2020/2021

2022/2023

DM270

MAT/07 (MATHEMATICAL PHYSICS)

DEPARTMENT OF MATHEMATICS "FELICE CASORATI"

MATHEMATICS

PERCORSO COMUNE

3°

2nd semester (01/03/2023 - 09/06/2023)

6

60 lesson hours

Italian

ORAL TEST

VIRGA EPIFANIO GUIDO (titolare) - 6 ECTS

Linear algebra. Multivariate differential and integral calculus. Elements of classical mechanics, thermodynamics, and electromagnetism.

Mathematical Physics studies natural phenomena through mathematical models apt to represent them. This course aims to expose students to the classical methods of this discipline; it is meant to provide them with ability to formalise elementary physical phenomena in mathematical terms so as to predict how they unfold, based on the properties of the solutions to the equations introduced to represent them.

Tensor algebra. Tensor analysis. Differential operators. Calculus in orthogonal (non Cartesian) coordinates. Continuum mechanics of one-dimensional bodies. Balance laws. Constitutive laws. Material indifference. Representation theorems for isotropic functions. Nonlinear equation of motion for a vibrating string. Virtual power principle. Weak form of the equation of motion. Rankine-Hugoniot jump conditions. Shock velocity. Uniqueness and regularity of the solution. Linearisation. Longitudinal and transverse waves. Tension. Initial-boundary-value problem for the case of uniform tension. Wave equation in one space dimension. d’Alembert’s general solution. Existence and uniqueness of a classical solution. Characteristic lines. Weak forms of the equation. Domain of influence. Domain of dependence. Non-homogeneous wave equation. Duhamel’s method. Wave equation with non uniform velocity. Second-oder linear equations. Classification of second-order differential operators. Hyperbolic equations. Parabolic equations. Elliptic equations. General representation of solution in the hyperbolic and parabolic cases. Laplace equation. Equilibrium of an elastic membrane. Poisson equation in electrostatics. Boundary-value problems of Dirchlet, Newman, and Robin. Green’s first identity. Uniqueness theorem for the solution to Poisson’s equation. Harmonic functions. Mean-value properties. Gauss’ mean-value theorem. Maximum principle for harmonic functions. Comparison theorem. Stability theorem. Hopf’s maximum principle. Green’s second identity. Gauss’ reverse theorem (Koebe’s theorem). Fundamental solution of the Laplace equation. Liouville’s theorem for harmonic functions. Green’s function for the aplace operator. Electrostatic interpretation. Counter-example to existence: Lebesgue’s “spike”. Geometric conditions on the domain for the existence of a solution to the Dirichlet problem. Green’s function properties. Gree’s function for ta ball. The heat equation. Fourier’s law. Thermodynamic restrictions. Parabolic boundary. Global Cauchy’s problem. Uniqueness theorem. Maximum principle. Fundamental solution to the heat equation. Similarity solutions. Thermal regularisation. Infinite speed of propagation of thermal data. Solution of the non-homogeneous problem. Tychonov’s counter-example to uniqueness of solution to the global Cauchy’s problem. Uniqueness theorem (in Tychonov’s class). Method of separation of variables. Fourier series. Convergence of Fourier series. Parseval’s identity. Harmonics’ decay in time. Time asymptotic.

Lectures and exercise classes

S. Salsa, Partial Differential Equations in Action: from Modelling to Theory, Third Ed., Spinger, Cham, 2016.

S. Salsa and G. Verzini, Partial Differential Equations in Action: Complements and Exercises, Springer, Cham, 2015.

Final exams will only be oral. Each test will be organized in two seprate steps: in one, the successful candidates should prove able to navigate the theoretical landscape of methods and results outlined in the course; in the other, they will be asked to solve a specific problem, similar in form and guise to those worked out in the exercise classes.