Lecture Notes Distributions and Partial Differential Equations by Thierry Ramond

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Lecture Notes Distributions and Partial Differential Equations written by Thierry Ramond, Université Paris Sud.

Lecture Notes Distributions and Partial Differential Equations written by Thierry Ramond cover the following topics.

Distributions in 1d
Test functions, Support of a smooth function, Plateau functions, Convergence in D(I), Definitions and examples, Definitions, Regular distributions, The Dirac mass, Hadamard principal value of 1/x, The order of a distribution, Definition, examples, Non-negative distributions, Derivatives of distributions, Product by a smooth function, Support of a distribution, Smooth partition of unity, Definitions, Some properties, Compactly supported distributions, Distributions supported at one point only, Sequences of distributions, Convergence in D′, Uniform Boundedness Principle, Exercises, Answers

Several variables calculus
Partial derivatives of functions, Definitions, Multiindices, The Chain rule, Stokes formula, Surface measure, Stokes Formula, Distributions on Rn, Surface measure: a distribution point of view, The surface measure on the sphere, A proof of Stokes’ formula, Exercises, Answers

The Laplacean
A Fundamental solution of the Laplacean, Green’s formula for the Laplacean, The Laplacean of a radial function, Computation of ∆( 1∥x∥n−2 ), Harmonic Functions, Exercises

Convolution of distributions
Differentiation and integration inside the bracket, Tensor products, Tensor products of functions, Tensor products of distributions, Convolution, The key lemma, Convolvable Pairs, Definition of the convolution of distributions, Main properties of the convolution, Particular cases : E′(Rn) ∗ C∞(Rn) and D′(Rn) ∗ C∞0(Rn), Application to constant coefficients PDE’s, Notations, Fundamental solutions, Singular support

The Fourier Transform
The space S(Rn), Definitions and examples, Convergence in S(Rn) and density results, The Fourier transform in S(Rn), Definition and first properties, Gaussians (1), The Inversion formula, Parseval and Plancherel, Convolution and the Fourier transform in S(Rn), The space S′(Rn) of tempered distributions, Definition, examples, Convergence in S′(Rn), The Fourier Transform in S′(Rn), Definition, Gaussians, Symmetries, The Fourier Transform on L1 and L2, The Fourier Transform of compactly supported distributions, Smoothness, Analyticity (1): Paley-Wiener’s theorem, Analyticity (2): Paley-Wiener-Schwartz’s theorem, Convolution and the Fourier transform in S′(Rn), E′(Rn) ∗ S(Rn), E′(Rn) ∗ S′(Rn), S(Rn) ∗ S′(Rn)

The wave equation
The partial Fourier transform, A fundamental solution of the wave equation, The support of the fundamental solution, The Cauchy problem

Sobolev spaces
Sobolev spaces on Rn, Definitions, Density of smooth functions, Multipliers of Hs(Rn), Some Sobolev Embedings, The Hs(Rn)/H−s(Rn) duality, Trace of an element in Hs(Rn), s > 1/2 , Sobolev spaces on Ω, Sobolev spaces of integer order on Rn, Sobolev spaces of positive integer order on Ω, Sobolev spaces of negative integer order on Ω, Poincaré’s Inequality, The Dirichlet Problem, An introduction to the finite elements method

Lebesgue Integration
Axioms, The dominated convergence theorem, Functions given by integrals with parameters, Fubini-Tonelli, Change of variable

Hilbert spaces
Scalar Products, Orthogonality, Riesz’s theorem

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