| Date | Speaker | Abstract |
|---|---|---|
| 9/28 | Tala Yasenpoor | Sobolev spaces and approximations. We start by a brief introduction on Holder and Sobolev spaces, continued by an introduction on weak derivatives. Then we'll discuss the motivation behind them, and show that weak derivatives, in case of existence, are unique and mention some of their most useful properties. Next, we'll develop some procedures to approximate functions in Sobolev spaces by smooth functions and discuss how we can extend a function in a sobolev space of an open set to a function on the sobolev space of R^n. |
| 10/05 | Mollifiers, approximation by smooth functions, extensions, and traces (Part 1). This week, we'll start with discussing some useful properties of mollifiers that'll help us in proving the approximation theorems later on. After that, we'll see how to extend functions in W^{1,p}U to functions in W^{1,p}R^n such that the extension preserves the weak derivatives across the boundary of U. And finally, we discuss the possibility of assigning boundary values along the boundary of U under some conditions using traces. | |
| 10/12 | Mollifiers, approximation by smooth functions, extensions, and traces (Part 2). We will finish up the remaining topics from last week. | |
| 10/19 | Paul Tee | GNS and Morrey's Inequalities. We begin our discussion of the Sobolev inequalities which describes the relationship of the Sobolev spaces W^k,p to various other spaces. GNS tells us for p in (1,n), the (weak) derivative of u controls u in a sense. This inequality will relate W^1,p to L^q where q is TBD in the talk. Morrey's inequality tells us for p in (n,infty), up to a measure 0 replacement, the weak derivative of u controls u in another sense. This inequality will relate W^1,p to C^0,gamma where gamma is TBD in the talk. We will briefly mention the borderline case of p=n. In light of the approximation theorems Tala presented over the past few weeks, most of the talk will be focused on the cases where u is smooth. |
| 10/26 | Rellich Lemma and Poincare Inequality. This week we discuss the important compactness result of Rellich, which tells us that W^1,p embeds compactly into L^q for q between 1 and p*:=np/n-p as in the GNS inequality. We apply this theorem to get Poincare inequality: for functions u in W^1,p, the difference between u and its average value on U is controlled by the weak derivative, where everything is measured in the p-norm. | |
| 11/02 | Cole Durham | Lax-Milgram Theorem, Energy Estimates, and First Existence Theorem. We will introduce the divergence and non-divergence forms of an elliptic partial differential operator. In order to solve boundary value problems for divergence form operators, we will first prove Lax-Milgram in its general form, then prove energy estimates and the First Existence Theorem for weak solutions to see how Lax-Milgram applies to the PDEs we are interested in. |
| 11/09 | Second and Third Existence Theorems, Regularity of Weak Solutions. In this talk we will see how our previous work and the Fredholm Alternative combine to give two more existence theorems. We then show (time permitting) that for f in H^m, a weak solution u to the equation Lu=f will belong to H^m+2. | |
| 11/16 | Regularity of weak solutions to Lu=f. We prove (under appropriate assumptions on the coefficients of L) that if f is in H^m(U), then a solution u to the problem Lu=f must lie in H^m+2(U). This will be done first on the interior of U, then extended up to the boundary. | |
| 11/30 | Paul Tee | Maximum Principles for Elliptic Operators. We will study maximum principles for 2nd order elliptic operators in nondivergence form. As opposed to the integral-based estimates shown by Cole in the previous month, these will instead be pointwise estimates. The outline will be as follows: we will start with a proof of weak maximum principle, then use it to prove the infamously technical Hopf Lemma, then using the Hopf Lemma to give a short proof of the strong maximum principle. |
| 12/7 | Harnack Inequality. Harnacks inequality says that solutions to elliptic operators are bounded in a strong sense. We will prove Harnacks inequality for elliptic operators with smooth 2nd order coefficients and no lower order coefficients. The proof will be elementary but tricky. If time permits we will mention maximum principles for parabolic operators. |