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# MATHEMATICAL METHODS AND MATHEMATICAL PHYSICS FOR APPLICATIONS

 Degree course Civil Engineering Curriculum IDRAULICA Learnings Orientamento unico Academic Year 2014/2015

## Module: MATHEMATICAL METHODS

 Degree course Civil Engineering Curriculum IDRAULICA Learnings Orientamento unico Academic Year 2014/2015 ECTS 6 Scientific Disciplinary Sector MAT/05 Year First year Time unit First semester Class hours 48 Educational activity Related and integrative training activities

## Single group

 Professor GIUSEPPINA BARLETTA Objectives The main purpose of the course is to provide the basic definitions and the main functional analysis results underlying the methods used to study linear and non linear differential equations (this notions are a necessary tool to treat and model many problems arising from the applied sciences).Also, the student should get a theoretical knowledge as to allow a good autonomy in the choice and use of analytical tools necessary for the study of various problems. Finally, he will be able to deepen his knowledge by himself. Programme Premises to the course. Introduction to variational methods for the study of differential equations: motivations, examples. Need for numerical resolution.Elements of Functional Analysis (I-II CFU)Metric spaces and normed spaces. Basic concepts. The normed space R^N:Young, Holder, Cauchy-Schwarz and Minkowski's inequalities. Successions in a metric space. Continuous and semicontinuous functions. Compact and complete metric spaces. Weierstrass theorem. Banach spaces. Functional spaces: main examples. Hilbert spaces. Parallelogram rule. Cauchy-Schwarz inequality. Fourier series in one or more variables (overview). Introduction to Lebesgue's theory of measure and integration. Theorems of Fubini and Tonelli. Main properties of the L^p and L^p_loc spaces. Notes on the theory of distributions in R and in R^N. Notations and definitions. Basic examples. Derivative of a distribution. Notes on the distributional convergence. Sobolev spaces. Poincaré inequality. Diving theorems for the Sobolev spaces. Trace inequalities. Variational formulation of non-linear problems (III-IV CFU)Elliptical equations. Classic, strong and weak (or variational) solutions. Fourier developments for some problems at the limits (outline). Variational formulation of a problem of diffusion, transport and reaction in the one-dimensional case with boundary Dirichlet, Neumann, mixed and Robin conditions. Variational formulation of the Poisson problem. Homogeneous and non-homogeneous Dirichlet conditions. Neumann problem. Mixed and Robin problems. General equations in the divergence form. Issues of regularity. Parabolic equations. Classic, strong and weak (or variational) solutions.Critical point theory and its applications (V CFU)Non-linear analysis. Strong differential (or Fréchet). Weak differential (or Gateaux). Link between strong and weak differentiability. Introduction to the theory of critical points for regular functions. Direct method in Calculus of Variations: semicontinuity, compactness and coercivity.Further Elements of Functional Analysis (VI CFU)Linear operators. Dual spaces. Riesz representation theorem. Bilinear forms, abstract variational problems. Lax-Milgram theorem. Symmetrical bilinear forms. Approximation and method of Galerkin: existence, uniqueness and stability of the discrete solution, convergence. Lemma of Céa. Examples.Lions-Magenes theorem. Books I-II CFUN. Fusco, P. Marcellini, C. Sbordone, Analisi Matematica due, Liguore Editore, 1996.(Spazi metrici e spazi di Banach)G. Di Fazio, M. Frasca, Metodi Matematici per lIngegneria, Monduzzi Editore, 2003.(Teoria della misura e dellintegrazione secondo Lebesgue)M. Codegone, Metodi Matematici per lIngegneria, Zanichelli, 1995. (Distribuzioni)III-IV CFUS. Salsa, Equazioni a Derivate Parziali: Metodi, Modelli a Applicazioni, Springer, 2004.L. C. Evans, Partial Differential Equations, A.M.S., Graduate Studies in Mathematics, 1998.V-VI CFUH. Brezis, Analisi Funzionale. Teoria e applicazioni, Liguori Editore 2002.D. Costa, An Invitation to Variational Methods in Differential Equations, Birkhäuser 2007.J.N. Reddy, Applied Functional Analysis and Variational Methods in Engineering, McGraw-Hill 1986.F. Clarke, Optimization and Nonsmooth Analysis, SIAMs Classics in Applied Mathematics, 1990.A. Quarteroni, Modellistica Numerica per Problemi Differenziali, Springer, 2008. Traditional teaching method Yes Distance teaching method No Mandatory attendance No Written examination evaluation No Oral examination evaluation Yes Aptitude test evaluation No Project evaluation No Internship evaluation No Evaluation in itinere No Practice Test No

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