Optimal Constants in the Theory of Sobolev Spaces and PDEs

Recent research activities on sharp constants and optimal inequalities have shown their impact on a deeper understanding of geometric, analytical and other phenomena in the context of partial differential equations and mathematical physics. These intrinsic questions have applications not only to a-priori estimates or spectral theory but also to numerics, economics, optimization, etc. Mathematics Subject Classification (2000): 35P15, 46E35, 15A42, 26D10, 28C20, 39B72 52A40. Introduction by the Organisers The problem of finding sharp constants in geometric and functional inequalities is an issue of recognized importance in modern analysis. Prototypical instances of this kind of problems include isoperimetric inequalities, spectral estimates in mathematical physics, Sobolev and Hardy inequalities. Knowledge of the optimal form of the relevant inequalities and of the corresponding extremals not only provides deeper insight into the inequalities themselves, but is often crucial for applications to related partial differential equations and problems of the calculus of variations, as witnessed, for example, by the solution to the Yamabe problem. Following classical results with their origins in the work of Steiner, Schwarz, Polya, and Szego, the proof of the isoperimetric inequality in the Euclidean space by De Giorgi was a breakthrough which paved the way to fundamental contributions to the study of optimal constants in the sixties and seventies of the last century by Federer, Maz’ya, Talenti, Aubin, Moser. Recent years have seen a renewed interest in investigations on these topics, which have benefited both from 326 Oberwolfach Report 08/2010 developments in classical methods such as symmetrizations and rearrangements, and also from new techniques, including scaling techniques and optimal mass transportation. It was the aim of the proposed workshop to bring together mathematicians who work on various aspects of functional inequalities from spectral theory, shape optimization, probability, partial differential equations and related fields. The workshop was attended by participants from various scientific communities including mathematical physics, nonlinear PDE, spectral theory, calculus of variations, optimal transportation and functional analysis. The common tie was the use of optimal estimates in different contexts. It came as a surprise even to the organizers how rich the theory of Sobolev and related inequalities is, and how varied the methods are. There were several contributions about the characterization of functions that provide sharp estimates, discussing qualitative properties such as symmetry or lack of symmetry, as well as quantitative aspects involving remainder terms in classical inequalities. To give an example, several talks dealt with inequalities like the Hardy-Sobolev inequality in the form