Regularity results and Harnack inequalities for minimizers and solutions of nonlocal problems: a unified approach via fractional De Giorgi classes

Abstract

We study energy functionals obtained by adding a possibly discontinuous potential to an interaction term modeled upon a Gagliardo-type fractional seminorm. We prove that minimizers of such non-differentiable functionals are locally bounded, Hölder continuous, and that they satisfy a suitable Harnack inequality. Hence, we provide an extension of celebrated results of M. Giaquinta and E. Giusti to the nonlocal setting. To do this, we introduce a particular class of fractional Sobolev functions, reminiscent of that considered by E. De Giorgi in his seminal paper of 1957. The flexibility of these classes allows us to also establish regularity of solutions to rather general nonlinear integral equations.

59 pages

References in corpus (4)

• Local behavior of fractional $p$-minimizers
• Nonlocal equations with measure data
• Plane-like minimizers for a non-local Ginzburg-Landau-type energy in a periodic medium
• Hölder estimates for viscosity solutions of equations of fractional $p$-Laplace type

Cited by in corpus (18)

• Integro-Differential Elliptic Equations
• Regularity for nonlocal problems with non-standard growth
• Harnack inequality for nonlocal problems with non-standard growth
• The Wiener criterion for nonlocal Dirichlet problems
• Global regularity results for non-homogeneous growth fractional problems
• Regularity estimates for fractional orthotropic $p$-Laplacians of mixed order
• A gradient estimate for nonlocal minimal graphs
• Nonlocal operators related to nonsymmetric forms II: Harnack inequalities
• Plane-like minimizers for a non-local Ginzburg-Landau-type energy in a periodic medium
• On the fractional NLS equation and the effects of the potential well's topology
• Local regularity for nonlocal equations with variable exponents
• Multiplicity and concentration results for local and fractional NLS equations with critical growth
• Variational solutions to the total variation flow on metric measure spaces
• Semilinear integro-differential equations, II: one-dimensional and saddle-shaped solutions to the Allen-Cahn equation
• Robust near-diagonal Green function estimates
• Planelike minimizers of nonlocal Ginzburg-Landau energies and fractional perimeters in periodic media
• Stable solutions to fractional semilinear equations: uniqueness, classification, and approximation results
• Local Hölder Regularity for Quasilinear Elliptic Equations with Mixed Local-Nonlocal Operators, Variable Exponents, and Weights