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of pdes probability and differential geometry a mathematical melting pot a blog about mathematics ok but mostly analysis of pdes probability and differential geometry blog about update blog restarted new name new blog address published december 29 2010 uncategorized 2 comments after an almost year long hiatus i am getting back into math blogging the new address is a bit long but at least easy to remember turns out pdeblog wordpress com was already taken please update your bookmarks 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 reviewing the regularity theory of elliptics pdes via the laplace equation part iii representation formulas published january 15 2010 partial differential equations 1 comment tags a priori estimates harmonic functions harnack s inequality integral representation formulas laplace equation mean value property 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 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 continue reading reviewing the regularity theory of elliptics pdes via the laplace equation part iii representation formulas reviewing the regularity theory of elliptics pdes via the laplace equation part ii published january 14 2010 partial differential equations leave a comment tags a priori estimates harmonic functions harnack s inequality laplace equation mean value property 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 joint mathematics meetings in san francisco blog published january 13 2010 quick posts leave a comment tags adriana salerno ams conference joint math meetings 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 reviewing the regularity theory of elliptics pdes via the laplace equation part i published january 6 2010 partial differential equations 3 comments tags harmonic functions harnack s inequality laplace equation mean value property 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 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 new year new posts published january 4 2010 quick posts 2 comments after 4 months of inactivity i am taking up again the task of updating the blog which has suffered of neglect due to my terrible time management skills 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 if you want to kill your productivity move to a new house published august 3 2009 differential geometry quick posts 2 comments i know things have been extremely slow lately but i was moving last week and well that means packing everything moving things to the new house cleaning the old house unpacking things at the new one well you get the picture 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 books of the trade published july 23 2009 quick posts 3 comments 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 solving the monge ampere equation continued and finished published july 18 2009 fully non linear equations monge ampere equation partial differential equations leave a comment 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 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 prescribing the ricci curvature of riemannian metrics published july 9 2009 differential geometry 4 comments i would like to spend the next few posts talking about a problem i read about in chapter 5 of the book einstein manifolds by arthur besse it turns out that the name arthur besse is made up as you can read in the preface of the book the question that we want to address is the following 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 connectio...