# TMA4305 Partial Differential Equations 2019

## Messages

- (2019-12-04) The solutions to today's exam problems are available. See “Old exams” in the menu.

## Lectures

Week | Chapter/section | Remarks and notes |
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34 | Ch 1–3 Notes | Read Ch 1 on your own. Also have a look at Ch 2 on your own – but we will return to it when needed (especially the higher dimensional calculus). For needed background on well-posedness theory for ODEs, consult for example my notes on dynamical systems, Ch 1, pp. 2–7. (No need to study the proofs.) |

Covered introductory and preliminary material, including some notation. The concept of well-posed problem [Jaques Hadamard 1902]: Solutions exist, are unique, and depend continuously on data.Basic theory of transport equations and conservations laws in one space dimension, balance law in multiple dimensions. |
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Notes: Collisions of characteristics (A5 for the screen, A4 for printing) |
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35 | Ch 3 end Ch 4: 4.1–4.4 | Start on the wave equation (Ch 4 in Borthwick): Characteristics and d'Alembert's solution (§4.2), and Duhamel's principle (§4.4) (if we get that far) |

Wednesday: Finished Ch 3 with a look at a traffic model. Also proved that the characteristic method applied to the IVP \(u_t+a(t,x,u)u_x=f(t,x,u)\) with \(u(0,x)=g(x)\) actually solves the problem. I used differentials to handle the algebra; much easier than transforming partial derivatives between coordinate systems. |
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Thursday: Wave equation, D'Alembert's solution, Duhamel's principle. | ||

36 | Ch 4: 4.6, 4.7 Notes | The wave equation in higher dimensions, energy method for uniqueness |

Wednesday: Solution formulas for the wave equation in 3 and 2 space dimensions. Huygens' principle in 3D, not in 2D. (A similar technique works for \(2k+1\) dimensions, with subsequent descent to \(2k\) dimensions.) |
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Thursday: Proof that the solution formula in three dimensions actually solves the problem. Energy methods for uniqueness and domains of dependence in the wave equation. |
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Notes: On the energy method for wave equations (A5 for the screen, A4 for printingThese notes may be thought of as an alternative to, and expansion of, §4.7) |
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37 | Notes Ch 6 | A bit more from the above notes on the energy method. Start chapter 6 (heat equation) |

Wednesday: Proved Theorem 2 from the end of the note (above) on the energy method for wave equations. The current version in that note is too complicated. I will update it to match the way I lectured it.Started on the heat equation, and derived the formula for the standard heat kernel in \(n\) space dimensions. |
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Thursday: The solution of the initial value problem for the heat equation, and Duhamel's principle for the non-homogeneous heat equation. As part of the proof I did a partial integration, moving the Laplacian from one term to another. This is really an application of Green's second identity ( B Theorem 2.11): The boundary integrals vanish when we integrate over a large enough region containing the support of \(f\). |
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38 | Notes Ch 9: 9.5 and more | Notes on the parabolic maximum principle (see below) Perhaps start working on Ch 9 and my associated notes. |

Wednesday: The weak maximum principle for the heat equation, on bounded and unbounded domains. Corollaries are continuous dependence of data, uniqueness. | ||

Notes: Weak maximum principle for the heat equation (A5 for the screen, A4 for printing)These notes are an alternative to §9.5 Updated 2019-09-18 with an even better version of the maximum principle on \(\mathbb{R}^n\) than the one in the lecture!Updated 2019-09-19 (removed an erroneous example) |
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Thursday: Started notes on the Laplace equation (below). Covered the mean value property, and proved its equivalence to harmonicity. Regularity (harmonic functions are \(C^\infty\)), and the strong and weak maximum principle. | ||

Notes: Harmonicfunctionology (A5 for the screen, A4 for printing)These notes are related to §§9.1–9.4 Latest substantial update 2019-10-13: More on regular boundary points, and Lebesgue's example. The latter is set in small print; no need to study the details. |
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39 | Notes Ch 9: 9.1–9.4 | Notes on the Laplace equation (continued) |

Wednesday: Solution of the heat equation in the (2-dimensional) unit disk, Poisson kernel (B §9.1). Then back to my notes; started on the Poisson equation. |
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Thursday: The fundamental solution \(\Phi\) for the Poisson equation and its basic properties; solution \(u=\Phi*f\) for the Poisson equation \(-\Delta u=f\) where \(f \in C^2_c(\mathbb{R}^n)\); and started on an introduction to Green's functions by deriving the formula \[ u(y) = -\int_{\Omega} \Phi(x-y)\,\Delta u(x)\,d^n x + \int_{\partial\Omega}\bigl(\Phi(x-y)\partial_nu(x)-u(x)\partial_n\Phi(x-y)\bigr) . \] Briefly stated how to get rid of the troublesome \(\partial_nu\) term in the boundary integral by adding a harmonic function to \(\Phi(x-y)\). | ||

