To acquire a good knowledge of the elementary theory of partial differential equations and of the basic methods of solution, with particular focus on the equations describing problems in mathematical physics.
Curriculum
teacher profile teaching materials
The continuity equation, or linear transport equation. Method of characteristics.
Microscopic derivation of the wave equation from a system of coupled harmonic oscillators. Uniqueness of the solution to the Cauchy problem associated with the wave equation. Traveling waves. D'Alembert's solution. Causally connected regions of space-time. Future and past light cones. Conservation of energy and uniqueness of the solution to the wave equation. Equipartition of energy. Duhamel's principle and the fundamental solution.
The notion of weak solution for the Cauchy problem associated with the wave equation. Symmetries and the method of images. Kirchhoff's solution to the wave equation in R^3: retarded potentials; Huygens' principle (strong and weak forms); forward and backward wave fronts. Uniqueness of solutions to the wave equation in R^n. Poisson's solution to the wave equation in R^2: violation of the strong Huygens principle.
Functional Analysis excursus: Lebesgue measure on R^n. Lebesgue measurable and integrable functions. Definition of the spaces L^p and L^\infty, their completeness, and weak convergence in L^p. Hilbert spaces. L^2 as an example of a separable Hilbert space. Countable orthonormal bases. The Fourier Transform (FT) in L^1. Riemann–Lebesgue lemma. Fourier transform of a convolution. FT in L^2. Plancherel's and Parseval's identities. Inversion theorem. Introduction to distributions: the space of test functions and its dual. Distributions of finite order. Distributions associated with functions in L^1_{loc} and L^2. L^1_{loc} and L^2 as subspaces of the space of distributions. Distributional derivatives, distributional derivatives of functions in L^1_{loc}; linear changes of variables in distributions; multiplication of a distribution by a smooth function. Schwartz space and tempered distributions. Regular tempered distributions. Fourier transform of tempered distributions. Operations and properties of the Fourier transform (Fourier transform of derivatives, translation operators, and polynomials).
The heat equation: processes of thermal and mechanical diffusion. Irreversible processes. Microscopic derivation of the heat equation: random walk and an introduction to Brownian motion (Wiener process). The Cauchy problem for the heat equation in a bounded domain. Maximum principle and uniqueness of the solution to the Cauchy problem associated with the heat equation. The fundamental solution: definition, explicit form, and remarkable properties. Solution of the global Cauchy problem associated with the nonhomogeneous heat equation, with arbitrary forcing term and arbitrary initial data, in terms of convolutions with the fundamental solution. The heat equation on an interval with homogeneous Dirichlet boundary conditions: method of reflections and separation of variables. Poisson summation formula. Shannon entropy associated with solutions of the heat equation. Jensen's inequality. Monotonicity of entropy as a function of time.
Historical introduction to the birth of Quantum Mechanics. Interference phenomena (double-slit experiment). Basic properties of wave phenomena. The photoelectric effect and its interpretation. Light: wave-particle duality. The free Schrödinger equation on L^2(R^d). Global existence and uniqueness theorem for the free Schrödinger equation with Schwartz-class initial data. Center of the wave packet associated with solutions of the free Schrödinger equation. Hilbert spaces: notions of linear operator, densely defined linear operator, operator norm, bounded extension lemma, Hermitian (or symmetric) operator. Definition and characterization of unitary operators. Momentum operator. Postulates of Quantum Mechanics. Correspondence rule for obtaining quantum observables from classical observables. Commutator of two operators. General uncertainty principle (with proof) and Heisenberg's uncertainty principle. One-dimensional quantum harmonic oscillator. Hermite functions. Definition of the spectr
[B21] P. Buttà: Note del Corso di Fisica Matematica.
[G07] G. Gallavotti: The elements of mechanics, Ipparco Editore 2007.
[LL01] E. H. Lieb, M. Loss: Analysis, 2nd edition, AMS Graduate Studies in Mathematics, 2001.
[S16] S. Salsa: Equazioni a derivate parziali, 3a edizione, Springer Unitext 2016.
