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Optimal regularity for nonlocal elliptic equations and free boundary problems
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abstract
In this article we establish for the first time the $C^s$ boundary regularity of solutions to nonlocal elliptic equations with kernels $K(y)\asymp |y|^{-n-2s}$. This was known to hold only when $K$ is homogeneous, and it is quite surprising that it holds for general inhomogeneous kernels, too. As an application of our results, we also establish the optimal $C^{1+s}$ regularity of solutions to obstacle problems for general nonlocal operators with kernels $K(y)\asymp |y|^{-n-2s}$. Again, this was only known when $K$ is homogeneous, and it solves a long-standing open question in the field. A new key idea is to construct a 1D solution as a minimizer of an appropriate nonlocal one-phase free boundary problem, for which we establish optimal $C^s$ regularity and non-degeneracy estimates.
Forward citations
Cited by 3 Pith papers
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Bernoulli problem for the fractional $p$-Laplacian
Minimizers of the fractional p-Laplacian one-phase Bernoulli problem exist, are locally Hölder continuous, solve the homogeneous equation in their positivity set, and satisfy the optimal free-boundary growth u(x) ≲ |x−x0|^s.
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$C^{\infty}$ Regularity for the free boundary of one-phase Fractional Laplacian problem
Flat free boundaries in the one-phase fractional Laplacian problem are C∞, not merely C^{1,α}.
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Boundary regularity for nonlocal elliptic equations over Reifenberg flat domains
For operators comparable to the fractional Laplacian of order 2s, solutions with zero exterior data on Reifenberg flat domains are C^{s-ε} up to the boundary.
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