Regularity estimates for the solution and the free boundary to the obstacle problem for the fractional Laplacian

Abstract

We use a characterization of the fractional Laplacian as a Dirichlet to Neumann operator for an appropriate differential equation to study its obstacle problem. We write an equivalent characterization as a thin obstacle problem. In this way we are able to apply local type arguments to obtain sharp regularity estimates for the solution and study the regularity of the free boundary.

References in corpus (1)

• An extension problem related to the fractional Laplacian

Cited by in corpus (127)

• Ten equivalent definitions of the fractional Laplace operator
• Nonlinear porous medium flow with fractional potential pressure
• Numerical Methods for Fractional Diffusion
• Next Order Asymptotics and Renormalized Energy for Riesz Interactions
• Existence of Compactly Supported Global Minimisers for the Interaction Energy
• Some New Monotonicity Formulas and the Singular Set in the Lower Dimensional Obstacle Problem
• Boundary regularity for fully nonlinear integro-differential equations
• Obstacle problems for integro-differential operators: Regularity of solutions and free boundaries
• Dirichlet Heat Kernel Estimates for Rotationally Symmetric Lévy processes
• Harnack inequality for fractional sub-Laplacians in Carnot groups
• Ground states solutions for a non-linear equation involving a pseudo-relativistic Schrödinger operator
• Symmetric Jump Processes and their Heat Kernel Estimates
• Some Liouville theorems for the fractional Laplacian
• A Liouville Theorem for the Fractional Laplacian
• Direct epiperimetric inequalities for the thin obstacle problem and applications
• External optimal control of nonlocal PDEs
• Integro-Differential Elliptic Equations
• On the measure and the structure of the free boundary of the lower dimensional obstacle problem
• Regularity in a one-phase free boundary problem for the fractional Laplacian
• Robin fractional problems with symmetric variable growth
• Sign-changing solutions for a class of zero mass nonlocal Schrödinger equations
• Optimal Regularity and the Free Boundary in the Parabolic Signorini Problem
• Hitchhiker's guide to the fractional Sobolev spaces
• The Navier-slip thin-film equation for 3D fluid films: existence and uniqueness
• Stability of Entropy Solutions for Levy Mixed Hyperbolic-Parabolic Equations
• Periodic solutions for critical fractional problems
• Convergence rates for the classical, thin and fractional elliptic obstacle problems
• Semilinear equations involving the fractional Laplacian on domains
• Some Free Boundary Problems involving Nonlocal Diffusion and Aggregation
• Inverse problem for a nonlocal diffuse optical tomography equation
• Fractional elliptic equations, Caccioppoli estimates and regularity
• Free boundary regularity for almost every solution to the Signorini problem
• Strong unique continuation and local asymptotics at the boundary for fractional elliptic equations
• Large solutions to elliptic equations involving fractional Laplacian
• Fractional Laplacian on the torus
• Nodal solutions for the fractional Yamabe problem on Heisenberg groups
• On the singular set in the thin obstacle problem: higher order blow-ups and the very thin obstacle problem
• On a Class of Nonlocal Obstacle Type Problems Related to the Distributional Riesz Fractional Derivative
• Higher Regularity for the fractional thin obstacle problem
• Higher regularity of the free boundary in the elliptic Signorini problem
• Heat Kernels for Non-symmetric Non-local Operators
• Infinitely many non-radial sign-changing solutions for a Fractional Laplacian equation with critical nonlinearity
• Fractional double phase Robin problem involving variable order-exponents and Logarithm-type nonlinearity
• An existence result for singular fractional Kirchhoff-Schrödinger-Poisson system
• Regularity of local minimizers of the interaction energy via obstacle problems
• Parabolic Obstacle Problems. Quasi-convexity and Regularity
• Variational inequalities for the spectral fractional Laplacian
• Schauder estimates for parabolic equations with degenerate or singular weights
• Parabolic Boundary Harnack Principles in Domains with Thin Lipschitz Complement
• The Variable Coefficient Thin Obstacle Problem: Carleman Inequalities
• Almost minimizers for certain fractional variational problems
• Heat kernels for non-symmetric diffusion operators with jumps
• Regularity of the minimiser of one-dimensional interaction energies
• Fractional Gaussian fields: a survey
• The Variable Coefficient Thin Obstacle Problem: Optimal Regularity and Regularity of the Regular Free Boundary
• Optimal control of nonlinear systems governed by Dirichlet fractional Laplacian in the minimax framework
• Nonexistence results for a class of fractional elliptic boundary value problems
• Optimal control of the coefficient for fractional and regional fractional {$p$}-{L}aplace equations: Approximation and convergence
• Fast operator splitting methods for obstacle problems
• Equations involving fractional Laplacian operator: Compactness and application
• Hydrodynamic Limit of Nonlinear Diffusions with Fractional Laplacian Operators
