The old site  (the one you are looking at now) will be up at least for a while, but all the old posts have been copied to the new blog, To reflect the fact that nearly all posts here have dealt with Partial Differential Equations the new site will go by the name  “PDE Blog”.

Let us recall Green’s identity, if are any functions smooth in and is a bounded domain with smooth boundary we have

this identity can be obtained with a couple of integration by parts involving the vector fields and .

Lets rewrite the identity as

thus, at least formally, if somehow we could find for every a function such that

then Green’s identity applied to both and in would give us an integral representation formula for harmonic functions

This can actually be carried out rigorously for all reasonable domains , but only in a few cases however, we know a useful expression for the functions . Happily for us, whenever is a ball there is a simple expression, and so, for any function harmonic in a neighborhood of a ball we have

Theorem 1

Let be a smooth harmonic function in some neighborhood of the ball , and let the dimension be , then

Here is just the surface area of the dimensional sphere, I will not derive it as this can be found in most introductory PDE textbooks (for instance, Evan’s book), suffice it to say that one needs to use the symmetries of Laplace’s equation (in particular under inversions) to manipulate the fundamental solution () and build the function .

Note that this integral representation for gives us as a corollary the mean value property (just take equal to the center of the ball, in this case ), and it tells us that for other points , the value is obtained via a weighted average of on the sphere, the average being balanced according to the position of with respect to the sphere. By the way, this weight function

is known as the Poisson kernel (for the ball). This representation formula also gives us directly another proof of Harnack’s inequality. It tells us even more, it says that if is continuous up to the boundary of , then is Lipscthiz on the boundary, which can be seen by just looking at the term . Finally, by differentiating the right hand side of the integral representation formula, we prove again the a priori estimate for the gradient I discussed last time:

taking absolute values we get , so the gradient is bounded by times the average of on the sphere of radius , which is actually a stronger estimate than what I proved last time, for the average is bounded by the supremum of , thus

(notice I had not specified the constant last time I wrote this estimate, here we see that we may take .)

Posted in Partial Differential Equations Tagged: A priori estimates, harmonic functions, Harnack's inequality, integral representation formulas, Laplace equation, mean value property ]]> http://themathingpot.wordpress.com/2010/01/15/reviewing-the-regularity-theory-of-elliptics-pdes-via-the-laplace-equation-part-iii-representation-formulas/feed/ 1 Nestor http://themathingpot.wordpress.com/2010/01/14/laplaceii/ http://themathingpot.wordpress.com/2010/01/14/laplaceii/#comments Thu, 14 Jan 2010 16:14:46 +0000 Nestor http://themathingpot.wordpress.com/?p=600 ]]> 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.

First, and perhaps the most important consequence, is the pointwise a priori estimate for the derivatives of a harmonic function in terms of its supremum:

Theorem 1 (A priori gradient estimate for harmonic functions)

Let be a harmonic function in a ball , then

Proof: Let be a harmonic function in , by the mean value property, as long as , therefore

(Recall is called the symmetric difference of sets), now the set lies in the union of two annuli with radii and , thus its volume is not larger than for a dimensional constant . We then have

since the direction of is arbitrary and is , dividing both sides by and taking we obtain the a priori estimate.

One may iterate this to estimate higher derivatives (thanks to the fact that the derivatives of a harmonic function are themselves harmonic). To obtain the estimate

I emphasize that these are a priori estimates, one needs to know is already smooth to prove them, what they say is that the derivatives of all orders are all controlled by the supremum of !. In particular, a family of uniformly bounded harmonic functions is compact in every . Usually, the first time you learn about this phenomenon is when studying Montel’s theorem in a complex analysis.

The a priori estimates and Harnack’s inequality also give a quick proof (which I will omit) of another classical result, but in potential theory, which I mentioned because it was due to Harnack himself:

Theorem 2 (Harnack’s convergence theorem) Let be a decreasing sequence of functions which are continuous in and harmonic in the interior. Then they converge uniformly in compact sets of to a smooth function which is harmonic.

Since I just mentioned a priori estimates, I should recall the (one of many ) proofs of the fact that being harmonic even in some weak sense forces a function to be smooth and harmonic in the classical sense. Let’s say for instance, harmonic in the sense of distributions (we will revisit this theorem for other weak notions of harmonicity):

Theorem 3 (Weak harmonic implies harmonic) Let be a bounded measurable function in such that

for any smooth function with compact support in . Then (after modifying it in at most a set of measure zero) is smooth in the interior of and harmonic.

Proof: Let be an approximation to the identity given by a kernel which is radially symmetric and supported in . Let , then for each and any compact the functions ( small enough depending on the distance between and ) are smooth with bounded derivatives of any orders, moreover, they are all uniformly bounded in by the boundedness assumption on . Now, using the symmetry of the kernel and Fubini’s theorem, one can see that for any

and since each is smooth we have . Furthermore, by the a priori estimates the functions are also uniformly bounded , for any . Then we know (by Arzela-Ascoli) that a suitable subsequence of with converges uniformly in to a function which lies in (for any ), since they also must converge to a.e.(by Lebesgue’s differentiation theorem) we conclude that agrees a.e. with a smooth harmonic function, as we wanted to prove.

I think that I will stop here for now. Tomorrow: I will review the Poisson kernel to give the potential theoretic proof of the Mean value property, Harnack’s inequality and the (a priori) gradient estimates for harmonic functions, and after that, it will be the Calderón-Zygmund estimates.

