B020977 - CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS

Italian

Introduction to CoV by examples, Indirect Methods, Tonelli's Direct Method: lower semicontinuity, existence, regularity.

{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.

Lebesgue measure theory, L^p spaces, Hilbert and Banach spaces, Ascoli-Arzela's compactness theorem, Hahn-Banach theorem, weak topologies.

Lessons.

Oral examination on a topic to be agreed upon with the Teacher.

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).
3. The Indirect Method in CoV (cp.{GiaHil96}):

(a) Extremals of Lagrangians, first (external) variation, weak and strong minimality

(b) First order necessary conditions: Euler-Lagrange equation and operator, examples

(c) Non-existence and regularity of minimizers: examples.

(d) Second order conditions: second variation, accessory integral and accessory Lagrangian

Necessary conditions: Legendre-Hadamard's condition for weak minimizers, Weierstrass' excess function, Weierstrass' necessary condition for strong minimizers

Sufficient conditions: convexity, Jacobi's theory for weak minimizers

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).

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.