Mutuazione: 20410410 FM310 - ISTITUZIONI DI FISICA MATEMATICA in Matematica L-35 R GIULIANI ALESSANDRO
Programme
Partial Differential Equations (PDEs) in Mathematical Physics: definitions and examples. Linear and nonlinear PDEs of order k. Second-order linear PDEs of elliptic, parabolic, and hyperbolic type.The continuity equation, or linear transport equation. Method of characteristics.
Microscopic derivation of the wave equation from a system of coupled harmonic oscillators. Uniqueness of the solution to the Cauchy problem associated with the wave equation. Traveling waves. D'Alembert's solution. Causally connected regions of space-time. Future and past light cones. Conservation of energy and uniqueness of the solution to the wave equation. Equipartition of energy. Duhamel's principle and the fundamental solution.
The notion of weak solution for the Cauchy problem associated with the wave equation. Symmetries and the method of images. Kirchhoff's solution to the wave equation in R^3: retarded potentials; Huygens' principle (strong and weak forms); forward and backward wave fronts. Uniqueness of solutions to the wave equation in R^n. Poisson's solution to the wave equation in R^2: violation of the strong Huygens principle.
Functional Analysis excursus: Lebesgue measure on R^n. Lebesgue measurable and integrable functions. Definition of the spaces L^p and L^\infty, their completeness, and weak convergence in L^p. Hilbert spaces. L^2 as an example of a separable Hilbert space. Countable orthonormal bases. The Fourier Transform (FT) in L^1. Riemann–Lebesgue lemma. Fourier transform of a convolution. FT in L^2. Plancherel's and Parseval's identities. Inversion theorem. Introduction to distributions: the space of test functions and its dual. Distributions of finite order. Distributions associated with functions in L^1_{loc} and L^2. L^1_{loc} and L^2 as subspaces of the space of distributions. Distributional derivatives, distributional derivatives of functions in L^1_{loc}; linear changes of variables in distributions; multiplication of a distribution by a smooth function. Schwartz space and tempered distributions. Regular tempered distributions. Fourier transform of tempered distributions. Operations and properties of the Fourier transform (Fourier transform of derivatives, translation operators, and polynomials).
The heat equation: processes of thermal and mechanical diffusion. Irreversible processes. Microscopic derivation of the heat equation: random walk and an introduction to Brownian motion (Wiener process). The Cauchy problem for the heat equation in a bounded domain. Maximum principle and uniqueness of the solution to the Cauchy problem associated with the heat equation. The fundamental solution: definition, explicit form, and remarkable properties. Solution of the global Cauchy problem associated with the nonhomogeneous heat equation, with arbitrary forcing term and arbitrary initial data, in terms of convolutions with the fundamental solution. The heat equation on an interval with homogeneous Dirichlet boundary conditions: method of reflections and separation of variables. Poisson summation formula. Shannon entropy associated with solutions of the heat equation. Jensen's inequality. Monotonicity of entropy as a function of time.
Historical introduction to the birth of Quantum Mechanics. Interference phenomena (double-slit experiment). Basic properties of wave phenomena. The photoelectric effect and its interpretation. Light: wave-particle duality. The free Schrödinger equation on L^2(R^d). Global existence and uniqueness theorem for the free Schrödinger equation with Schwartz-class initial data. Center of the wave packet associated with solutions of the free Schrödinger equation. Hilbert spaces: notions of linear operator, densely defined linear operator, operator norm, bounded extension lemma, Hermitian (or symmetric) operator. Definition and characterization of unitary operators. Momentum operator. Postulates of Quantum Mechanics. Correspondence rule for obtaining quantum observables from classical observables. Commutator of two operators. General uncertainty principle (with proof) and Heisenberg's uncertainty principle. One-dimensional quantum harmonic oscillator. Hermite functions. Definition of the spectr
Core Documentation
[C18] W. Craig: A Course on Partial Differential Equations, Graduate Studies in Mathematics, AMS 2018[B21] P. Buttà: Note del Corso di Fisica Matematica.
[G07] G. Gallavotti: The elements of mechanics, Ipparco Editore 2007.
[LL01] E. H. Lieb, M. Loss: Analysis, 2nd edition, AMS Graduate Studies in Mathematics, 2001.
[S16] S. Salsa: Equazioni a derivate parziali, 3a edizione, Springer Unitext 2016.
