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1996-09-04
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In the first week and yesterday I lectured from my notes, covering some of the basics of metric spaces, concluding (so far) with the result on differentiating under the integral sign.
Today I started lecturing from Evans' lecture notes, covering some of the basics on notation including multi-indices and Taylor's formula in several variables, whereupon we classified PDEs as linear, semilinear, quasilinear, or fully nonlinear. Then we solved the transport equation and started looking at the Laplace equation, deriving the most general radially symmetric harmonic function.
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1996-09-11
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We have covered the fundamental solution for the Laplace operator and shown how it solves the Poisson equation; then we have proved the mean-value property for harmonic functions and shown that, conversely, any continuous function satisfying the mean-value property is in fact infinitely differentiable and harmonic. We finished with the strong maximum principle.
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1996-09-18
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Skipped all the estimates on derivatives (Thm 7 p. 32) and the proof that harmonic functions are analytic. Lectured on the other various properties of harmonic functions (including Harnack's inequality), and defined Green's function. We skipped the discussion of Green's function in the half space, but did it on the unit ball instead, with the Poisson integral formula resulting.
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1996-10-02 (two weeks have passed)
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Finished up the discussion of the Poisson integral formula with proving it gives the proper boundary conditions. Then we gave the "energy" proof of uniqueness for the nonhomogeneous Dirichlet problem, and discussed Dirichlet's principle.
The heat equation was next: We derived the fundamental solution in a fashion somewhat different from the lecture note: By utilising the scale invariance of the equation and requiring that the space integral of the solution be independent of time. We showed how this leads to a solution of the heat equation with initial data in Euklidean space, and discussed how Duhamel's principle leads to a solution of the nonhomogeneus heat equation (but we skipped the proof that it really solves the problem). We explained the mean value theorem for solutions of the heat equation, but skipped the proof (which does not seem to be very instructive). We used this to show the strong maximum principle and hence uniqueness of the initial-boundary value problem for the heat equation on bounded domains. We also showed the maximum principle for solutions on Euklidean space satisfying a growth condition. Finally, we went through parts of the proof of regularity for solutions of the heat equation.
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1996-10-09
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Amplified on our partial exposition of the regularity proof by a brief discussion of mollifiers. Skipped the small section on estimates for solutions of the heat equation. Discussed "energy methods" for the heat equation.
Derived d'Alembert's solution formula for the one-dimensional wave equation. Used the method of spherical means to give a solution formula for the wave equation in odd dimensions.
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1996-10-16
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Wrote up the above-mentioned solution formula for the special case n=3 (Kirchoff's formula), which is much simpler than the general case. Used the method of recursive descent to obtain a solution formula for the wave equation in two dimensions (the same method works for any even dimension, but we skipped that). Brief look at Huygens' principle and energy methods. We skipped the nonhomogeneous wave equation.
Started on Chapter 3 (first order nonlinear PDEs) where I introduced complete integrals and envelopes, before giving a hurried and very brief introduction to the method of characteristics for the quasilinear equation au_x+bu_y=c (where a, b, c are functions of x,y,u but none of the derivatives of u).
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1996-10-22
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On the solution of the general first order quasilinear equation in two variables, ala Fritz John (pp. 9-18). Evans treats this (pp. 104-107) as a special case of the more general and more difficult nonlinear equations.
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1996-10-30
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We are done with the method of characteristics for general nonlinear first order PDEs (section 3.2). Skipped over the Hamilton-Jacobi equation (section 3.3), and started on conservation laws (3.4). Derived the integral formulation of a conservation law (not to be confused with the weak formulation mentioned in the book), and used it to derive the Rankine-Hugoniot jump condition for shocks. Studied Burgers' equation briefly. Introduced the Lax shock condition, and very briefly mentioned the "rubber band" (convex/concave envelope) method of solving a Riemann problem.
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1996-11-06
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Conservation laws finished with more on the weak formulation and Rankine-Hugoniot condition, expansion waves, the vanishing viscosity method, shock profiles and the entropy condition for nonconvex flux functions; the Riemann problem and its solution by the "rubber band" (convex or concave envelope) method.
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1996-11-13
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Lectured on noncharacteristic Cauchy problems, the Cauchy-Kowalevskaya theorem, and the Fredholm uniqueness theorem for second order quasilinear PDEs. Then we specialised to two variables and presented the classification of such equations into elliptic, parabolic, and hyperbolic equations. Notes handed out (4 A4 pages) are available as a DVI file (requires the two PostSCript files wchar-1.eps and wchar-2.eps) or PostSCript file.