Intrinsic scaling properties for nonlocal operators

Kaßmann M, Mimica A. Intrinsic scaling properties for nonlocal operators. Journal of the European Mathematical Society. 2017;19(4):983-1011.We study integrodifferential operators and regularity estimates for solutions to integrodifferential equations. Our emphasis is on kernels with a critically low singularity which does not allow for standard scaling. For example, we treat operators that have a logarithmic order of differentiability. For corresponding equations we prove a growth lemma and derive a priori estimates. We derive these estimates by classical methods developed for partial differential operators. Since the integrodifferential operators under consideration generate Markov jump processes, we are able to offer an alternative approach using probabilistic techniques

Solvability of nonlocal systems related to peridynamics

Kaßmann M, Mengesha T, Scott J. Solvability of nonlocal systems related to peridynamics. Communications on Pure and Applied Analysis . 2019;18(3):1303-1332.In this work, we study the Dirichlet problem associated with a strongly coupled system of nonlocal equations. The system of equations comes from a linearization of a model of peridynamics, a nonlocal model of elasticity. It is a nonlocal analogue of the Navier-Lame system of classical elasticity. The leading operator is an integro-differential operator characterized by a distinctive matrix kernel which is used to couple differences of components of a vector field. The paper's main contributions are proving well-posedness of the system of equations and demonstrating optimal local Sobolev regularity of solutions. We apply Hilbert space techniques for well-posedness. The result holds for systems associated with kernels that give rise to non-symmetric bilinear forms. The regularity result holds for systems with symmetric kernels that may be supported only on a cone. For some specific kernels associated energy spaces are shown to coincide with standard fractional Sobolev spaces

Remarks on the Nonlocal Dirichlet Problem

Grzywny T, Kaßmann M, Lezaj L. Remarks on the Nonlocal Dirichlet Problem. Potential Analysis. 2020.We study translation-invariant integrodifferential operators that generate Levy processes. First, we investigate different notions of what a solution to a nonlocal Dirichlet problem is and we provide the classical representation formula for distributional solutions. Second, we study the question under which assumptions distributional solutions are twice differentiable in the classical sense. Sufficient conditions and counterexamples are provided

On Regularity for Beurling–Deny Type Dirichlet Forms

Kaßmann M. On Regularity for Beurling–Deny Type Dirichlet Forms. Potential Analysis. 2003;19(1):69-87.Characteristic examples of Beurling–Deny type Dirichlet forms are considered. The forms are identified with bilinear forms of integro-differential operators that arise as generators of jump-diffusion processes. The aim of this article is to prove Harnack inequalities for these operators and consequently Hölder regularity of weak H 1-solutions. Moser's iteration technique is used

A note on integral inequalities and embeddings of Besov spaces

Kaßmann M. A note on integral inequalities and embeddings of Besov spaces. JIPAM. J. Inequal. Pure Appl. Math. 2003;4(5):Article 107, 3

Variational solutions to nonlocal problems

Kaßmann M. Variational solutions to nonlocal problems. In: Spectral structures and topological methods in mathematics. EMS Ser. Congr. Rep. EMS Publ. House, Z\"{u}rich; 2019: 183-196

The theory of De Giorgi for non-local operators

Kaßmann M. The theory of De Giorgi for non-local operators. Comptes Rendus Mathematique. 2007;345(11):621-624

A new formulation of Harnack's inequality for nonlocal operators

Kaßmann M. A new formulation of Harnack's inequality for nonlocal operators. Comptes Rendus Mathematique. 2011;349(11-12):637-640.We provide a new formulation of Harnack's inequality for nonlocal operators. In contrast to previous versions we do not assume harmonic functions to have a sign. The version of Harnack's inequality given here generalizes Harnack's classical result from 1887 to nonlocal situations. As a consequence we derive Holder regularity estimates by an extension of Moser's method. The inequality that we propose is equivalent to Harnack's original formulation but seems to be new even for the Laplace operator. (C) 2011 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved

L-harmonische Funktionen und Sprungprozesse

Kaßmann M. L-harmonische Funktionen und Sprungprozesse. Mitteilungen der Deutschen Mathematiker-Vereinigung. 2006;14(2):80-87

A priori estimates for integro-differential operators with measurable kernels

Kaßmann M. A priori estimates for integro-differential operators with measurable kernels. Calculus of Variations and Partial Differential Equations. 2009;34(1):1-21