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# Introduction to Partial Differential Equations

SUMMARY OF LECTURES

Lecture 1. Introduction, notations and general definitions; linear, semilinear, quasilinear, fully nonlinear PDEs. The transport equation, method of characteristics. The Laplace and Poisson equations, harmonic functions. Physical interpretation. Radial solutions of the Laplace equation. Fundamental solution.

Lecture 2. Explicit solution of the Poisson equation in R^n. Mean value theorem for harmonic functions. Converse mean value theorem.

Lecture 3. Application of the mean value formula: Newton's theorem. Weak and strong maximum principles for harmonic functions.

Lecture 4. Application of the maximum principle: positivity of solutions of the Laplace equation in a bounded domain. Uniqueness of the solution of the Poisson equation in a bounded domain. Smoothness of the solutions of the Laplace equation, bounds on the derivatives. Liouville theorem. Representation formula for the bounded solutions of the Poisson equation in R^n.

Lecture 5. Bounds on the derivatives of harmonic functions (proof). Analyticity of harmonic functions. Harnack inequality.

Lecture 6. Green's function and representation formula for the solution of the Poisson equation in a bounded domain. Calculation of the Green's function in the half-space.

Lecture 7. Characterization of mean-value subharmonic functions. The harmonic lifting. Convergence theorems for harmonic functions. Set of relative subfunctions, harmonicity of the supremum. (see Gilbarg-Trudiger, Sec. 2.8).

Lecture 8. Existence of solutions for the Laplace equation in a bounded domain (method of subharmonic functions).

Lecture 9. Solution of first exercise of the second exercise sheet. Energy methods: uniqueness of the solution of the Dirichlet problem, Dirichlet principle. The heat equation: motivations, derivation of the fundamental solution.

Lecture 10. Properties of the fundamental solution of the heat equation. Nonhomogeneous problem, Duhamel principle.

Lecture 11. Mean-value formula for the solution of the heat equation. Maximum principle.

Lecture 12. Application of the strong maximum principle for the heat equation: infinite propagation speed. Uniqueness of the solution on bounded domains. Maximum principle on unbounded domains. Uniqueness of the solution on unbounded domains. Smoothness of the solution of the heat equation.

Lecture 13. Convergence of the solution of the heat equation to the solution of the Laplace equation. Energy methods: uniqueness of the solution. Backward uniqueness. The wave equation: physical interpretation, solution in one dimension (on the line and on the half line).

Lecture 14. The Euler-Poisson-Darboux equation for the average of the solution of the wave equation on a sphere. Solution of the wave equation in dimension three: the Kirchoff formula. Solution of the wave equation in dimension 2.

Lecture 15. Solution of the wave equation in any dimension (without proof). Nonhomogeneous problem, Duhamel formula. Examples. Energy methods: uniqueness, backward wave cone.

Lecture 16. Lp spaces, definitions, results. The Fourier transform in L1. Properties of the Fourier transform. Plancherel's theorem, the Fourier transform in L2.

Lecture 17. Proof of Plancherel theorem. More properties of the Fourier transform in L2: invariance of the L2 scalar product, FT of derivatives, inverse FT. Application to PDEs: Bessel potentials.

Lecture 18. Application of the Fourier transform: the Schroedinger equation, conservation of the L2 norm. Solution of the wave equation. Equipartition of energy.

Lecture 19. Quasilinear PDEs of first order. Homogeneous case with constant coefficients. Nonlinear case with costant coefficient: local existence and uniqueness of the solution.

Lecture 20. Quasilinear PDEs with non constant coefficients. Method of characteristics. Local existence and uniqueness of the solution. Application: Burgers equation. Solution for short times, crossing of characteristics.

Lecture 21. Weak solutions for conservation laws. Rankine-Hugoniot condition. Global weak solution for the Burgers equation. Rarefaction waves. Uniqueness of weak solutions: entropy condition.

Lecture 22. Entropy condition and entropy solutions. Introduction to elliptic, second order PDEs. Weak derivatives.

Lecture 23. Interior approximation of Sobolev spaces by smooth functions. Global approximation of Sobolev spaces by smooth functions.

Lecture 24. Proof of the global approximation theorem. Global approximation up to the boundary (without proof).

Lecture 25. Characterization of Sobolev spaces via the Fourier transform. Introduction to the theory of distributions. Introduction to the variational formulation of second order elliptic PDEs.

Lecture 26. Weak solutions of second order PDEs: definition. Basics of the theory of Hilbert spaces. Lax-Milgram theorem (statement).

Lecture 27. Gagliardo-Nirenberg-Sobolev inequality for C^{1}_{c} functions. Extension to Sobolev functions. Embedding of W^{1,p}_{c} in Lq. Poincare' inequality.

Lecture 28. Proof of Lax-Milgram theorem. First existence theorem for second order elliptic PDEs.

Exam sessions:

12.02.2018, from 15.00 until 17.00, room N16.

16.04.2018

Recommended books:

Partial Differential Equations. L. C. Evans, American Mathematical Society.

Elliptic Partial Differential Equations of Second Order. D. Gilbarg and N. S. Trudinger, Springer.

Analysis. E. H. Lieb and M. Loss. American Mathematical Society.

Lecture Notes. (12.02.2018)

Introduction to Partial Differential Equations

## Lecturer

Prof. Dr. Marcello Porta