arXiv subjects
Publications and source records attributed to Marvin Weidner.
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.
We study parabolic equations governed by integro-differential operators with nonlocal components in some directions and local components in the remaining directions. The setting contains the purely nonlocal, as well as the purely local case. Our approach is based on an energy method allowing for jumping measures that are singular or supported on cusps. In addition, the jumping measure may depend on the direction. The emphasis of our study is on the weak Harnack inequality and Hölder regularity estimates for solutions of such equations. The main regularity estimates are robust in the sense that the constants can be chosen independently of the order of differentiability of the operators.
In this paper we establish optimal regularity estimates and smoothness of free boundaries for nonlocal obstacle problems governed by a very general class of integro-differential operators with possibly singular kernels. More precisely, in contrast to all previous known results, we are able to treat nonlocal operators whose kernels are not necessarily pointwise comparable to the one of the fractional Laplacian. Such operators might be very anisotropic in the sense that they "do not see" certain directions at all, or might have substantial oscillatory behavior, causing the nonlocal Harnack inequality to fail.
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.
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 Hölder regularity and a weak Harnack inequality are proved in a similar setup.
The aim of this article is to prove that diffusion processes in $\mathbb{R}^d$ with a drift can be approximated by suitable Markov chains on $n^{-1}\mathbb{Z}^d$. Moreover, we investigate sufficient conditions on the conductances which guarantee convergence of the associated Markov chains to such Markov processes. Analogous questions are answered for a large class of nonsymmetric jump processes. The proofs of our results rely on regularity estimates for weak solutions to the corresponding nonsymmetric parabolic equations and Dirichlet form techniques.
The aim of this article is to develop the regularity theory for parabolic equations driven by nonlocal operators associated with nonsymmetric forms. Hölder regularity and weak Harnack inequalities are proved using extensions of recently established nonlocal energy methods. We are able to connect the theory of nonsymmetric nonlocal operators with the important results of Aronson-Serrin in the local linear case. This connection is exemplified by nonlocal-to-local convergence results identifying the limiting class of operators as second order differential operators with drift terms.
We prove a full Harnack inequality for local minimizers, as well as weak solutions to nonlocal problems with non-standard growth. The main auxiliary results are local boundedness and a weak Harnack inequality for functions in a corresponding De Giorgi class. This paper builds upon a recent work on regularity estimates for such nonlocal problems by the same authors.
We study robust regularity estimates for local minimizers of nonlocal functionals with non-standard growth of $(p,q)$-type and for weak solutions to a related class of nonlocal equations. The main results of this paper are local boundedness and Hölder continuity of minimizers and weak solutions. Our approach is based on the study of corresponding De Giorgi classes.
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.
In this paper, we study the existence of minimizers of the Sobolev quotient for a class of nonlocal operators with an orthotropic structure having different exponents of integrability and different orders of differentiability. Our method is based on the concentration-compactness principle which we extend to this class of operators. One consequence of our main result is the existence of a nontrivial nonnegative solution to the corresponding critical problem.