Comments, lectures

• Lecture 8, October 19: The lecture consisted of a discussion of Sobolev spaces. We started by introducing the notion of  a weak derivative and defining $W^{k,p}(U)$, the associated norm etc.  After going through some basic properties of the Sobolev spaces, we proved that $W^{k,p}(U)$, with its associated norm, is a Banach space. After this introduction, most of the lecture was concerned with the topic of approximating functions belonging to Sobolev spaces with smooth functions. To this end, we introduced the notion of a mollifier and demonstrated how convolution with a mollifier can be used to approximate a function in a Sobolev space with a smooth function. Using a partition of unity, combined with convolution with a mollifier, we sketched an argument which, given a function $u\in W^{k,p}(U)$, yields a sequence of $u_m\in W^{k,p}(U)\cap C^\infty(U)$ converging to $u$ in $W^{k,p}(U)$. In this case, we had to assume $U$ to be bounded. Assuming, in addition, the boundary to be $C^1$, it is possible to approximate with functions that are smooth up to and including the boundary. The relevant sections in Evans' book are 5.1-5.3.
• Lecture 9, October 26: The lecture started with a discussion of extensions of elements of $W^{k,p}(U)$ to $W^{k,p}(\mathbb{R}^n)$. We then discussed the problem of restricting a $W^{1,p}(U)$ function to the boundary. The remainder of the lecture was devoted to Sobolev inequalities. We sketched partial proofs of the embeddings of $W^{1,p}(\mathbb{R}^n)$ into $L^{p^*}(\mathbb{R}^n)$ and into $C^{0,\gamma}(\mathbb{R}^n)$, depending on whether $p<n$ or $p>n$. Thee relevant sections in Evans' book are 5.4, 5.5 and parts of 5.6.
• Lecture 10, November 2: The lecture started with a discussion of how to get from Sobolev inequalities for functions on $\mathbb{R}^n$ to Sobolev inequalities for functions on bounded open subsets $U$ with $C^1$ boundary $\partial U$. The general Sobolev inequalities were then stated. This was followed by a discussion of what it means for one Banach space to be compactly embedded in another. We also stated the Rellich-Kondrachov Compactness Theorem. The next topic was boundary value problems for second order elliptic PDE's. We started with an informal discussion of $-\Delta u+u=f$. This naturally led to the Riesz representation theorem, which we recalled. Since this result is not sufficient for all the problems of interest, we stated and proved the Lax-Milgram Theorem. We ended the lecture by introducing the class of boundary value problems we are interested in, defining uniform ellipticity and providing a weak formulation for boundary value problems.
• Lecture 11, November 9: The lecture started with a repetition of the formulation of the boundary value problems of interest. We also recalled the notion of uniform ellipticity and the weak formulation of the boundary value problems of interest. Next we considered the question of whether the weak formulation satisfies the conditions of the Lax Milgram Theorem. In general, this turns out not to be the case. However, there is always a $\gamma\geq 0$ such that $B_\gamma[u,v]=B[u,v]+\gamma (u,v)_{L^{2}(U)}$ does satisfy the criteria. In particular, for every $\mu\geq\gamma$ and every $f\in L^2(U)$ there is thus a unique solution $u\in H^1_0(U)$ to $Lu+\mu u=f$ in $U$ and $u=0$ on $\partial U$. In order to take the step from this observation to solvability of the original problem, we introduced the formal adjoint of $L$, the adjoint bilinear form and the notion of a weak solution of the adjoint problem. Given this terminology, we formulated the consequences of the Fredholm alternative for the boundary value problems of interest. We ended the lecture by discussing the interior regularity of weak solutions.
• Lecture 12, November 16: The lecture started with a statement of the interior and boundary regularity results in the smooth setting. We then discussed, preliminarily, the spectrum of uniformly elliptic operators. After this, the discussion turned to maximum principles. We stated and proved the weak maximum principle. We also stated the strong maximum principle and Harnack's inequality. After this we turned to the topic of eigenvalues and eigenfunctions in the case of symmetric operators (i.e., when the operator is equal to its formal adjoint) satisfying the conditions of the Lax Milgram Theorem. First, we stated general results, such as the fact that the eigenvalues are real, have finite multiplicity and can be arranged so that they diverge to infinity. Moreover, there is an orthonormal basis of eigenfunctions of $L^2(U)$. This essentially follows from results in functional analysis concerning compact operators. We then stated and proved the following results concerning the principal eigenvalue (i.e., the smallest eigenvalue): it can be calculated using Rayleigh's formula; the minimum is attained for a strictly positive function; and the eigenspace corresponding to the principal eigenvalue is 1-dimensional.
• Lecture 13, November 23: The lecture notes are to be found here Download here. There are some typos in the lecture notes. It's an exercise to correct the typos.
• Lecture 14, November 30: The lecture notes are to be found here Download here.