Archive for the 'Partial Differential Equations' Category

Reviewing the regularity theory of elliptics PDEs via the Laplace equation. Part III. (representation formulas)

This is the third of a series of posts dealing with the regularity theory of elliptic equations. My motivation in writing these is outlined in the first post. The previous post is here.

Let us recall Green’s identity, if {u,v} are any functions smooth in {\bar{\Omega}} and {\Omega} is a bounded domain with smooth boundary we have

\displaystyle \int_\Omega u\Delta v - v \Delta u dx = \int_{\partial \Omega}u\frac{\partial v}{\partial \nu}-v\frac{\partial u}{\partial \nu}dS

this identity can be obtained with a couple of integration by parts involving the vector fields {u \nabla v} and {v \nabla u}.

Lets rewrite the identity as

\displaystyle \int_\Omega u\Delta vdx = -\int_\Omega v \Delta u dx + \int_{\partial \Omega}u\frac{\partial v}{\partial \nu}-v\frac{\partial u}{\partial \nu}dS

thus, at least formally, if somehow we could find for every {x \in \Omega} a function {v_x(y)} such that

\displaystyle \Delta v_x(y)=\delta_x(y)

\displaystyle  v_x(y) \equiv 0 \mbox{ on } \partial \Omega

then Green’s identity applied to both {u} and {v_x} in {\Omega} would give us an integral representation formula for harmonic functions

Continue reading ‘Reviewing the regularity theory of elliptics PDEs via the Laplace equation. Part III. (representation formulas)’

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Reviewing the regularity theory of elliptics PDEs via the Laplace equation. Part II.

This is the second of a series of posts dealing with the regularity theory of elliptic equations. My motivation in writing these is outlined in the first post.

Some consequences of Harnack’s inequality the Mean value property

The mean value property is characteristic of harmonic functions, but the fact that harmonic functions control their pointwise values by their local average is a general fact that is characteristic of elliptic equations (as we will see later, less sharp but more general theorems for nonlinear elliptic equations still have this flavor and are at the very heart of the regularity theory of fully nonlinear elliptic PDEs). Let me mention a few of its consequences, I already talked last time about Harnack’s inequality, as it follows from the mean value theorem, the mean value theorem (at least for harmonic functions) is more fundamental.

Continue reading ‘Reviewing the regularity theory of elliptics PDEs via the Laplace equation. Part II.’

Reviewing the regularity theory of elliptics PDEs via the Laplace equation. Part I.

There is a tedious, simple but hopefully fruitful exercise I always wanted to do. It is to review all the different proofs of the Harnack inequality and regularity of solutions to elliptic equations that I know, but only for the Laplace equation. First, because it is a good way to really get your hands on some of the ideas of several deep theorems (like those of De Giorgi-Nash-Moser and Krylov-Safonov) in the simplest possible setting. Second, because looking at all the different proofs it is possible to trace the evolution of analysis and PDEs through the last century (and a bit before that) and appreciate the level maturity reached in several fields: potential theory, singular integrals, calculus of variations, fully non linear elliptic PDE and free boundary problems. The `simple’ and `elementary’ Laplace equation lies at the intersection of all these fields, so every new breakthrough reflected on our understanding of this equation, each new proof emphasizing a different approach or point of view. Each of the proofs that I will discuss are based on one of the following:

  • The mean value property (the proof you learn in your typical complex variables or introductory PDE course).
  • The Poisson Kernel for the ball (the proof from potential theory).
  • The Calderón-Zygmund theorem (ok not exactly a `Harnack inequality’, but it should be on this list anyway) which uses the machinery of singular integrals.
  • The De Giorgi-Nash-Moser theorem, which follows the variational point of view and it is best suited for quasilinear equations or equations in divergence form.
  • The Aleksandrov-Bakelman-Pucci estimate and the Krylov-Safonov’s `Harnack’s inequality’, which follows the comparison principle point of view and it is best suited for fully non linear equations or equations in non-divergence form.

So I am going to review each theorem and its proof but only for Laplace’s equation: {\Delta u = 0}. To start off easy, I am going to do first the proof via the mean value property.

