This doctoral thesis is devoted to the analysis of some minimization problems that involve nonlocal functionals. We are mainly concerned with the s-fractional perimeter and its minimizers, the s-minimal sets. We investigate the behavior of sets having (locally) finite fractional perimeter and we establish existence and compactness results for (locally) s-minimal sets. We study the s-minimal sets in highly nonlocal regimes, that correspond to small values of the fractional parameter s. We introduce a functional framework for studying those s-minimal sets that can be globally written as subgraphs. In particular, we prove existence and uniqueness results for minimizers of a fractional version of the classical area functional and we show the equivalence between minimizers and various notions of solution of the fractional mean curvature equation. We also prove a flatness result for entire nonlocal minimal graphs having some partial derivatives bounded from either above or below. Moreover, we consider a free boundary problem, which consists in the minimization of a functional defined as the sum of a nonlocal energy, plus the classical perimeter. Concerning this problem, we prove uniform energy estimates and we study the blow-up sequence of a minimizer---in particular establishing a Weiss-type monotonicity formula.

MINIMIZATION PROBLEMS INVOLVING NONLOCAL FUNCTIONALS: NONLOCAL MINIMAL SURFACES AND A FREE BOUNDARY PROBLEM / L. Lombardini ; coordinatore: V. Mastropietro ; tutor: E. Valdinoci, A. Farina. - : . DIPARTIMENTO DI MATEMATICA "FEDERIGO ENRIQUES", 2019 Jan 07. ((31. ciclo, Anno Accademico 2018. [10.13130/lombardini-luca_phd2019-01-07].

MINIMIZATION PROBLEMS INVOLVING NONLOCAL FUNCTIONALS: NONLOCAL MINIMAL SURFACES AND A FREE BOUNDARY PROBLEM

L. Lombardini
2019-01-07

Abstract

This doctoral thesis is devoted to the analysis of some minimization problems that involve nonlocal functionals. We are mainly concerned with the s-fractional perimeter and its minimizers, the s-minimal sets. We investigate the behavior of sets having (locally) finite fractional perimeter and we establish existence and compactness results for (locally) s-minimal sets. We study the s-minimal sets in highly nonlocal regimes, that correspond to small values of the fractional parameter s. We introduce a functional framework for studying those s-minimal sets that can be globally written as subgraphs. In particular, we prove existence and uniqueness results for minimizers of a fractional version of the classical area functional and we show the equivalence between minimizers and various notions of solution of the fractional mean curvature equation. We also prove a flatness result for entire nonlocal minimal graphs having some partial derivatives bounded from either above or below. Moreover, we consider a free boundary problem, which consists in the minimization of a functional defined as the sum of a nonlocal energy, plus the classical perimeter. Concerning this problem, we prove uniform energy estimates and we study the blow-up sequence of a minimizer---in particular establishing a Weiss-type monotonicity formula.
VALDINOCI, ENRICO
MASTROPIETRO, VIERI
Nonlocal minimal surfaces, regularity theory, subgraphs, nonparametric, Bernstein, free boundary
Settore MAT/05 - Analisi Matematica
MINIMIZATION PROBLEMS INVOLVING NONLOCAL FUNCTIONALS: NONLOCAL MINIMAL SURFACES AND A FREE BOUNDARY PROBLEM / L. Lombardini ; coordinatore: V. Mastropietro ; tutor: E. Valdinoci, A. Farina. - : . DIPARTIMENTO DI MATEMATICA "FEDERIGO ENRIQUES", 2019 Jan 07. ((31. ciclo, Anno Accademico 2018. [10.13130/lombardini-luca_phd2019-01-07].
Doctoral Thesis
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/2434/607164
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