Posted in Partial Differential Equations Tagged: A priori estimates, harmonic functions, Harnack's inequality, Laplace equation, mean value property ]]> http://themathingpot.wordpress.com/2010/01/14/laplaceii/feed/ 0 Nestor http://themathingpot.wordpress.com/2010/01/13/joint-mathematics-meetings-in-san-francisco-blog/ http://themathingpot.wordpress.com/2010/01/13/joint-mathematics-meetings-in-san-francisco-blog/#comments Thu, 14 Jan 2010 02:47:30 +0000 Nestor http://themathingpot.wordpress.com/?p=596 ]]> …and now for a little advertisement:

My friend and former UT graduate student Adriana Salerno (currently at Bates) will be running the 2010 AMS Joint Math meetings blog. She was also in charge of the blog in previous years (you can check them out here and here). I recommend you check it out in the next few days to see what has been going on at the meetings (specially if, just like me, you don’t happen to be in San Francisco this week).

• 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: . 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 be a function in the unit ball of . If and is a sphere contained in and centered at , then equals the average of on

It is not hard to prove with some calculus, one basically looks at the function `Average of on the sphere of radius centered at ‘= and shows that , and since by continuity , the theorem follows. To show one sees (by say, a change of variables) that and this last integral is zero thanks to Stokes’ theorem and the fact that . 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 in we have the inequality

Proof. Let , then the ball of radius centered at (call it ) is completely contained in , thus by the mean value property

but is also contained in and since is nonnegative we have , 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)

I am not going to take off from where I left last time (namely, the posts about the Minkowski problem, which I will finish, someday), but instead will start the year with some shorter, lighter posts. I plant to start with a few posts about varifolds vs currents vs BV sets, and also about the Harnack inequality, maybe later I will write a bit about topics from phase transitions such as the Stefan problem or the Cahn-Hilliard equation

My goal for this week is presenting Aleksandrov’s solution to  the Minkowski problem (see an earlier post I did introducing this problem). So I am going to leave you a problem as a preview, it is a sort of discrete version of the Minkowski problem:

Let   be a family of non-coplanar unit vectors in and let be positive numbers such that

Then, show that there exists a convex closed polyhedron with exactly faces with normal vectors given by and corresponding areas . Plus, this polyhedron is unique up to translation.

This is fact is not surprising (since it is not hard to check that any polyhedron has this property), but the proof is far from trivial. As you may guess, the proof cannot be constructive, it will use a continuity argument to show that there must be at least one such polyhedron. I will present this in my next post.

Don’t be scared, this has not been turned into a photo blog!  Today I received my copy of Singular Integrals I ordered on-line recently, which is funny given that I have been reading that book for a long time now (at least now I won’t have to borrow a copy from the library or from a friend).

In any case, this reminded me of something I heard once: basically, that throughout the early years of your career as a mathematician, there will be a  list of books (that will depend strongly on your research interests) that you must read completely and in full detail to the point where you are be able to reproduce their contents on command. So, wondering what that list should be for me, I piled up some books from my book shelf and took a picture.

…perhaps I should be reading some of those books instead of writing this blog post

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

There is a constant (depending only on and the norm of ) such that

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

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 be a solution of the elliptic equation

for

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

where depends only on , and 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 be a solution to our equation, there is a universal and constant controlled by such that  for and we have

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 controlled by the previous two, and for , we have

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 Hessian equation, was proven  in a  paper by Caffarelli, Nirenberg and Spruck.

Given a compact smooth manifold without boundary , when is it possible to find a Riemannian metric in satisfying for a given Ricci candidate ?

It is of course too ambitious to try to answer this question in full generality but we can start by showing some examples of Ricci candidates for which this equation does not have a solution.

Trying to solve for amounts to solving a second order, quasilinear PDE on , however, the main difficulty here is that the operator is not elliptic.

A motivation for considering this problem comes from the question of existence of metrics with constant sectional curvature on - manifolds (compact and without boundary). This of course has to do with the celebrated theorem of Richard Hamilton on the description of - manifolds with positive Ricci curvature:

Theorem (Hamilton, 1982): Let be a connected, compact smooth dimensional manifold without boundary and suppose that admits a metric such that is positive definite everywhere. Then also admits a metric with constant sectional curvature.

We will discuss some of the ideas involved in the proof of this theorem in future posts. A consequence of this result is that is diffeomorphic to the quotient of the -sphere by a discrete group .

Back to our original problem, recall that given a Riemannian metric , the full curvature tensor is defined by

Where

Here is the Levi-Civita connection of .

The Ricci tensor is then defined as

.

Here are two basic properties of :

1) is symmetric in and , i.e.

2) In local coordinates looks as follows

We are using the summation convention (i.e we sum over repeated indices). The Christoffel symbols are defined by

Where are entries of the matrix . This says that in local coordinates we can write schematically where is a function that depends linearly on the entries of . If we then want to solve locally, property 2) tells us that we have to look at a system of the form

Property 1) tells us that an admissible Ricci candidate has to be symmetric.

One encounters obstructions for solving this system right away. One of the main difficulties has to do with the fact that the Ricci tensor satisfies the differential Bianchi identity

Where is the divergence operator respect to and is the scalar curvature of (the trace of the Ricci tensor). This says that if we define a 1-form by , then in order for to satisfy we must have

To write in coordinates, we start with

From

(which is just the definition of in coordinates) and from the expression in local coordinates of the symbols , we easily see that

As discussed in Besse’s book, Dennis DeTurck came up with examples of symmetric tensors that cannot satisfy the Bianchi identity respect to any metric. One of his examples is the following

Consider in a symmetric tensor of the form

The existence of a metric such that implies as we saw before that . In particular, from our expression for we must have

This implies that on the hyperplane , the metric must satisfy which is impossible for a Riemannian metric. It follows that for any point in the hyperplane the equation has no solution near . Notice that at these points the tensor is singular.

In the next post we will interpret the existence of examples like the one we have just discussed as a consequence of the non-ellipticity of the system .