An introduction to Sobolev Spaces and Partial Differential Equations

  

Program

05.11 Measures. Measurable sets and functions. Lebesgue measure. Monotone Convergence Theorem. Dominated Convergence Theorem. Hoelder's inequality.

12.11 L^p spaces. Interpolation inequalities in L^p spaces. Approximation by smooth functions. Riesz representation theorem. Strong and weak convergence. Relations between weak, strong, and a.e. convergence. Weak derivatives. Sobolev spaces. Basic properties. Approximation by smooth functions. Trace inequality. Trace operator. 

18.11 Sobolev inequality. Morrey inequality. Continuous and compact inclusions, Rellich's theorem.

03.12 Hilbert space. Riesz-Frechet representation theorem. Homogenoeous and non-homogeneous solutions to the Neumann and the Dirichlet problems for the Laplace operator: weak solutions, existence and uniqueness.

10.12 Maximal regularity estimates (formal) for the homogeneous Neumann problem: in the interior and up to the boundary.

17.12 and students' seminars on January 2015. Connections to Calculus of Variations. The Lax-Milgram theorem; existence and uniqueness for second-order elliptic operators. Smoothness of solutions to the Dirichlet problem. Minty-Browder Theorem: existence and uniquenesss for the p-Laplace equation. Gagliardo-Nirenberg interpolation inequalities.

References:
* H. Brezis. Functional Analysis, Sobolev Spaces and PDEs. Springer.
* L.C. Evans. Partial Differential Equations. Graduate Studies in Mathematics 19, AMS.


Pagina iniziale della didattica

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