• On a frequency function approach to the unique continuation principle
• Numerical Methods for the Fractional Laplacian: a Finite Difference-quadrature Approach
• Dirichlet heat kernel estimates for fractional Laplacian under non-local perturbation
• Saddle solutions for the fractional Choquard equation
• Almost minimizers for the thin obstacle problem
• Extension problem and Harnack's inequality for some fractional operators
• Regularity of solutions of the fractional porous medium flow
• Harnack's inequality for fractional nonlocal equations
• Fractional Laplacian phase transitions and boundary reactions: a geometric inequality and a symmetry result
• Regularity of radial extremal solutions for some non local semilinear equations
• Fractional Laplacian in Conformal Geometry
• Heat Kernel Estimate for $\Delta +{\Delta }^{\alpha /2}$ in $C^{1,1}$ open sets
• Existence results for the fractional Nirenberg problem
• Parabolic problem with fractional time derivative with nonlocal and nonsingular Mittag-Leffler kernel
• Existence and regularity results for the penalized thin obstacle problem with variable coefficients
• Liouville theorems for elliptic equations involving regional fractional Laplacian with order in $(0,1/2]$
• A perturbation result for the $Q_{gamma}$ curvature problem on ${\mathbb{S}}^n$
• Rigidity results for non local phase transitions in the Heisenberg group $H$
• Potential theory of subordinate Brownian motions with Gaussian components
• Fractional Cahn-Hilliard, Allen-Cahn and porous medium equations
• A non-local one-phase free boundary problem from obstacle to cavitation
• Boundary estimates and a Wiener criterion for the fractional Laplacian
• Strong competition versus fractional diffusion: the case of Lotka-Volterra interaction
• Nonlocal fully nonlinear double obstacle problems
• Fractional semilinear Neumann problems arising from a fractional Keller--Segel model
• The obstacle problem for nonlinear integro-differential operators
• Existence, non-existence and blow-up behavior of minimizers for the mass-critical fractional nonlinear Schrödinger equations with periodic potentials
• Aperiodic fractional obstacle problems
• Non-Local Tug-of-War and the Infinity Fractional Laplacian
• Regularity theory for the fractional harmonic oscillator
• Higher regularity of the free boundary in the parabolic Signorini problem
• Classification of isolated singularities of positive solutions for Choquard equations
• The Dirichlet Problem for the Logarithmic Laplacian
• Stabilization of fractional-evolution systems
• Certain fractional Laplacian equations that do not have smooth solutions
• Symmetry of Positive Solutions for the Fractional Schr$\ddot{\text{o}}$dinger Equations with Choquard-type Nonlinearities
• Optimal Regularity for The Signorini Problem and its Free Boundary
• Ground state solution of fractional Schrödinger equations with a general nonlinearity
• Optimal regularity for the Signorini problem
• On the Interior $W^{1,p}$ estimate of extension problem of fractional elliptic equation
• Ground states of nonlinear fractional Schrödinger equation involving critical growth
• Arbitrary many positive solutions for a nonlinear problem involving the fractional Laplacian
• Separation of a Lower Dimensional Free Boundary in a Two Phase Problem
• Extension problem and fractional operators: semigroups and wave equations
• Oscillation of harmonic functions for subordinate Brownian motion and its applications
• Solitons and Gibbs measures for nonlinear Schroedinger equations
• Random Homogenization of Fractional Obstacle Problems
• Frequency of Harmonic functions in Carnot groups and for operators of Baouendi type
• Left-Inverses of Fractional Laplacian and Sparse Stochastic Processes
• The obstacle problem for a fractional Monge--Ampère equation
• Fractional Kirchhoff equation with a general critical nonlinearity
• Thin obstacle problem: estimates of the distance to the exact solution
• Optimal regularity and nondegeneracy of a free boundary problem related to the fractional Laplacian
• Weyl and Marchaud derivatives: a forgotten history
• On the Strong unique continuation property of a degenerate elliptic operator with Hardy type potential
• The concentration-compactness principles for $W^{s,p(\cdot ,\cdot )}({\mathbb{R}}^N)$ and application
• Non-local diffusion with free boundaries
• Quantitative estimate of the continuum approximations of interacting particle systems in one dimension
• Nonlocal equations with gradient constraints
• Liouville-type theorems, radial symmetry and integral representation of solutions to Hardy-Hénon equations involving higher order fractional Laplacians
• The $C^{1,\alpha }$ boundary Harnack principle in a slit domain and its application to the Signorini problem
• On the Stability of the $s$-Nonlocal $p$-Obstacle Problem and their Coincidence Sets and Free Boundaries
• Fractional Parabolic Theory as a High-Dimensional Limit of Fractional Elliptic Theory
• On a weak maximum principle for a class of fractional diffusive equations
• Kroger's upper bound types for the Dirichlet eigenvalues of the fractional Laplacian
• Infinitely many solutions for a class of fractional Orlicz-Sobolev Schrödinger equations