# 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.