Gastvorlesung zur Analysis 2010: "Regularity Theory in the Calculus of Variations" (SS 2010)

Prof. Dr. Jan Kristensen, University of Oxford


Jeweils in den Wochen vom 14.-19. Juni und 12.-23. Juli.


The calculus of variations is one of the oldest subjects of mathematics, yet it remains very active and is still evolving fast. Besides its mathematical importance and its links to other branches of mathematics, including geometry and partial differential equations, it is widely used in engineering, physics, economy and biology.

This course aims to give an introduction to the modern theory of existence and regularity of minimizers of multiple integrals in the calculus of variations. The emphasis will be on explaining the basic principles behind some proofs for partial regularity of minimizers of variational integrals

defined on Sobolev mappings u : Ω ⊂ RnRN . Our starting point is Weyl’s Lemma for harmonic functions, and the integrands F will be related to the Dirichlet integrand in a sense that is natural in the variational context.

Prior knowledge of elementary functional analysis, Lebesgue and Sobolev spaces is assumed.

Lectures 1 & 2. Existence of minimizers by the direct method: lower semicontinuity and coercivity as motivation for convexity conditions.
Weyl’s Lemma, Hilbert’s 19th problem, and a brief survey of basic regularity theory in the scalar case and of counter–examples to full interior regularity in the multi-dimensional vectorial case.
The Euler-Lagrange equation for strongly convex integrals: the difference-quotient method and Caccioppoli inequalities. Brief discussion of related recent results.
Weyl’s Lemma for linear elliptic systems and quadratic functionals: regularity by way of the difference-quotient method and brief review of Schauder estimates. The A-harmonic approximation lemma of Duzaar and Steffen. Campanato–Meyers’ characterization of Hölder continuity in terms of excess decay.

Lectures 3 & 4. Regularity under smallness condition on the excess. Strategy for proof of partial regularity and implementation via A-harmonic approximation lemma. Caccioppoli’s second inequality from strong quasiconvexity. Regularity versus compactness. Brief discussion of related recent results.

Lectures 5 & 6. Estimates for the size of the singular set. Brief review of the notions of porosity and Hausdorff dimensions for sets. The convex case: size estimates from higher differentiabil- ity. Characterization of Sobolev maps by use of excess functionals. A Carleson condition for the excess. Uniform porosity for the singular set of a Lipschitzian minimizer of a strongly quasiconvex integral.