Final syllabus

• 5.1. H?lder spaces: Everything.
• 5.2. Sobolev spaces: Everything.
• 5.3. Approximation: Everything.
• 5.4. Extensions: Everything.
• 5.5. Traces: You need to understand why we cannot simply evaluate an \(L^p\) function on the boundary, and that the regularity provided by being in a Sobolev space helps. You need to understand what having zero trace means, and how this relates to the \(W^{k,p}_0\) spaces.
• 5.6. Sobolev inequalities: You need to know the statements of every theorem here. You should have an idea of the proofs ¨C they all follow a similar idea, first proving it for smooth functions using the Fundamental Theorem of Calculus, then using approximation by smooth functions.
• 5.7. Compactness: You need to know the statement of the Rellich¨CKondrachov Theorem, as well as the remark after its proof. You should understand that the proof essentially consists of using the Arzela¨CAscoli Theorem, after mollifying the elements of the sequence. You need to understand what the theorem means: A bound on the Sobolev norm of a sequence of functions will imply that the sequence (or at least some subsequence) will converge in the \(L^q\) norm, for some q.
• 5.8. Additional Topics: We have used Poincar¨¦'s inequality and difference quotients. You need to know Poincar¨¦'s inequality.
• 5.9. Other spaces of functions: You should understand that the negative-order Sobolev spaces \(W^{-k,p}\) are dual spaces of \(W^{k,p}_0\), and that we can identify \(W^{-1,p}\) with functions in \(L^p\). We have also used time-dependent functions (Section 5.9.2), but somewhat informally; read up on this if you need a more solid understanding.

• 6.1. Definitions: You need to know this by heart. You need to be able to derive the weak formulation of a given PDE, and vice versa, be able to deduce which PDE a bilinear form comes from.
• 6.2. Existence of weak solutions: You need to know the statements, and have an idea of the proofs, of Theorems 1¨C3. You need to know the Lax¨CMilgram  Theorem by heart, and be able to apply it to a given problem. You need understand how we derive energy estimates and why it's important (Theorem 2), and understand how Theorem 3 (The First Existence Theorem) follows from the previous two theorems. You need to know the statements of Theorems 4¨C6.

• 6.3. Regularity: You need to know the statements of Theorems 2 and 4 (Theorems 1 and 3 are just special cases of Thms 2, 4). You also need to understand the main idea in the proof of Theorem 1, namely using \(u_{x_kx_k}\) as a test function in the weak formulation. You also need to understand why the proof is so intricate ¨C we need to use difference quotients because we don't know a priori that u is differentiable, and we need to localize away from the boundary.

• 6.4. Maximum principles: You need to know what the weak and strong maximum principles are, that we only prove them for classical (smooth) solutions, and you need to know the idea of the proof of the weak maximum principle.

Chapter 7

• 7.1. Second-order parabolic equations: You need to ble able to check that an equation is parabolic, and what we mean by a weak solution. You should understand the construction and uniqueness of weak solutions as I outlined in the lectures; this is Theorems 1¨C4.