Ing Ind - Inf (Mag.)(ord. 270) - MI (487) MATHEMATICAL ENGINEERING - INGEGNERIA MATEMATICA
095963 - ADVANCED PARTIAL DIFFERENTIAL EQUATIONS
To present mathematical models and advanced techniques in the theory of partial differential equations, of frequent use in the applied sciences.
Risultati di apprendimento attesi
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.
Basic knowledge of the most common equations of mathematical Physics: Heat and Laplace equation (separation of variables, maximum principles, fundamental solution). Wave equation (d'Alembert, Kirchhoff formulas). Scalar conservation laws (characteristics, rarefaction waves, shocks, 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.
Modalità di valutazione
Written examination: to the student it is required to solve problems concerning boundary value problems, control problems and nonlinear equations via fixed point methods. Highly insufficient grades (<12/30) in the written examination are invited to skip the oral discussion.
Oral discussion on the whole program: the student must be able to explain the main concepts and prove the main results with property of language.
There is a mid-term written examination on parts 1,2,3 of the program with the aim of monitoring the preparation state of the class.
S. Salsa, Partial Differential Equations in action, from modelling to theory, Editore: Springer 3Â° edition, Anno edizione: 2016, ISBN: 978-3-319-31237-8
S. Salsa G. Verzini, Partial Differential Equations, complements and exercises, Editore: springer, Anno edizione: 2015, ISBN: 978-3-319-15415-2
Files on Control Theory (on line, from the course web site).
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