First proof: mean value property

The mean value property says basically this

Let {u} be a {C^2} function in the unit ball {B_1} of {\mathbb{R}^n}. If {\Delta u = 0} and {S} is a sphere contained in {B_1} and centered at {x_0}, then {u(x_0)} equals the average of {u} on {S}

It is not hard to prove with some calculus, one basically looks at the function `Average of {u} on the sphere of radius {r} centered at {x_0}‘={f(r)} and shows that {f'(r)=0}, and since by continuity {f(0)=u(x_0)}, the theorem follows. To show {f'(r)=0} one sees (by say, a change of variables) that {\frac{d}{dr} \frac{1}{|S|}\int_{S_r}u(x)d\sigma =\frac{1}{|S}\int_{S_r}u_nd\sigma} and this last integral is zero thanks to Stokes’ theorem and the fact that {\Delta u = 0}. Moreove, integrating the result with respect to the radius of the sphere one gets the same statement where instead of average over a sphere we have an average over a ball.

With this, one may prove easily Harnack’s inequality for harmonic functions, which I will state formally for the first time

Theorem 1 For any nonnegative harmonic function {u} in {B_1} we have the inequality

\displaystyle  u(x) \leq 2^nu(0) \;\;\; \mbox{ for all } x \in B_{1/2}

Proof. Let {x \in B_{1/2}}, then the ball of radius {1/4} centered at {x} (call it {B}) is completely contained in {B_1}, thus by the mean value property

u(x)=\int_B u(y)dy

but B is also contained in B_{1/2} and since u is nonnegative we have \frac{1}{|B|}\int_B u(y)dy \leq \frac{2^n}{|B_{1/2}|} \int_{B_{1/2}}u(y)dy=2^nu(0), again by the mean value property. This finishes the proof.

That is for today, in the next post I will explain some of the consequences of this theorem and maybe move on to the proof with potential theory methods.

(Note: this post was made using Luca Trevisan’s Latex to WordPress program, which is very useful although I am still getting used to using it. It allows you to prepare your post in a latex editor and then translate it into HTML code which WordPress can read, I strongly recommend it)

Solving the Monge-Ampere equation (continued… and finished).

I have been postponing this post for over a week due to lack of time, but finally here it is. This post ought to finish a series of past posts (here and here) where I have been describing the proof of existence of classical solutions to the Dirichlet problem for the Monge-Ampere equationvia the continuity method. Via the method of continuity  we reduced the question of existence of classical solutions to the problem of proving good a priori estimates for classical solutions, namely, we were trying to prove

Theorem (A priori estimate for the Monge-Ampere equation) Let u be a smooth solution of

det(D^2u)=\psi \mbox{ in } B_1

u= 0 \mbox{ on } \partial B_1

There is a constant C>0 (depending only on n and the C^3 norm of \psi) such that

||u||_{C^{2,\alpha}}\leq C

Last time we got almost there, using the maximum principle and the right barriers we proved the estimate

||u||_{C^{1,1}(B_1)}\leq C

Which is still not strong enough for our needs, so in order to finish the proof of the a priori estimate, we are going to use a powerful interior estimate for concave elliptic equations, proved independently by L.C. Evans and N. Krylov in the 80’s:
Theorem (Evans-Krylov) Let u be a C^{1,1} solution of the elliptic equation

F(D^2u)=f(x) for x \in B_1

if F is concave (or convex), then we have the following interior estimate

||u||_{C^{2,\alpha_0}(B_{\frac{1}{2}})}\leq C||u||_{C^{1,1}(B_1)}

where C depends only on F, and \alpha_0 is a universal constant.

This a well known theorem, a couple of places where one can read it are the book of Gilbarg and Trudinger (last edition), or the book of Caffarelli and Cabre. More recently, Caffarelli and Silvestre have come up with a shorter proof, still based on the original ideas of Evans and Krylov, this proof is available in arxiv. Maybe I will talk about the proof in some other post, but for now I am just going to quote the result.