Attendance
In-person attendance is strongly recommended.Type of evaluation
Two midterms. One written exam, oral exam teacher profile teaching materials
The continuity equation, or linear transport equation. Method of characteristics.
Microscopic derivation of the wave equation from a system of coupled harmonic oscillators. Uniqueness of the solution to the Cauchy problem associated with the wave equation. Traveling waves. D'Alembert's solution. Causally connected regions of space-time. Future and past light cones. Conservation of energy and uniqueness of the solution to the wave equation. Equipartition of energy. Duhamel's principle and the fundamental solution.
The notion of weak solution for the Cauchy problem associated with the wave equation. Symmetries and the method of images. Kirchhoff's solution to the wave equation in R^3: retarded potentials; Huygens' principle (strong and weak forms); forward and backward wave fronts. Uniqueness of solutions to the wave equation in R^n. Poisson's solution to the wave equation in R^2: violation of the strong Huygens principle.
Functional Analysis excursus: Lebesgue measure on R^n. Lebesgue measurable and integrable functions. Definition of the spaces L^p and L^\infty, their completeness, and weak convergence in L^p. Hilbert spaces. L^2 as an example of a separable Hilbert space. Countable orthonormal bases. The Fourier Transform (FT) in L^1. Riemann–Lebesgue lemma. Fourier transform of a convolution. FT in L^2. Plancherel's and Parseval's identities. Inversion theorem. Introduction to distributions: the space of test functions and its dual. Distributions of finite order. Distributions associated with functions in L^1_{loc} and L^2. L^1_{loc} and L^2 as subspaces of the space of distributions. Distributional derivatives, distributional derivatives of functions in L^1_{loc}; linear changes of variables in distributions; multiplication of a distribution by a smooth function. Schwartz space and tempered distributions. Regular tempered distributions. Fourier transform of tempered distributions. Operations and properties of the Fourier transform (Fourier transform of derivatives, translation operators, and polynomials).
The heat equation: processes of thermal and mechanical diffusion. Irreversible processes. Microscopic derivation of the heat equation: random walk and an introduction to Brownian motion (Wiener process). The Cauchy problem for the heat equation in a bounded domain. Maximum principle and uniqueness of the solution to the Cauchy problem associated with the heat equation. The fundamental solution: definition, explicit form, and remarkable properties. Solution of the global Cauchy problem associated with the nonhomogeneous heat equation, with arbitrary forcing term and arbitrary initial data, in terms of convolutions with the fundamental solution. The heat equation on an interval with homogeneous Dirichlet boundary conditions: method of reflections and separation of variables. Poisson summation formula. Shannon entropy associated with solutions of the heat equation. Jensen's inequality. Monotonicity of entropy as a function of time.
Historical introduction to the birth of Quantum Mechanics. Interference phenomena (double-slit experiment). Basic properties of wave phenomena. The photoelectric effect and its interpretation. Light: wave-particle duality. The free Schrödinger equation on L^2(R^d). Global existence and uniqueness theorem for the free Schrödinger equation with Schwartz-class initial data. Center of the wave packet associated with solutions of the free Schrödinger equation. Hilbert spaces: notions of linear operator, densely defined linear operator, operator norm, bounded extension lemma, Hermitian (or symmetric) operator. Definition and characterization of unitary operators. Momentum operator. Postulates of Quantum Mechanics. Correspondence rule for obtaining quantum observables from classical observables. Commutator of two operators. General uncertainty principle (with proof) and Heisenberg's uncertainty principle. One-dimensional quantum harmonic oscillator. Hermite functions. Definition of the spectr
[B21] P. Buttà: Note del Corso di Fisica Matematica.
[G07] G. Gallavotti: The elements of mechanics, Ipparco Editore 2007.
[LL01] E. H. Lieb, M. Loss: Analysis, 2nd edition, AMS Graduate Studies in Mathematics, 2001.
[S16] S. Salsa: Equazioni a derivate parziali, 3a edizione, Springer Unitext 2016.