40 | Notes | Poisson integral formula in 2 or more dimensions, Perron's method (from updated “harmonicfunctionology” notes, see above). |

Wednesday: The Dirichlet problem in balls, solved by Poisson's integration formula. | ||

Thursday: The Dirichlet problem in bounded domains. Showed that the pointwise supremum of all subsolutions is harmonic. (Note that the note was updated after the lecture; see above.) | ||

41 | Notes 7.1–7.4 | Finish “Harmonicfunctionology” notes with a look at barriers and boundary values; then back to the book, where we start work on the more functional analytic part of the course. We start with a quick introduction (review for some) of function spaces. |

Wednesday: Finished the “Harmonicfunctionology” note, including some material that I ought to include: A barrier for two dimensions, and an example in three dimensions. (Now in the notes.) | ||

Thursday: Chapter 7: Lebesgue measure and integral, \(L^p\) spaces. Also \(L^p_{\operatorname{loc}}(\Omega)\), not mentioned in the book. | ||

42 | 7.5–7.6 Start Ch 10 | Finish Ch 7. Move on to weak derivatives and weak solutions to PDEs, Sobolev spaces. |

Wednesday: §7.5 (orthonormal bases) real quick, §7.6 (selfadjointness for the Laplace operator) even quicker; but I left the discussion on eigenvalues and -functions for later. §10.1 (test functions and weak derivatives), skipped §10.2 (for now), beginning of §10.3 (Sobolev spaces). |
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Thursday: §10.3, §10.4 (Sobelev spaces and Sobolev regularity). | ||

43 | 10.2, 10.5, 11.1–3 Notes | Sobolev spaces and boundary traces. Weak solutions (perhaps including a brief intro to distribution theory). |

Wednesday: Weak solutions of continuity equations – The Rankine–Hugoniot shock condition, weak solutions of the Posisson equation, Dirichlet's principle. | ||

Thursday: Dirichlet's principle contiunued, the Poincaré inequality, existence of weak solutions by minimising the Dirichlet functional. Boundary traces. | ||

Notes: Boundary traces. (A5 for the screen, A4 for printing)Update 2019-10-24: More direct proof. (And again the same day (v2): Fixed the norm estimates.) |
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44 | Ch 11 11.4–5 | Elliptic regularity, spectral theory for the Dirichlet Laplacian. |

Wednesday: §11.4 – Elliptic regularity. | ||

Thursday: §11.5 – The spectral theorem for the Dirichlet Laplacian | ||

45 | Ch 11, 12 11.6, 11.8, 12.2–3 | The min-max theorem, Rellich's theorem, Euler–Lagrange equations, distribution theory, applications to PDE |

Wednesday: Finishing remarks on the Spectral theorem for the Dirichlet Laplacian Notes, §11.6 – compactness and Rellich's theorem |
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Thursday: §11.8 Euler–Lagrange equations (generalising the Dirichlet principle) Started Ch 12 on distributions |
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Notes: Compactness in Banach spaces Handwritten note (I may type it up later) meant to cover a possible gap in students' background knowledge . |
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46 | Ch 12 | Distribution theory, applications to PDE |

Wednesday: §12.4 Fundamental solutions, convolutions. I defined the support of a distribution as the smallest closed set so that \((u,\phi)=0\) whenever the support of \(\psi\) is disjoint from the set. I also stated (without proof) that distributions with compact support have finite order, in that \((u,\psi)\) can be bounded by a finite number of derivatives of \(\psi\). Moreover, such a distribution is a finite linear combination of derivatives of functions. (This is mostly useful background material. I really should write it up, briefly.) |
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Thursday: A bit more on convolutions, then §12.6 time-dependent distributions, applied to conservation laws (rederived the Rankine–Hugoniot shock condition) and the 1-D wave equation / wave kernel | ||

47 | Summaries | Final week of lectures. Summaries / repetition, maybe go through some exam problems |

Wednesday: Summarised some basics regarding the divergence theorem and partial integration; conservation laws; the wave equation; just barely got started on the heat equation. | ||

Thursday: Continued summarising: More on the heat equation; then harmonicfunctionology (mean value property, sub- and superharmonic functions, weak and strong maximum principles, the fundamental solution, Green's function, Poisson kernels, the Dirichlet principle, Poincaré's inequality, … and then I ran out of time.End of lectures. |