For the first talk of this seminar I presented the proof of the Isoperimetric inequality using mass transport. The proof in this specific form was found independently by N. Trudinger and R. McCann, and it is certainly strongly influenced by work of M. Gromov which I am currently not competent to describe. Modulo the use of the Brenier-McCann theorem from Optimal transport, this is the shortest proof of the isoperimetric theorem  (in general dimensions) that I am aware of .

Further, this method of proof has rather deep ramifications, exemplified by  (sharp) quantitative refinements of the isoperimetric inequality for general non-isotropic area functionals, cf. work of A. Figalli, F. Maggi and A. Pratelli.

The Isoperimetric inequality says that given a domain  with a rectifiable boundary,

is a dimensional constant such that equality holds if and only if is a ball.

Proof. Let be such that , then by the Brenier-McCann theorem (cf. Villani’s book) we know there is a continuous convex function such that is a measure preserving map from to , we thus have

now, since for all we can apply the arithmetic-geometric  mean inequality to conclude that in .  Integrating this relation and using the inequality above we arrive at

and by the divergence theorem this is the same as

since maps into we know that , then

Finally, we take into account that    was chosen so that , thus the last inequality is the same as

note that the constant is such that we have equality if .

Conversely, if we have equality then in particular we must have equality in the arithmetic-geometric mean inequality, implying all eigenvalues of must be the same, and so must all equal . We conclude is just a translation and thus itself is a translation of and that finishes the proof.

Antoine’s talk discussed the thin film equation, I know very little about this subject (in fact all I know I learned from Antoine’s talk last week) so take some of the things I say with a grain of salt. The thin film equation models the spreading of a drop or thin layer of liquid on top of a flat, impermeable surface and there is a vast literature on the subject, see for instance this survey by Andrea Bertozzi. Incidentally, Terry Tao discussed the closely related shallow water wave equation in a blog plost yesterday.

The thin film equation is the fourth-order non-linear equation

where typically and one has boundary conditions (which I will explain in a bit)

As an introduction to his talk Antoine went through a rather insightful and brief heuristic derivation of this model, when this models the evolution of a very thin film of liquid spreading on top of a flat, horizontal surface. That this is more or less described by the above equation can be derived as follows: Let us think of the region as the portion of space occupied by a thin drop at time , so represents the height of the drop at (a point on the flat surface) at time , and it is through that we choose to keep track of the configuration of the liquid.

The velocity of the fluid particles should not vary too much in the vertical direction (since the drop is assumed to be very thin), thus the flow of particles happens roughly only in horizontal directions. The same can be said about the fluid density, so the height itself is a good measurement of mass density, under these circumstances then, conservation of mass gives us

where denotes the average horizontal speed of the fluid lying over .

Under the physical assumption of a thin liquid this approximation is a valid one (and after all, the thinner the liquid the better this approximation works). With this in mind, the contact angle condition above

says that the angle at which the liquid meets the planar surface is constant, this is an equilibrium condition stemming from the balance of interfacial forces of the drop and the flat surface (i.e. the constant is dictated by the chemistry of liquid and the surface and it is given a priori). By the way, the boundary which is where this contact condition is given is a free boundary which in this model is usually called the contact line.

To complete this heuristic derivation of the model, we need to specify how is the vector field related to the drop represented by . The liquid in such a thin film is likely irrotational, so it’s not surprising there is a form of Darcy’s law, which in this case is

where typically and is the pressure, which by the Young-Laplace law is proportional to the mean curvature of the film surface. As our drop of liquid is assumed to be very flat,

gives a good approximation to the Young-Laplace law. This explains the appearance of the term , where , moreover since we were assuming (no flux at the fixed boundary) this justifies referring to the condition

as a “no flux through the boundary” condition.

Recapitulating, the equations are ( is fixed)

A remarkable feature of these equations is that the fact that has a regularizing effect, namely: linear higher order parabolic equations (in particular, 4th order equations like ) do not have a comparison principle and in particular positive initial data could later on change sign. However, the non-linear equation preserves positivity as long as vanishes fast enough as . This was discovered first by Bernis and Friedman in 1990.

Theorem [Bernis-Friedman '90]

If , and with then the unique solution to (1) stays positive for all times.

