Scope and detailed program
Aim: To present mathematical models and advanced techniques in the theory of partial differential equations, of frequent use in the applied sciences.
1. Sobolev spaces. Definitions of the Hilber-type spaces and their duals. Traces. Poincaré inequalities. Immersion theorems. Time dependent Sobolev spaces. Bochner Theorem. Integration by parts.
2. Elliptic equations in divergence form. Variational formulation of the most common boundary value problems. Lax-Milgram Theorem and analysis of the well posedness. Stability estimates. Weak maximum principles. Hilbert triplets. Fredholm’s Alternative Theorem for bilinear forms. Eigenvalues and eigenfunctions. The Stokes system, equilibrium of a plate. Method of sub and supersolutions for semilinear equations.
3. Evolution equations.Abstract evolution problems: existence, uniqueness, stability. Parabolic equations .Weak formulation of the most common initial-boundary value problems. Weak maximum principles. Wave equation: Weak formulation of the most common initial-boundary value problems. Faedo-Galerkin method and analysis of the well posedness.
4. Fixed point techniques. The contraction Theorem. Leray and Leray-Shauder Theorems. Application to the steady Navier-Stokes equations.
5. Optimization and control. Minimization of functionals in Banach and Hilbert spaces. Variational inequalities and optimality conditions. Convexity and semicontinuity. Projections. Quadratric functionals. Control problems governed by linear elliptic and parabolic equations. Well posedness and optimality conditions. Lagrange multipliers and the adjoint problem.
6. Systems of conservation laws. Hyperbolic systems. Characteristics, Riemann invariants, Weak solutions, Rankine-Hugoniot condition. Rarefaction waves, shoks, entropy condition. The Riemann problem. Application ti the p-system.
Basic knowledge of the most common equations of mathematical Physisc: Heat and Laplace equation (separation of variables, maximum principles, fundamental solution). Wave equation (d'Alembert, Kirchhoff formulas). Scalar conservation laws (characteristics, rarefaction waves, shoks, Rankine-Hugoniot and Entropy condition, solution of the Riemann problem).
Basic calculus with Distributions.
Lebesgue integration and spaces of p-summable functions. Banach and Hilbert spaces. Riesz Representation and Lax-Milgram Theorems. Fourier series. Open mapping Theorem. Weak convergence. Banach-Alaoglu Theorem. Spectral theory of compact self adjoint operators in Hilbert spaces.