The Evans-Krylov theorem is an interior result, we need also control at the boundary, that is provided by a result of Krylov

Theorem (Krylov) Let u be a solution to our equation, there is a universal \beta and constant C controlled by |u|_\infty such that  for x \in B_1 and y \in \partial B_1 we have

|D^2u(x)-D^2u(y)|\leq C|x-y|^\beta

This actually a corollary of Krylov’s theorem, which is a more general and remarkable result about equations in non-divergence form with measurable coefficients, but again I want to focus on the Monge-Ampere equation, I will talk about Krylov’s theorem some other time, a good place to read about it is the last chapter of Kazdan’s book. With these two tools its a standard argument to show that for a constant C controlled by the previous two, and for \alpha=min(\alpha_0,\beta), we have

|D^2u(x)-D^2u(y)|\leq C|x-y|^\alpha

I won’t do it in detail, but the proof is not too hard: basically, if the two points are closer to each other than to the boundary, then the Evans-Krylov estimate (properly scaled) gives us the inequality above, otherwise, the two points are closer to the boundary than to each other, so by the estimate of Krylov we get the same inequality in this case, and thats it!

…and that finishes the proof!. I did not present the most general result to simplify the presentation(at least for the weaker a priori estimates, which is where I did most of the details), but one can work in a more general domain (as long as it is convex) and have arbitrary boundary conditions. A much more general result, which includes not only the Monge-Ampere equation but also the k-Hessian equation, was proven  in a  paper by Caffarelli, Nirenberg and Spruck.

Solving the Monge-Ampere equation (continued)

In a previous post I began the proof of the following theorem:

Let \psi \in C^\alpha be a positive function in B_1, then there exists a unique function u \in C^{2,\alpha}(B_1) such that

det(D^2u(x))=\psi \mbox{ in } B_1

u(x) = 0 \mbox{ on } \partial B_1

To prove the theorem, we looked at the function

\psi_t = (1-t)+t\psi(x).

we noted that it is easy to solve the problem explicitely for \psi=\psi_0.  Then thanks to the inverse function theorem (the Banach space version) we saw that the set A of t‘s for which we can solve the equation A is open (recall that A was defined as the set of those t in [0,1] for which we can solve the equation above with right hand side  equal to \psi_t).

Since A is open and non-empty, if we show that its also closed the theorem would be proved.

Part II:  Showing A is closed.

(see previous post of the series for Part I)

That A is closed will be shown to be a consequence of the following a priori estimate:

Theorem (A priori estimate for the Monge-Ampere equation) Let u be a smooth solution of

det(D^2u)=\psi \mbox{ in } B_1

u= 0 \mbox{ on } \partial B_1

There is a constant C>0 (depending only on n and the C^3 norm of \psi) such that

||u||_{C^{2,\alpha}}\leq C

Proving the a priori estimate is usually the hardest step, so first let’s see that the a priori estimate in fact implies that A is closed: let t_k be a sequence in A converging to a number t \in [0,1], by the a priori estimate above the sequence of solutions u^{(t_k)}=u^{(k)} is uniformly bounded in C^{2,\alpha}, thus by the Arzela-Ascoli theorem a subsequence converges to a function u \in C^{2,\alpha}_0, since \psi_{t_k} converges to \psi_t we see this u is a solution of Problem (t), i.e. t \in A. This proves A is closed and therefore the theorem.

Of course, now we “just” have to prove the a priori estimate. ..

The broad outline of its proof is: first we prove a series of a priori estimates, each stronger than the previous one, and at each given step we  use the estimates already achieved, this will bound the first and second derivatives, the main tool here is the maximum principle used as in the classical technique of Bernstein. Then, the Holder regularity of the second derivatives will be consequence of deep regularity results from the theory of fully non-linear equations, among them the Evans-Krylov theorem. Now lets go through the argument in detail, and to try to make this just a bit readable I will number the steps

Step 1) The easiest estimate is the L^\infty bound

(a priori estimate (1))

||u||_{L^\infty(B_1)} \leq C

where C depends only on the L^\infty norm of \psi. To prove it, consider the auxiliary function \phi(x)=\frac{M}{2n}(|x^2|-1), this is a smooth function that vanishes on \partial B_1 and D^2\phi = M^n, picking M large enough (and controlled by the supremum of |\psi|) we have

det(D^2\phi) \geq \psi = det(D^2u)

then the maximum principle implies that

u \geq \phi \geq -\frac{M}{2n} in B_1

plus u is a convex function  vanishing on \partial B_1, so we have u \leq 0 as well. This proves the a priori estimate (1).