Mutuazione: 20410410 FM310 - ISTITUZIONI DI FISICA MATEMATICA in Matematica L-35 R GIULIANI ALESSANDRO
Programme
Partial Differential Equations (PDEs) in Mathematical Physics: definitions and examples. Linear and nonlinear PDEs of order k. Second-order linear PDEs of elliptic, parabolic, and hyperbolic type.The continuity equation, or linear transport equation. Method of characteristics.
Microscopic derivation of the wave equation from a system of coupled harmonic oscillators. Uniqueness of the solution to the Cauchy problem associated with the wave equation. Traveling waves. D'Alembert's solution. Causally connected regions of space-time. Future and past light cones. Conservation of energy and uniqueness of the solution to the wave equation. Equipartition of energy. Duhamel's principle and the fundamental solution.
The notion of weak solution for the Cauchy problem associated with the wave equation. Symmetries and the method of images. Kirchhoff's solution to the wave equation in R^3: retarded potentials; Huygens' principle (strong and weak forms); forward and backward wave fronts. Uniqueness of solutions to the wave equation in R^n. Poisson's solution to the wave equation in R^2: violation of the strong Huygens principle.
Functional Analysis excursus: Lebesgue measure on R^n. Lebesgue measurable and integrable functions. Definition of the spaces L^p and L^\infty, their completeness, and weak convergence in L^p. Hilbert spaces. L^2 as an example of a separable Hilbert space. Countable orthonormal bases. The Fourier Transform (FT) in L^1. Riemann–Lebesgue lemma. Fourier transform of a convolution. FT in L^2. Plancherel's and Parseval's identities. Inversion theorem. Introduction to distributions: the space of test functions and its dual. Distributions of finite order. Distributions associated with functions in L^1_{loc} and L^2. L^1_{loc} and L^2 as subspaces of the space of distributions. Distributional derivatives, distributional derivatives of functions in L^1_{loc}; linear changes of variables in distributions; multiplication of a distribution by a smooth function. Schwartz space and tempered distributions. Regular tempered distributions. Fourier transform of tempered distributions. Operations and properties of the Fourier transform (Fourier transform of derivatives, translation operators, and polynomials).
The heat equation: processes of thermal and mechanical diffusion. Irreversible processes. Microscopic derivation of the heat equation: random walk and an introduction to Brownian motion (Wiener process). The Cauchy problem for the heat equation in a bounded domain. Maximum principle and uniqueness of the solution to the Cauchy problem associated with the heat equation. The fundamental solution: definition, explicit form, and remarkable properties. Solution of the global Cauchy problem associated with the nonhomogeneous heat equation, with arbitrary forcing term and arbitrary initial data, in terms of convolutions with the fundamental solution. The heat equation on an interval with homogeneous Dirichlet boundary conditions: method of reflections and separation of variables. Poisson summation formula. Shannon entropy associated with solutions of the heat equation. Jensen's inequality. Monotonicity of entropy as a function of time.
Historical introduction to the birth of Quantum Mechanics. Interference phenomena (double-slit experiment). Basic properties of wave phenomena. The photoelectric effect and its interpretation. Light: wave-particle duality. The free Schrödinger equation on L^2(R^d). Global existence and uniqueness theorem for the free Schrödinger equation with Schwartz-class initial data. Center of the wave packet associated with solutions of the free Schrödinger equation. Hilbert spaces: notions of linear operator, densely defined linear operator, operator norm, bounded extension lemma, Hermitian (or symmetric) operator. Definition and characterization of unitary operators. Momentum operator. Postulates of Quantum Mechanics. Correspondence rule for obtaining quantum observables from classical observables. Commutator of two operators. General uncertainty principle (with proof) and Heisenberg's uncertainty principle. One-dimensional quantum harmonic oscillator. Hermite functions. Definition of the spectr
Core Documentation
[C18] W. Craig: A Course on Partial Differential Equations, Graduate Studies in Mathematics, AMS 2018[B21] P. Buttà: Note del Corso di Fisica Matematica.
[G07] G. Gallavotti: The elements of mechanics, Ipparco Editore 2007.
[LL01] E. H. Lieb, M. Loss: Analysis, 2nd edition, AMS Graduate Studies in Mathematics, 2001.
[S16] S. Salsa: Equazioni a derivate parziali, 3a edizione, Springer Unitext 2016.
