Introduction to CoV by examples, Indirect Methods, Semiclassical theory: the Direct Method on the space of Lipschitz functions. Tonelli's Direct Method: lower semicontinuity, existence, regularity. First order Sobolev spaces theory.
{Dac08} B. Dacorogna,
Direct methods in the calculus of variations.
Second edition. Applied Mathematical Sciences, 78. Springer, New York, 2008. xii+619 pp.
{EvaGar92} L.C. Evans, R. Gariepy, Measure theory and fine properties of functions. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992. viii+268 pp.
{GiaHil96} M. Giaquinta, S. Hildebrandt, Calculus of variations. I. The Lagrangian formalism.
Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences],
310. Springer-Verlag, Berlin, 1996. xxx+474 pp.
{Giu03} E. Giusti, Direct methods in the calculus of variations. World Scientific Publishing Co., Inc., River Edge, NJ, 2003. viii+403 pp.
{Leo09} G. Leoni, A first course in Sobolev spaces.
Graduate Studies in Mathematics, 105. American Mathematical Society, Providence, RI, 2009. xvi+607 pp.
Learning Objectives
Knowledge in details of the direct method of the CoV by analyzing pivotal instances. Therefore several topics necessary to analyze relaxation, semicontinuity and regularity issues shall be dealt with. In particular, to this aim students will acquaint a deep knowledge of first order Sobolev spaces.
At the end of the course students will have the expertise to set up and discuss the solution of variational problems via the direct method. In addition, students will have a strong background for improving the study of regularity issues for solutions to elliptic PDE's.
Lectures: Presentation of the theory described in the course program, with teacher-student direct interaction, to ensure a full understanding of the subject.
Moodle learning platform: online teacher-student interaction, posting of additional notes.
Further information
Partecipation to lessons is not compulsory.
Office hours to be agreed upon with students.
Type of Assessment
Oral examination on a topic to be agreed upon with the Teacher. The topic can be chosen either among those analyzed during the course or among those qualified as research material.
The effective understanding of the ideas of the Direct Method and Sobolev space theory and the ability of critical thinking shall be investigated both via questions on results taught during the course and related to the topic of examination, and via the analysis of specific model problems.
In the assessment, special attention is paid to communication skills, critical thinking and appropriate use of mathematical language.
Course program
1. Calculus of Variations: an introduction by examples.
2. Background on Real Analysis: the maximal function operator, weak estimate, Hardy-Littlewood's maximal theorem, Lebesgue points, convolutions, Fundamental Lemma of the Calculus of Variations (several statements). Du Bois-Raymond lemma.
3. The direct Method in CoV (cp.{Giu03}): Rademacher's theorem, Mac Shane lemma. super- and sub-solutions, maximum principle, the bounded slope conditions, barriers.
4. The Direct Method of CoV (cp. {Dac08}{Giu03}).
(a) Motivations. Weierstrass theorem: semicontinuity e compactness, relaxation, examples.
(b) Weak formulation of variational problems. Sobolev spaces W^{1,p}, p\in[1,\infty] (cp {EvaGar92}, {Giu03}, {Leo09}): weak derivatives, definition and uniqueness.
Examples: the one-dimensional case, comparison between W^{1,\infty} and Lipschitz functions, Rademacher's theorem.
Meyers-Serrin's theorem and its corollaries: chain rule, truncations, locality of weak derivatives, W^{1,p}(R^n)=W^{1,p}_0( R^n), alternative characterizaions of W^{1,p}.
Immersion theorems: Morrey's theorem, Sobolev-Gagliardo-Nirenberg's theorem, immersion for W^{1,n}(R^n).
Extension and approximation of W^{1,p} maps on Lipschitz open sets by bi-Lipschitz maps.
Rellich-Kondrakov's theorem, Poincare's and Sobolev-Poincare's inequality. Trace theory (hints).
(c) Necessary conditions for lower semicontinuity in the strong topology of L^1 in the scalar case: approximation of affine maps with Lipschitz maps with gradients taking only two values, Jensen's inequality, convexity and Serrin's theorem in the autonomous case.
(d) Necessary conditions for lower semicontinuity in the weak topology of W^{1,p} in the vectorial case: Riemann-Lebesgue's lemma, Jensen's type inequality for functions with periodic gradient on the unitary cube, quasi-convexity, poly-convexity, rank-1 convexity and the relations among them, examples and counter-examples.
(e) Sufficient conditions for lower semicontinuity in the weak topology of W^{1,p} in the vectorial case: Morrey's theorem in W^{1,\infty}.
The theorem by Acerbi & Fusco and Marcellini in W^{1,p}, p\in[1,\infty): biting Lemma and equi-integrability, Mac-Shane's lemma, decomposition lemma, approximation of W^{1,p} functions with Lipschitz ones.
(f) Solution of Hilbert's IXth problem: existence and uniqueness for minimum problems by the Direct Method, weak form of the Euler-Lagrange equation. Elliptic regularity in the interior for minimizers of C^2 Lagrangians: difference quotients method, W^{2,2}_loc regularity for minimizers of C^1 functionals, Morrey's theorem, Schauder's theorem, De Giorgi's theorem.