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Second Semester 
Teaching style
Lingua Insegnamento

Informazioni aggiuntive

Course Curriculum CFU Length(h)
[60/65]  MATHEMATICS [65/40 - Ord. 2020]  MATEMATICA PURA 9 72


1. Knowledge and understanding. The student will perfect his/her knowledge of classical functional analysis and its applications to partial differential equations of both linear type. He/She will learn the basic properties of abstract Banach spaces, of the main function spaces, and of operators between them.
2. Applying knowledge and understanding. The student will learn functional-analytic resolution methods for several types of linear second-order partial differential equations: Laplace-Poisson equations, general elliptic equations with several bounday conditions, heat equation, wave equations.
3. Making judgements. The student will learn how to classify linear second-order partial differential equations (elliptic, parabolic, hyperbolic) and how to apply to each type the most convenient resolutive method, placing the problem in the correct functional-analytic framework.
4. Communications skills. Attending lectures, reading the suggested texts (in English), and preparing for the final test, the student will become familiar with the formal language of current mathematical research, and will learn how to explain results in a rigorous way.
5. Learning skills. The student will be encouraged, especially in the final part of the course, towards a creative and autonomous study: on the basis of meaningful examples and with convenient bibliographical references, he/she will be able to expand in an independent way his/her knowledge of partial differential equations to more general and complex cases.


Elements of functional analysis (Banach spaces, duality, weak topology, reflexivity). Lebesgue measure and integration, Lebesgue spaces. Banach’s contraction theorem. Linear operators on finite-dimensional spaces. Elements of general topology (compactness, connectedness, metric spaces). Ordinary differential equations.


1. Hilbert spaces and linear operators (18). Definitions of scalar product, Hilbert space; parallelogram identity; uniform convexity; reflexivity. Approximation properties, metric projection, orthogonal projection. Dual of a Hilbert space: Riesz representation theorem, orthogonal complement. Bilinear forms. Stampacchias, Lax-Milgram theorems. Orthonormal bases: decomposition theorem, Parsevals identity. Adjoint of an operator: orthogonality relations, self-adjoint operators on Hilbert spaces. Compact, finite rank operators. Schauders theorem. Recalls on bounded linear operators: open map, closed graph theorems. Unbounded operators: domain, density. Riesz-Fredholm theory: Riesz lemma, Riesz theorem (characterization of finite- dimensional spaces), Fredholms alternative theorem. Resolvent and spectrum of an operator: eigenvalues and eigenvectors, spectrum of compact, self-adjoint operators, spectral decomposition theorem.
2. Sobolev spaces (18). Recalls on L^p-spaces: convolution, Youngs theorem, mollifiers, translation operator. Ascoli-Arzelà, Kolmogorov-Riesz-Fréchet theorems. Definitions of weak derivative, W^1,p-space. Properties of W^1,p: separability, uniform convexity, reflexivity, characterization. The space H^1 = W^1,2.
Definition and properties of W^k,p (k > 1). Operations on weak derivatives. Extension theorems:
extension by reflection, extension theorem for regular domains. Density: Friedrichs theorem, density theorem for regular domains. Embedding theorems: Sobolev (Gagliardo-Nirenberg) inequality, continuous embedding of W^1,p into L^q-spaces, Morrey's theorem. Compact embeddings: Rellich-Kondrachov theorem. Embeddings of W^k,p. The space W^1,p_0 (H^1_0): definition and characterization, Poincaré's inequality, equivalent norm, dual space.
3. Partial differential equations/1: Stationary problems (18). Introduction to linear second order partial differential equations: definition of classical solution, classification, boundary conditions (Dirichlet and Neumann). Elliptic equations: definition of weak solution for the homogeneous Dirichlet problem with Laplacian operator, existence and uniqueness (via Lax-Milgram theorem), regularity (Nirenberg's translation method, cases of R^N, R^N+, and of a regular domain with bounded boundary), return to the classical solution, maximum principle. Existence of solutions to elliptic equations via Fredholms alternative. Spectrum of the Laplacian: characterization of eigenvalues, spectral decomposition. Hints on general linear operators, non-homogeneous Dirichlet problem, Neumann problem, p-Laplacian operator, Laplace's, Poisson's equation.
4. Partial differential equations/2: Evolutive problems (18). Monotone operators in Hilbert spaces: definitions of monotone, maximal monotone operator, properties, resolvent, Yosida approximation. Evolution problems in Hilbert spaces, driven by maximal monotone operators: definition of solution, existence and uniqueness in the bounded case. Hille-Yosida theorem: from the regularized to the unbounded case, conservation properties. The self-adjoint case. Regularity of the solution. Parabolic equations: existence and uniqueness of the solution to the heat equation with Cauchy-Dirichlet conditions, regularity, resolution by spectral decomposition, maximum principle. Hyperbolic equations: existence and uniqueness of the solution the wave equation with Cauchy-Dirichlet conditions, regularity, solution by spectral decomposition. Cauchy-Neumann conditions.

Teaching Methods

The course (9 CFU/72 hours) will consist in lectures. Some proofs and applications will be left to the student as practice, and then checked by the teacher. Active participation will be encouraged. Blackboard, slides, and occasionally calculus software will be employed. The course notes, aimed at complementing the suggested texts, will be made available in due time. Lectures will be given in presence or online, according to national and university rules.

Verification of learning

The exam will aim at evaluating mainly the following items: knowledge of the course’s contents, independent though, exposition skills. The exam will consist in two tests:
1. Optional test. An oral (seminar) or written (paper) exposition of a part of the course’s program, or connected to it. For this test the student may choose from a list of suggested topics and the teacher’s bibliographical and scientific advice. The test will be evaluated with a mark between 1 and 30.
2. Mandatory test. An oral exam (45 minutes’ duration), consisting in 3 questions covering all the course’s program except (if this is the case) the part covered by the previous test. On the student’s request, this test may be divided in 2 exams, covering units 1–2 and 3–4, respectively. This test will also be evaluated with a mark between 1 and 30.
The final mark will be computed as the mean value of the marks of both tests, and the result will be considered positive if the mean of the two marks will be between 18 (sufficient) and 30 (excellent). The ’lode’ will acknowledge specially brilliant performances. Examinations will be held in presence or online, according to national and university rules.


1. H. Brezis, Functional analysis, Sobolev spaces and partial differential equations, Springer, New York (2011)
2. L.C. Evans, Partial differential equations, American Mathematical Society, Providence (1998)
3. Teacher's notes

More Information

Notes and more material (including the detailed program, required proofs for the examination, and suggested topics for the optional test) will be uploaded in due time on the teacher's webpage. Our University provides support for students with specific learning disability (SLD).

Questionnaire and social

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