To present mathematical models and advanced techniques in the theory of partial differential equations, of frequent use in the applied sciences.

At the end of the course the student is expected to:

1. know the basic theory of Hilbertian Sobolev spaces: density and approximation by smooth functions, traces, compactness, embeddings, Bochner integral. The theory at the base of finite elements methods and the spectral properties and the alternative for general bilinear forms in Hilbert spaces. The most important fixed point theorems. The adjoint and the multiplier method for the optimization and control of quadratic functionals in Hilbert spaces. The basic theory of 1-d systems of conservation laws. 

2. be able to derive the variational formulation of the most common boundary and initial boundary value problems for stationary and evolutive PDEs and analyze their well-posedness. To apply the fixed point method to nonlinear equations like the stationary Navier-Stokes system or rather general semilinear equations. To analyze and solve quadratic control problems governed by elliptic equations, interpreting the results in terms of projections and computing the gradient of the functional. Explain the construction of rarefaction and shock waves for 1-d systems of conservation laws. Solve the Riemann problem for the p-system.

1. Sobolev spaces. Definitions of the Hilbert-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-Schauder 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. Quadratic 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, shocks, entropy condition. The Riemann problem. Application to the p-system.