Attendance
In-person attendance is strongly recommended.Type of evaluation
Two midterms. One written exam, oral exam teacher profile teaching materials
The continuity equation, or linear transport equation. Method of characteristics.
Microscopic derivation of the wave equation from a system of coupled harmonic oscillators. Uniqueness of the solution to the Cauchy problem associated with the wave equation. Traveling waves. D'Alembert's solution. Causally connected regions of space-time. Future and past light cones. Conservation of energy and uniqueness of the solution to the wave equation. Equipartition of energy. Duhamel's principle and the fundamental solution.
The notion of weak solution for the Cauchy problem associated with the wave equation. Symmetries and the method of images. Kirchhoff's solution to the wave equation in R^3: retarded potentials; Huygens' principle (strong and weak forms); forward and backward wave fronts. Uniqueness of solutions to the wave equation in R^n. Poisson's solution to the wave equation in R^2: violation of the strong Huygens principle.
Functional Analysis excursus: Lebesgue measure on R^n. Lebesgue measurable and integrable functions. Definition of the spaces L^p and L^\infty, their completeness, and weak convergence in L^p. Hilbert spaces. L^2 as an example of a separable Hilbert space. Countable orthonormal bases. The Fourier Transform (FT) in L^1. Riemann–Lebesgue lemma. Fourier transform of a convolution. FT in L^2. Plancherel's and Parseval's identities. Inversion theorem. Introduction to distributions: the space of test functions and its dual. Distributions of finite order. Distributions associated with functions in L^1_{loc} and L^2. L^1_{loc} and L^2 as subspaces of the space of distributions. Distributional derivatives, distributional derivatives of functions in L^1_{loc}; linear changes of variables in distributions; multiplication of a distribution by a smooth function. Schwartz space and tempered distributions. Regular tempered distributions. Fourier transform of tempered distributions. Operations and properties of the Fourier transform (Fourier transform of derivatives, translation operators, and polynomials).
The heat equation: processes of thermal and mechanical diffusion. Irreversible processes. Microscopic derivation of the heat equation: random walk and an introduction to Brownian motion (Wiener process). The Cauchy problem for the heat equation in a bounded domain. Maximum principle and uniqueness of the solution to the Cauchy problem associated with the heat equation. The fundamental solution: definition, explicit form, and remarkable properties. Solution of the global Cauchy problem associated with the nonhomogeneous heat equation, with arbitrary forcing term and arbitrary initial data, in terms of convolutions with the fundamental solution. The heat equation on an interval with homogeneous Dirichlet boundary conditions: method of reflections and separation of variables. Poisson summation formula. Shannon entropy associated with solutions of the heat equation. Jensen's inequality. Monotonicity of entropy as a function of time.
Historical introduction to the birth of Quantum Mechanics. Interference phenomena (double-slit experiment). Basic properties of wave phenomena. The photoelectric effect and its interpretation. Light: wave-particle duality. The free Schrödinger equation on L^2(R^d). Global existence and uniqueness theorem for the free Schrödinger equation with Schwartz-class initial data. Center of the wave packet associated with solutions of the free Schrödinger equation. Hilbert spaces: notions of linear operator, densely defined linear operator, operator norm, bounded extension lemma, Hermitian (or symmetric) operator. Definition and characterization of unitary operators. Momentum operator. Postulates of Quantum Mechanics. Correspondence rule for obtaining quantum observables from classical observables. Commutator of two operators. General uncertainty principle (with proof) and Heisenberg's uncertainty principle. One-dimensional quantum harmonic oscillator. Hermite functions. Definition of the spectr
[B21] P. Buttà: Note del Corso di Fisica Matematica.
[G07] G. Gallavotti: The elements of mechanics, Ipparco Editore 2007.
[LL01] E. H. Lieb, M. Loss: Analysis, 2nd edition, AMS Graduate Studies in Mathematics, 2001.
[S16] S. Salsa: Equazioni a derivate parziali, 3a edizione, Springer Unitext 2016.
Mutuazione: 20410410 FM310 - ISTITUZIONI DI FISICA MATEMATICA in Matematica L-35 R GIULIANI ALESSANDRO
Programme
Partial Differential Equations (PDEs) in Mathematical Physics: definitions and examples. Linear and nonlinear PDEs of order k. Second-order linear PDEs of elliptic, parabolic, and hyperbolic type.The continuity equation, or linear transport equation. Method of characteristics.
