Elliptic problems: regularity of stable solutions and a nonlocal Weierstrass extremal field theory

Abstract

(English) This PhD dissertation deals with qualitative questions from the theory of elliptic Partial Differential Equations (PDE) and integro-differential equations. We are primarily interested in a distinguished class of solutions satisfying appropriate minimality conditions. The first part of the thesis provides a regularity theory for stable solutions to semilinear problems involving variable coefficients. Here, stability refers to the nonnegativity of the principal eigenvalue of the linearized equation. For variational problems, this amounts to the nonnegativity of the second variation, a necessary condition for minimality. Our main achievement is to show the boundedness of stable solutions in C11 domains in the optimal range of dimensions n < 10. This result is new even for the Laplacian, for which a C3 assumption on the domain was needed. The second part furnishes natural sufficient conditions for the minimality of critical points in a general nonlocal framework. Namely, we construct a calibration for nonlocal energy functionals, under the assumption that the critical point is embedded in a family of sub/supersolutions whose graphs produce a foliation. As a consequence, we deduce that the solution is a minimizer with respect to competitors taking values in the foliated region. Our result extends, for the first time, the classical Weierstrass extremal field theory in the Calculus of Variations to a nonlocal setting. To find a calibration for the most basic fractional functional, the Gagliardo-Sobolev seminorm, was an important open problem that we have solved.