Regularity in metric spaces
Using arguments developed by De Giorgi in the 1950's, it is possible to prove the regularity of the solutions to a vast class of variational problems in the Euclidean space. The main goal of the present thesis is to extend these results to the more abstract context of metric spaces with a measure. In particular, working in the axiomatic framework of Gol'dshtein – Troyanov, we establish both the interior and the boundary regularity of quasi-minimizers of the p-Dirichlet energy. Our proof works for quite general domains, assuming some natural hypotheses on the (axiomatic) D-structure. Furthermore, we prove analogous results for extremal functions lying in the class of Sobolev functions in the sense of Hajłasz – Koskela, i.e. functions characterized by the single condition that a Poincaré inequality be satisfied. Our strategy to prove these regularity results is first to show that, in a very general setting, the (Hölder) continuity of a function is a consequence of three specific technical hypotheses. This part of the argument is the essence of the De Giorgi method. Then, we verify that for a function u which is a quasi-minimizer in an axiomatic Sobolev space or an extremal Sobolev function in the sense of Hajłasz – Koskela, these technical hypotheses are indeed satisfied and u is thus (Hölder) continuous. In addition to that, we establish the Harnack's inequality for these extremal functions, and we show that the Dirichlet semi-norm of a piecewise-extremal function is equivalent to the sum of the Dirichlet semi-norms of its components.
Keywords: Analysis on metric spaces ; Sobolev spaces ; Quasi-minima ; Interior and boundary regularity ; De Giorgi's method ; Analyse sur les espaces métriques ; Espaces de Sobolev ; Fonctions quasi-minimisantes ; Régularité intérieure et au bord ; Méthode de De GiorgiThèse École polytechnique fédérale de Lausanne EPFL, n° 3571 (2006)
Section de mathématiques
Faculté des sciences de base
Institut de géométrie, algèbre et topologie
Public defense: 2006-7-7
Record created on 2006-05-22, modified on 2016-08-08