Microscopic derivation of the wave equation from a system of coupled harmonic oscillators. Uniqueness of the solution to the Cauchy problem associated with the wave equation. Traveling waves. D'Alembert's solution. Causally connected regions of space-time. Future and past light cones. Conservation of energy and uniqueness of the solution to the wave equation. Equipartition of energy. Duhamel's principle and the fundamental solution.
The notion of weak solution for the Cauchy problem associated with the wave equation. Symmetries and the method of images. Kirchhoff's solution to the wave equation in R^3: retarded potentials; Huygens' principle (strong and weak forms); forward and backward wave fronts. Uniqueness of solutions to the wave equation in R^n. Poisson's solution to the wave equation in R^2: violation of the strong Huygens principle.
Functional Analysis excursus: Lebesgue measure on R^n. Lebesgue measurable and integrable functions. Definition of the spaces L^p and L^\infty, their completeness, and weak convergence in L^p. Hilbert spaces. L^2 as an example of a separable Hilbert space. Countable orthonormal bases. The Fourier Transform (FT) in L^1. Riemann–Lebesgue lemma. Fourier transform of a convolution. FT in L^2. Plancherel's and Parseval's identities. Inversion theorem. Introduction to distributions: the space of test functions and its dual. Distributions of finite order. Distributions associated with functions in L^1_{loc} and L^2. L^1_{loc} and L^2 as subspaces of the space of distributions. Distributional derivatives, distributional derivatives of functions in L^1_{loc}; linear changes of variables in distributions; multiplication of a distribution by a smooth function. Schwartz space and tempered distributions. Regular tempered distributions. Fourier transform of tempered distributions. Operations and properties of the Fourier transform (Fourier transform of derivatives, translation operators, and polynomials).
The heat equation: processes of thermal and mechanical diffusion. Irreversible processes. Microscopic derivation of the heat equation: random walk and an introduction to Brownian motion (Wiener process). The Cauchy problem for the heat equation in a bounded domain. Maximum principle and uniqueness of the solution to the Cauchy problem associated with the heat equation. The fundamental solution: definition, explicit form, and remarkable properties. Solution of the global Cauchy problem associated with the nonhomogeneous heat equation, with arbitrary forcing term and arbitrary initial data, in terms of convolutions with the fundamental solution. The heat equation on an interval with homogeneous Dirichlet boundary conditions: method of reflections and separation of variables. Poisson summation formula. Shannon entropy associated with solutions of the heat equation. Jensen's inequality. Monotonicity of entropy as a function of time.
Historical introduction to the birth of Quantum Mechanics. Interference phenomena (double-slit experiment). Basic properties of wave phenomena. The photoelectric effect and its interpretation. Light: wave-particle duality. The free Schrödinger equation on L^2(R^d). Global existence and uniqueness theorem for the free Schrödinger equation with Schwartz-class initial data. Center of the wave packet associated with solutions of the free Schrödinger equation. Hilbert spaces: notions of linear operator, densely defined linear operator, operator norm, bounded extension lemma, Hermitian (or symmetric) operator. Definition and characterization of unitary operators. Momentum operator. Postulates of Quantum Mechanics. Correspondence rule for obtaining quantum observables from classical observables. Commutator of two operators. General uncertainty principle (with proof) and Heisenberg's uncertainty principle. One-dimensional quantum harmonic oscillator. Hermite functions. Definition of the spectr
Core Documentation
[C18] W. Craig: A Course on Partial Differential Equations, Graduate Studies in Mathematics, AMS 2018[B21] P. Buttà: Note del Corso di Fisica Matematica.
[G07] G. Gallavotti: The elements of mechanics, Ipparco Editore 2007.
[LL01] E. H. Lieb, M. Loss: Analysis, 2nd edition, AMS Graduate Studies in Mathematics, 2001.
[S16] S. Salsa: Equazioni a derivate parziali, 3a edizione, Springer Unitext 2016.
Attendance
In-person attendance is strongly recommended.Type of evaluation
Two midterms. One written exam, oral exam