Step 2) Now we go for a Lipschitz bound for u, which we will refer to as the

(a priori estimate (2))

||\nabla u||_{L^\infty(B_1)} \leq C

Let’s differentiate the equation det(D^2u)=\psi in the direction of the unit vector \xi, and we obtain

L u_\xi= a_{ij} (u_\xi)_{ij}=\psi_\xi

Here a_{ij}(x) correspond to the entries of the cofactor matrix of D^2u(x), what the left hand side above amounts to is the linearization of det(D^2u) at u, which is the elliptic operator L (we did this before, in the previous post). Notice also that I can talk freely about third derivatives of u because any strictly convex, C^{2,\alpha} solution of the Monge-Ampere equation is actually C^\infty (that this is true is another of the many consequences of  the Schauder theory for second order  linear elliptic equations with Holder continuous coefficients, and its not too hard to check), so the equation above holds classically. In particular, by the maximum principle (again) we conclude that for some C depending on the supremum of |\psi|:

\sup\limits_{B_1} |u_e| \leq \sup \limits_{\partial B_1}|\nabla u|+C

but u \equiv 0 on \partial B_1 thus there \nabla u = u_n n, where n is the outer normal to \partial B_1. Here we bring back our old friend, the function \phi defined in step 1), as we know u\geq \phi on B_1 and they agree on the boundary, therefore

u_n \leq \phi_n on \partial B_1

also, since u \leq 0 and vanishes on \partial B_1 we also have the inequality 0 \leq u_n. Therefore the |u_n| is  controlled by M times a dimensional constants, i.e.  the supremum of |\psi| controls |\nabla u|. Putting the last three identities/inequalities together we get the a priori estimate (2).

Step 3) Notice that up to now we have only used the ellipticity of the equation and not the particular structure of the Monge-Ampere equation (even the differentiation step of Step 2) is a fact that holds for general elliptic equations). This  will change now that we are going to prove the

(a priori estimate (3))

||D^2 u||_{L^\infty(B_1)} \leq C

We start off as in step 2 by differentiating along a fixed, arbitrary direction \xi,  except now we do it twice and thus we get something much more complicated than before. We get around this noting that the map M \to det(M) is a convex function on positive definite matrices, and get an inequality that allows us to drop the lower order terms:

Lu_{\xi \xi} \geq \psi_{\xi\xi} \geq -C

and this says that  \sup \limits_{B_1} u_{\xi\xi} is controlled by the same supremum on \partial B_1 plus a constant C controlled by \sup |\psi|.  Then we only have to estimate things on the boundary (just as for the first derivatives).

So lets pick a point x_0 \in \partial B_1 and estimate the entries of D^2u(x_0). Since on \partial B_1 we have u \equiv 0 all the first, and second order angular derivatives of u at x_0 are zero, and this says a lot about the Hessian of u on the boundary, its just a matter of relating partial derivatives in polar and Cartesian coordinates to see then (after doing an adequate rotation of our coordinate system) that the entries of the Hessian satisfy the relations:

u_{ij}(x_0)=u_n(x_0)\delta_{ij} for i,j<n

here “n” stands both for the number n and the exterior normal direction \partial B_1 since in our rotated coordinate system the vector e_n is normal to \partial B_1 at the point x_0.Since we already have an a priori bound for u_n this gives us an a priori bound for the second partial derivatives on the boundary, except for the partial derivatives of the form u_{in}, 0\leq i \leq n.

Here we use the fact that our equation is rotation invariant (that is, the determinant is invariant under rotations of the coordinate system). Since u_{ij} (i,j<n) is diagonal, we can compute at once det(D^2u(x_0))

det(D^2u(x_0))=\psi(x_0)=u_n^{n-1}u_{nn}

I claim that there exists a \delta_0>0  such that u_n(x_0)>\delta_0 on \partial B_1, and \delta_0 is bounded away from zero by a quantity controlled by \inf \psi. Therefore there exists a  C controlled by \psi, such that

u_{nn}\leq C\psi(x_0)\leq C\sup |\psi|

Plus, u_{nn} is non-negative (since u is convex). Thus we have given a priori estimates for all the pure second order derivatives of u on \partial B_1, and by the remarks made at the beginning of this step we have proved the a priori estimate (3)

Step 4) Putting the 3 previous estimates together, we have:

a priori estimate (4)

||u||_{C^{1,1}(B_1)}\leq C

Next, we have to push this to a C^{2,\alpha} estimate in  B_1. This will be the content of the next post (I am sure we can agree that this post is long enough). In the next post I  will talk about the Evans-Krylov theorem and the boundary behavior of solutions to elliptic equations. Hopefully that will wrap up this series of posts on the Dirichlet problem for Monge-Ampere,  after this I might go back to the Minkowski problem.

