Regularity theory for general stable operators

By Xavier Ros-Oton, J. Serra. 2014

207 citations recorded by the source corpus.

We establish sharp regularity estimates for solutions to $Lu=f$ in $\Omega\subset\mathbb R^n$, being $L$ the generator of any stable and symmetric L\'evy process. Such nonlocal operators $L$ depend on a finite measure on $S^{n-1}$, called the spectral measure. First, we study the interior regularity of solutions to $Lu=f$ in $B_1$. We prove that if $f$ is $C^\alpha$ then $u$ belong to $C^{\alpha+2s}$ whenever $\alpha+2s$ is not an integer. In case $f\in L^\infty$, we show that the solution $u$ is $C^{2s}$ when $s\neq1/2$, and $C^{2s-\epsilon}$ for all $\epsilon>0$ when $s=1/2$. Then, we study the boundary regularity of solutions to $Lu=f$ in $\Omega$, $u=0$ in $\mathbb R^n\setminus\Omega$, in $C^{1,1}$ domains $\Omega$. We show that solutions $u$ satisfy $u/d^s\in C^{s-\epsilon}(\overline\Omega)$ for all $\epsilon>0$, where $d$ is the distance to $\partial\Omega$. Finally, we show that our results are sharp by constructing two counterexamples.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/5fbd655e9d6a7ea679d7255ddedcfa423e80ec71

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Citations in the corpus

The Bernstein Technique for Integro-Differential Equations (2020, background): “able symmetric operators, where the kernels are given by K(z) := 1 2|z|n+2s a z |z| +a − z |z| , (1.17) under appropriate hypotheses on the positive function a. See, for instance, (1.3) in [28]. Integro-diﬀerential operators with such kernels naturally arise in the L´evy-Khintchine probabilisti”
A new look at the fractional Poisson problem via the logarithmic Laplacian (2019, background): “Moreover, Grubb [20] and Ros-Oton and Serra [25, 26] provide estimates on the interior regularity.”
The Schauder estimate for kinetic integral equations (2018, background): “for stable-like kernels of the form K(w) = |w|−d−2sa(w/|w|), they are equivalent. This follows easily by computing the Fourier symbol of the operator associated with K (see for example [17], and also [16]). For non-stable-like processes the situation is less clear. We do not know any example of a ke”
Nonlocal elliptic equations in bounded domains: a survey (2015, background): “[46] X. Ros-Oton, J. Serra, Boundary regularity for fully nonlinear integro-differential equations, preprint arXiv (2014).”
Obstacle problems for integro-differential operators: Higher regularity of free boundaries (2019, background): “The only known results in this direction are due to the second author and Serra [30–33] for k = 1, or to Grubb [21, 22] for k = ∞, and are actually very delicate to establish.”
A Simple Solver for the Fractional Laplacian in Multiple Dimensions (2018, methodology): “While we use standard regularity assumptions to prove convergence of the finite-difference quadrature approximation, such regularity does not hold in general for solutions to (3), particularly near the boundary ∂Ω [45].”
Regularity theory for general stable operators: parabolic equations (2015, methodology): “or (1.5). Still, even in the case of translation invariant equations, the interior regularity for general stable operators was open. In case of elliptic equations, this problem was recently solved in [RS14b], where the second author and Serra obtained sharp regularity estimates in H¨older spaces for”
Entire $s$-harmonic functions are affine (2014, background): “ville theorem holds as well and was proved by K. Bogdan et al. in [4]. Applications of Liouville theorems for nonlocal operators in the study of nonlocal elliptic systems of equations can be found in [1,8,13,14]. The aim of this paper is to classify all s-harmonic functions in RN, thereby obtaining ”