The essence of their proof comes down to an Entropy method: they observed that if one lets be defined by

then for any classical solution of (1) one has the energy equality

Therefore is a monotone quantity (thus the name entropy). In particular, if it is bounded for the initial data then it ought to be bounded for all later times, and since as the function cannot stick too much too zero or else the Entropy would blow up!.

This shows that if then the solution with initial data remains positive for all later times, and taking they conclude that the solution with initial data stays non-negative for all later times.

For the full details of the argument, see the paper of Bernis and Friedman. They consider explicitly only dimension , but many of their results hold in higher space dimensions as well. Certainly, their idea of using an Entropy to obtain positivity ( pointwise information) is very useful in many other contexts.
The solutions produced by the Bernis-Friedman method all satisfy the zero contact angle condition, namely in

The non-zero contact angle condition corresponds to the so-called partial wetting regime. For this case an approach developed by Felix Otto relies on the fact that the dynamics are described by the gradient flow induced via the Wasserstein distance by the Energy functional

However, this approach is only known to work for . Antoine explained yet a third possible method to produce solutions which can handle for . It also uses a variational formulation, the idea is to take the functional above and approximate it by a regularized energy

In this case, the dynamics are described by the equation

where and is a kind of bump function which vanishes for . It turns this approximation has a physical interpretation: the new pressure term models the disjoining/cojoining intermorlecular forces affecting the triple junction of the fluid, air and solid (flat surface) phases. As , this approximation allowed Antoine to obtain a solution that satisfies a non-zero contact angle condition

Theorem[Mellet '11]

If for , and is and then the approximating solutions converge uniformly to a continuous function which solves in the sense of distributions

moreover, for almost every if is the smallest or the largest point in then

and for the other points all we can say is that the contact angle condition is the same on the end points of each bounded, connected component of .

This last property of the solution has to do with the lack of continuity of the energy functional with respect to just uniform convergence, but I will not go into that in detail here for lack of time.

]]> https://partialdifferentialequations.wordpress.com/2011/03/15/msri-workshop-day-2-a-week-later-free-boundary-problems-involving-thin-films/feed/ 0 Nestor https://partialdifferentialequations.wordpress.com/2011/03/08/msri-workshop-day-1-unique-continuation-for-nonlinear-pdes/ https://partialdifferentialequations.wordpress.com/2011/03/08/msri-workshop-day-1-unique-continuation-for-nonlinear-pdes/#comments Tue, 08 Mar 2011 16:51:45 +0000 Nestor http://partialdifferentialequations.wordpress.com/?p=659 ]]> This week MSRI is a rather busy place, as the flagship workshop of the program on Free boundary problems is taking place. Luis Silvestre (who happens to be my mathematical older brother) gave a talk this monday on his very recent work with Scott Armstrong, which deals with unique continuation for nonlinear elliptic PDEs, this post is an overview of his talk. A preprint of their work is available on the arxiv.

As usual, any potential mistakes and inaccuracies in the presentation below are due to the author of this post and shouldn’t reflect on what was a terrific lecture by Luis.

The main question stems from the analytic continuation property for Laplace’s equation , which is a very well known classical fact. A fact which however, is not “so easy” to prove. Classically, it says the following, let us put for a second then “unique continuation” refers to the following fact:

Note that since is linear, this is equivalent to the following

Luis claims that (as far as he knows) there are really only three ways to prove this:

1) By using the analyticity of , which follows by the regularity of the equation (and this, in a sense, is kinda cheating).

2) One can also prove this via Carleman estimates (discovered by Carleman in 1939). See for instance works by Daniel Tataru, he has several expository notes about this in his webpage.

3) Using monotonicity formulas, such as Almgren’s frequency formula (this is an approach developed in series of collaborations by N. Garofalo and F. Lin).

In what sense is the first method kind of cheating? Well, analyticity is really strong and does not hold in many interesting problems. In contrast, the last two methods can handle linear equations with variable coefficients whose solutions are not necessarily analytic. Namely if consider the linear operator

Then unique continuation holds if is Lipschitz and uniformly elliptic and are (say) bounded. Moreover, if is slightly less than Lipschitz then unique continuation does not hold, as there are known counterexamples.

So for linear equations the story is rather complete. What about non-linear equations?

There are some 2-dimensional results, including for the -Laplacian: if unique continuation does not hold, if it has been proven (again, only in 2d). For the -Laplacian is not known even in 2d.

We will preoccupy ourselves with the case of Fully non-linear equations

where is a weak solution in the Crandall-Lions “viscosity” sense. If we assume enough things about (say is convex and ) then unique continuation is true. Why?