Quick post: A theorem and an open problem about Gauss curvature

While I finish my next post on the Dirichlet problem for the Monge-Ampere equation, I thought I could mention two neat things I have learned from reading Jerry Kazdan’s survey on Prescribing the curvature of a Riemannian manifold, a short book that I recommend strongly (its a bit out of date, but hey!, geometry has advanced quite a bit in the last couple of years!)

First, a cute solution to an easily-stated problem, as  observed by Wallach and Warner in 1970

Theorem:  Given a compact smooth 2-d manifold and \Omega is a 2-form such that \int_M \Omega = 2\pi \chi(M), then there exists a smooth Riemannian metric g on M such that \Omega = KdA, where K is the Gauss curvature of g and A is the area form of g.

Proof: Pick your favorite metric g_0 on M, we will  prove that the metric we are looking for is in fact conformal to g_0, for any smooth function u, define g_u = e^{2u}g_0. Using the well known formulas for the Gauss curvature and area form of g_u, one arrives at the identity

K_udA_u=K_odA_0-(\Delta_0 u) dA_0

where the sub-indexes 0 and u refer to the object corresponding to the metric g_0 and g_u, in particular, \Delta_0 is the Laplace operator for (M,g_0). Then one wants to pick u such that

K_odA_0-\Delta_0 u dA_0=\Omega

or (\Delta_0 u) dA_0 = K_0dA_0-\Omega

but, using the metric g_0 we can write \Omega = f(x)dA_0 for some f (or equivalently, using the Hodge star operator given by g_0) thus the u we want is the solution of the equation

\Delta_0 u = K_0-f (**)

but since \int_M \Omega =\int_MfdA_0= 2\pi\chi(M)=\int_MK_0dA_0 we have that \int_MK_0-fdA_0=0 thus by standard elliptic theory (or Fredholm theory, etc) there exists a smooth function u solving equation (**). Thus the metric g_u is the one we were looking for and the theorem is proved.

Now, an easily stated problem whose solution is not likely to be as short, and has led to a lot of research in the last 2 decades. I  read about it for the first time in an interview with Louis Nirenberg, which can be found here.

Open problem:  Given a two dimensional Riemannian manifold, can we embed it isometrically (even just locally) in three dimensional Euclidean space?

Recall that Gauss curvature solely determines the local geometry of a 2d manifold, so its not surprising that this problem is equivalent to the following problem involving the  (…tataaaaa! )  Monge-Ampere equation:

Open problem restated: Given a C^\infty function K(x,y), find (even just locally) a function u(x,y) which solves the equation

\frac{u_{xx}u_{yy}-u_{xy}^2}{\left (1+u_x^2+u_y^2 \right)^2} = K(x,y)

but mind you, this is not your grandpa’s Monge Ampere equation, for it is not an elliptic equation unless K is strictly positive. For general K, it is an equation that varies  between hyperbolic or elliptic according to the sign of K, so you run into real problems in the set of points where K vanishes .  If on the other hand K is strictly negative, Kazdan says that the equation above is solved using tools from non-linear hyperbolic equations, of which sadly I know nothing. The case K >0 is then dealt with the techniques from the last two and the next post, which I should finish in a day or two.

Solving the Monge-Ampere equation

In my previous post, while talking about the Minkowski problem, I introduced the Monge-Ampere equation. I would like to talk about it for a post or two before continuing with the Minkowski problem, in part because the Dirichlet problem for Monge-Ampere is arguably an easier problem than the Minkowski problem.

To make the presentation cleaner, allow me to work only in the unit ball in Euclidean space, this situation contains all of the important issues one deals with to solve the equation, and it will let us avoid some of the technicalities that give little insight. In any case, all of the arguments can be tweaked without too much effort (but perhaps losing some clarity of exposition) to make them work in any smooth, strictly convex domain \Omega, the same remark goes for dealing with non-zero boundary data,  I will make more detailed remarks about this in the next post. Here we go then.

