolds

Several well known properties of the evolutions in Rn

are no longer true in the case when the Riemannianmanifold M has negative sectional curvature.

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Example 6. When (M, g) is the Euclidean space equipped with the canonicalmetric, Ambrosio and Soner have proved in [1] that the distance function |d| isalways a supersolution to the mean curvature equation for level sets. This is nolonger the case of a general Riemannian manifold.Conjecture: If M has nonnegative sectional curvature then |d| is always asupersolution. If M has negative curvature then there always exists Γ0 suchthat |d| is not a supersolution.

Example 7. When M = Rn, Evans and Spruck [15, Theorem 7.3] showed thatif Γ0, Γ0 are compact level sets and Γt, Γt are the corresponding generalizedevolutions by mean curvature, then

dist(Γ0, Γ0) ≤ dist(Γt, Γt)

for all t > 0. This result fails for manifolds of negative curvature. For instance,let M = (x, y, z) ∈ R3 : x2 + y2 = 1 + z2 be a hyperboloid of revolutionembedded in R3. Let

Γ0 = (x, y, z) ∈ M : z = 0,

andΓ0 = (x, y, z) ∈ M : z = 1.

ThenΓt = Γ0 for all t > 0, and dist(Γ0, Γ0) > 0,

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butdist(Γt, Γt) = dist(Γ0, Γt) → 0 as t →∞.

Conjecture: Evans-Spruck’s [15, Theorem 7.3] result holds true for all mani-folds of nonnegative sectional curvature, but fails for all manifolds of negativecurvature.

Example 8. In the case M = Rn it is well known that equation (4.1) preservesLipschitz properties of the initial data. Namely, if g is L-Lipschitz and u isthe unique solution of (4.1) then u(t, ·) is L-Lipschitz too, for all t > 0; see[19, Chapter 3]. Since the proof of Theorem 7.3 in [15] remains valid for anymanifold provided that one assumes the Lipschitz preserving property of (4.1),the preceding example also shows that (4.1) does not preserve Lipschitz constantswhen M is a hyperboloid of revolution.

Conjecture: The equation (4.1) has the Lipschitz preserving property if andonly if M has nonnegative sectional curvature.

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12. General theory on manifolds

For general theory on viscosity solutions to PDE onmanifolds, we refer to

D. Azagra, J. Ferrera, B. Sanz, Viscosity solu-tions to second order partial differential equations I,preprint, 2006 (math.AP/0612742v2),

for stationary equations, and to the Appendix in

D. Azagra, M. Jimenez-Sevilla, F. Macia,Generalized motion of level sets by functions oftheir curvatures on Riemannian manifolds, preprint,arXiv:0707.2012v2 [math.AP],

for evolution equations.

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13. References

References

[1] L. Ambrosio, Luigi; H. M. Soner, Level set approach to mean curvature flowin arbitrary codimension. J. Differential Geom. 43 (1996), no. 4, 693–737.

[2] B. Andrews,Gauss curvature flow: the fate of the rolling stones. Invent.Math. 138 (1999), no. 1, 151–161.

[3] D. Azagra, J. Ferrera, F. Lopez-Mesas, Nonsmooth analysis and Hamilton-Jacobi equations on Riemannian manifolds, J. Funct. Anal. 220 (2005) no.2, 304-361.

[4] D. Azagra, J. Ferrera, B. Sanz, Viscosity solutions to second order partialdifferential equations I, preprint, 2006 (math.AP/0612742v2).

[5] D. Azagra, M. Jimenez-Sevilla, F. Macia, Generalized motion of levelsets by functions of their curvatures on Riemannian manifolds, preprint,arXiv:0707.2012v2 [math.AP]

[6] K.A. Brakke, The motion of a surface by its mean curvature. MathematicalNotes, 20. Princeton University Press, Princeton, N.J., 1978.

[7] Y. G. Chen, Y. Giga, S. Goto, Uniqueness and existence of viscosity solu-tions of generalized mean curvature flow equations. J. Differential Geom. 33(1991), no. 3, 749–786.

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[8] M.P. do Carmo, Riemannian Geometry, Mathematics: Theory & Applica-tions, Birkhauser Boston, 1992.

[9] M.G. Crandall, H. Ishii, P.-L. Lions, User’s guide to viscosity solutions ofsecond order partial differential equations, Bull. Amer. Math. Soc. 27 (1992)no. 1, 1-67.

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[11] R. Deville, J.A. Jaramillo, Almost classical solutions of Hamilton-Jacobiequations, preprint.

[12] R. Deville, E. Matheron, Infinite games, Banach space geometry and theeikonal equation, Proc. Lond. Math. Soc. (3) 95 (2007), no. 1, 49–68.

[13] A. Fathi, A. Siconolfi, Existence of C1 critical subsolutions of the Hamilton-Jacobi equation. Invent. Math. 155 (2004), no. 2, 363–388.

[14] A. Fathi, A. Siconolfi, PDE aspects of Aubry-Mather theory for quasiconvexHamiltonians. Calc. Var. Partial Differential Equations 22 (2005), no. 2,185–228.

[15] L. C. Evans, J. Spruck, Motion of level sets by mean curvature. I. J. Differ-ential Geom. 33 (1991), no. 3, 635–681.

[16] L. C. Evans, J. Spruck, Motion of level sets by mean curvature. II. Trans.Amer. Math. Soc. 330 (1992), no. 1, 321–332.

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[17] W.J. Firey, Shapes of worn stones. Mathematika 21 (1974), 1–11.

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[25] M. Gursky and J. Viaclovsky, Prescribing symmetric functions of the eigen-values of the Ricci tensor, to appear in Annals of Math.

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[26] G. Huisken, Flow by mean curvature of convex surfaces into spheres. J.Differential Geom. 20 (1984), no. 1, 237–266.

[27] G. Huisken, Contracting convex hypersurfaces in Riemannian manifolds bytheir mean curvature. Invent. Math. 84 (1986), no. 3, 463–480.

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[35] T. Sakai, Riemannian Geometry, Translations of Mathematical Mono-graphs, vol 149, Amer. Math. Soc. 1992

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