Since convex, the Evans-Krylov theorem says that is for some . Then, since is we can differentiate the equation and see that the derivatives solve a linear equation with Lipschitz coefficients and thus we can apply the linear result to the linearized equation.

Therefore this case is a trivial application of the linear result and the regularity for the equation. In particular, since it reduces to the linear case, it proves both formulations of unique continuation which are not necessarily equivalent for non-linear equations since its no longer true that . However, if we remove any of the assumptions (convexity, regularity of ) the proof no longer works: convexity was needed to guarantee existence of classical solutions by the Evans-Krylov Theorem, and regularity was required to apply the linear theorem after taking derivatives.

If we could remove any of them it would be interesting because (for instance) we could then prove unique continuation for the Pucci operators:

Here denotes the projection of a symmetric matrix to the subspaces where it is positive or negative define.

The result of Silvestre and Armstrong consists in removing the condition that is convex and prove unique continuation for a single as long as is . namely:

Theorem [Armstrong-Silvestre '10 (arxiv)]

Let be a function in the space of symmetric matrices giving rise to a uniformly elliptic equation and let be a viscosity solution of in some open set . Then

To fix ideas, let us see how Harnack inequality says that if unique continuation does not hold then the counterexample is going to be very hard to build…

For suppose you had a that violates unique continuation, if had a sign in some neighborhood of a point of then by Harnack inequality

thus, the that violates unique continuation must change sign in every neighborhood of every point in !.

Let us give a sketch of their proof: has non-empty interior, so it contains some little ball . Let us translate this ball inside until it touches the boundary of the positivity set from outside.

Now note that since might not be convex, all the regularity we know of is (by Krylov-Safonov plus translation invariance of the equation). However, the assumption that this ball touches from outside is very strong, and in particular as we are about to see will be classical in a neighborhood of that point, which effectively allows us to work around the lack of an Evans-Krylov theorem for non-convex equations. To see is classical near the contact point we proceed in a couple of steps:

Step 1. We are going to show that grows better than quadratically away from the point . Let , then solves

for some which is bounded measurable and . Therefore, we can apply the boundary Harnack principle for non-divergence form equations (cf. Gilbarg-Trudinger section 9.9, second edition) to get

So looks almost as if it were near . Thus

Now observe that since and vanishes in we must have . Since this works for every directional derivative of , this guarantees that grows like   near . In particular, exists and must be .

Step 2. The important thing is that the power is greater than , thus, if we rescale

then in and also solve in . Then, if is very small, is going to be very flat in so we may apply a deep result of O. Savin that then says that is in . Rescaling back, we get

This then has essentially worked around the lack of convexity in , which is what we used first to get and if is then we can argue as before and get the unique continuation by linearizing and applying the result for linear operators.

Remark: this proof will not work to prove that general unique continuation holds, i.e. if we instead have two viscosity solutions of the same equation in which agree on a set with non-empty interior then we cant prove that they agree (if one of them is smooth enough though, these arguments will suffice).

This leads to the next part of Luis’ talk, where he announced a new partial regularity result obtained with Scott Armstrong and Charlie Smart*. Namely, what is the size of the set of possible singularities for fully non-linear equations? As this result is still not published, I will only quote their main result, and go back to it in another post once their preprint becomes available in the next couple of weeks

Theorem [Armstrong, Silvestre, Smart '11]

If in , is uniformly elliptic and , then there exists such that is outside of a closed set of dimension at most .

*As Scott duly observed, his authorship of a several joint papers where the two other authors have last names starting with “S” might lead to rather amusing citations.

The old site will be up at least for a while, but all the old posts have been copied to this 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 .)

]]> https://partialdifferentialequations.wordpress.com/2010/01/15/reviewing-the-regularity-theory-of-elliptics-pdes-via-the-laplace-equation-part-iii-representation-formulas/feed/ 1 Nestor https://partialdifferentialequations.wordpress.com/2010/01/14/laplaceii/ https://partialdifferentialequations.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.

]]> https://partialdifferentialequations.wordpress.com/2010/01/14/laplaceii/feed/ 2 Nestor https://partialdifferentialequations.wordpress.com/2010/01/13/joint-mathematics-meetings-in-san-francisco-blog/ https://partialdifferentialequations.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 again since is contained in and 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.