The Dirichlet problem for the Monge-Ampere equation: Consider a strictly convex domain \Omega \subset \mathbb{R}^n, and a smooth, positive function \psi \bar{\Omega} \to \mathbb{R}. Then find a convex function u twice differentiable in \Omega such that:

det (D^2u(x)) = \psi(x) \mbox{ for all } x \in B_1

u(x)= 0 \mbox{ on } \partial B_1

That a solution exists can be seen via the method of continuity (although, as I pointed out, there are other approaches). The principle behind this method is that if one can show that the set of those positive \psi \in C^\infty(\bar{\Omega}) for which there is a solution is  open, closed and not empty then by connectivity there is a solution for any such positive \psi \in C^\infty . This is a very simple and universal method that has proven to be succesful in many situations in Mathematics (e.g. in many applications of PDEs to differential geometry), but in these post I will talk about its application to the Monge-Ampere equation.

From now on we fix \psi, and define

\psi_t(x)=(1-t)+t\psi(x)

this is a smooth function for every t, and strictly positive for every t \in [0,1]. Thus we can consider the Dirichlet problem above with \psi_t as the right hand side, lets call it “Problem (t)“. Problem (1) is the actual problem we want to solve, and Problem (0) t=0  we can solve explicitely, the solution is given by u_0(x)=\frac{1}{2}(|x|^2-1) which satisfies:

det(D^2u_0(x))=1 = \psi_0(x)\mbox{ in } B_1

u_0(x)= 0 \mbox{ on } \partial B_1

Consider the set

A=\{t \in [0,1] | \mbox{ Problem } (t) \mbox{ can be soved } \}

the example above shows that 0 \in A. Then the strategy is to show that A is at the same time open and closed in [0,1], which would imply the Dirichlet problem has a solution for any \psi. Here is where the real work begins, and we divide it in two parts.

Part I: Showing A is open.

This will be dealt with machinery from classical analysis, namely the inverse and implicit function theorems for Banach spaces. Let us consider the map

T : C^{2,\alpha}_0 \to C^\alpha

given by T(u)=det(D^2u), this is a continuous but non-linear map between these two Banach spaces, we will apply the inverse function theorem to this map. Suppose t \in A and let u^{(t)} be the solution to the correspoding problem, then we can assume without loss of generality that u^{(t)} \in C^{2,\alpha}_0 and that it is a strictly convex function. Then the differential to T at u^{(t)} can be computed by the chain rule, and it turns out that it is just the second order differential operator

L(v)= \sum \limits_{ij}a_{ij}(x)v_{ij}(x)

where a_{ij}(x) are the entries of the cofactor matrix of D^2u^{(t)}(x).  The fact that u^{(t)} is strictly convex means that L is an elliptic operator, moreover it’s coefficients are C^{\alpha}.Now I claim that L is a one to one bounded linear map* from C^{2,\alpha}_0 to C^\alpha.

This means that the differential of T at  u^{(t)} is regular, and by the inverse function theorem T is a diffeomorphism between a neighborhood of u^{(t)} in C^{2,\alpha}_0 and a neighborhood of T(u) in C^{\alpha}. But then we can find \delta>0 such that any s such that |t-s|<\delta the function \psi_s lies in this neighborhood, and thus one can find a u^{(s)} \in C^{2,\alpha}_0 such that det(D^2u^{(s)})=\psi_s,  i.e. Problem (s) has a solution and thus s \in A, which shows A is open.

*Remark: The fact that L is one to one is a consequence of the Schauder theory for linear 2nd order equations, but that is something that could be discussed by itself in several posts! The theory says explicitely that given a differential operator as L, strictly elliptic and with C^\alpha coefficients, then for any f \in C^\alpha(\Omega) there exists a unique u \in C^{2,\alpha}(\Omega) that vanishes on \partial \Omega and such that

Lu=f \mbox{ in } \Omega

That takes care of Part I. Showing that A is closed is usually the more interesting part and it changes from problem to problem (showing A is open is always more or less the same). I will do the second part in the next post