Nonlocal operators related to nonsymmetric forms II: Harnack inequalities

Local boundedness and Harnack inequalities are studied for solutions to parabolic and elliptic integro-differential equations whose governing nonlocal operators are associated with nonsymmetric forms. We present two independent proofs, one being based on the De Giorgi iteration and the other one on the Moser iteration technique. This article is a continuation of a recent work by the same authors, where H\"older regularity and a weak Harnack inequality are proved in a similar setup

Optimal regularity for nonlocal elliptic equations and free boundary problems

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

Gradient Riesz potential estimates for a general class of measure data quasilinear systems

We study the gradient regularity of solutions to measure data elliptic systems with Uhlenbeck-type structure and Orlicz growth. For any bounded Borel measure, pointwise estimates for the gradient of solutions are provided in terms of the truncated Riesz potential. This allows us to show a precise transfer of regularity from data to solutions on various scales.Comment: 36 page

Robust H\"older Estimates for Parabolic Nonlocal Operators

In this work we study parabolic equations determined by nonlocal operators in a general framework of bounded and measurable coefficients. Our emphasis is on the weak Harnack inequality and H\"older regularity estimates for solutions of such equations. We allow the underlying jump measures to be singular with a singularity that depends on the coordinate direction. This approach also allows to study several classes of non-singular jump measures that have not been dealt with so far. The main estimates are robust in the sense that the constants can be chosen independently of the order of differentiability of the operators.Comment: 39 pages, 3 figure

Potential theory for nonlocal drift-diffusion equations

The purpose of this paper is to prove new fine regularity results for nonlocal drift-diffusion equations via pointwise potential estimates. Our analysis requires only minimal assumptions on the divergence free drift term, enabling us to include drifts of critical order belonging merely to BMO. In particular, our results allow to derive new estimates for the dissipative surface quasi-geostrophic equation.Comment: 38 page

Energy methods for nonsymmetric nonlocal operators

Weidner M. Energy methods for nonsymmetric nonlocal operators. Bielefeld: Universität Bielefeld; 2022.The goal of this thesis is to develop the regularity theory for nonlocal parabolic equations. We focus on problems driven by nonlocal operators associated with nonsymmetric bilinear forms. In contrast to the symmetric case, nonsymmetric nonlocal operators have not yet been studied systematically. Our main results contain Hölder regularity estimates, full Harnack inequalities, and two-sided heat kernel estimates, which we obtain by extending recently established nonlocal energy methods. We are able to connect the theory of nonsymmetric nonlocal operators with the important results of Aronson-Serrin, Ladyzhenskaya-Solonnikov-Ural'tceva, and Aronson in the local linear case. Moreover, we develop a new technique for proving upper off-diagonal heat kernel estimates for nonlocal operators, which is based on the original ideas by Aronson in the local case. As an application of our regularity results, we establish Markov chain approximations for a large class of nonsymmetric diffusions and jump processes

Upper heat kernel estimates for nonlocal operators via Aronson’s method

Kaßmann M, Weidner M. Upper heat kernel estimates for nonlocal operators via Aronson’s method. Calculus of Variations and Partial Differential Equations. 2023;62(2): 68.**Abstract** In his celebrated article, Aronson established Gaussian bounds for the fundamental solution to the Cauchy problem governed by a second order divergence form operator with uniformly elliptic coefficients. We extend Aronson’s proof of upper heat kernel estimates to nonlocal operators whose jumping kernel satisfies a pointwise upper bound and whose energy form is coercive. A detailed proof is given in the Euclidean space and extensions to doubling metric measure spaces are discussed