arXiv subjects
Publications and source records attributed to Moritz Kassmann.
We study function spaces and extension results in relation with Dirichlet problems involving integrodifferential operators. For such problems, data are prescribed on the complement of a given domain in the Euclidean space. We introduce a function space that serves as a trace space for nonlocal Dirichlet problems and study related extension results.
The paper deals with homogenization of Levy-type operators with rapidly oscillating coefficients. We consider cases of periodic and random statistically homogeneous micro-structures and show that in the limit we obtain a Levy-operator. In the periodic case we study both symmetric and non-symmetric kernels whereas in the random case we only investigate symmetric kernels. We also address a nonlinear version of this homogenization problem.
We study translation-invariant integrodifferential operators that generate Lévy 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.
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-Lamé 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.
We study nonlocal operators acting on functions in the Euclidean space. The operators under consideration generate anisotropic jump processes, e.g., a jump process that behaves like a stable process in each direction but with a different index of stability. Its generator is the sum of one-dimensional fractional Laplace operators with different orders of differentiability. We study such operators in the general framework of bounded measurable coefficients. We prove a weak Harnack inequality and Hölder regularity results for solutions to corresponding integro-differential equations.
We study quadratic functionals on $L^2(\mathbb{R}^d)$ that generate seminorms in the fractional Sobolev space $H^s(\mathbb{R}^d)$ for $0 < s < 1$. The functionals under consideration appear in the study of Markov jump processes and, independently, in recent research on the Boltzmann equation. The functional measures differentiability of a function $f$ in a similar way as the seminorm of $H^s(\mathbb{R}^d)$. The major difference is that differences $f(y) - f(x)$ are taken into account only if $y$ lies in some double cone with apex at $x$ or vice versa. The configuration of double cones is allowed to be inhomogeneous without any assumption on the spatial regularity. We prove that the resulting seminorm is comparable to the standard one of $H^s(\mathbb{R}^d)$. The proof follows from a similar result on discrete quadratic forms in $\mathbb{Z}^d$, which is our second main result. We establish a general scheme for discrete approximations of nonlocal quadratic forms. Applications to Markov jump processes are discussed.
We study positive solutions to the heat equation on graphs. We prove variants of the Li-Yau gradient estimate and the differential Harnack inequality. For some graphs, we can show the estimates to be sharp. We establish new computation rules for differential operators on discrete spaces and introduce a relaxation function that governs the time dependency in the differential Harnack estimate.
We prove Schauder estimates in generalized Hölder spaces $C^ψ(\mathbb{R}^d)$. These spaces are characterized by a general modulus of continuity $ψ$, which cannot be represented by a real number. We consider linear operators $\mathcal{L}$ between such spaces. The operators $\mathcal{L}$ under consideration are integrodifferential operators with a functional order of differentiability $φ$ which, again, is not represented by a real number. Assuming that $\mathcal{L}$ has $ψ$-continuous coefficients, we prove that solutions $u \in C^{φψ}(\mathbb{R}^d)$ to linear equations $\mathcal{L} u = f \in C^ψ(\mathbb{R}^d)$ satisfy a priori estimates in $C^{φψ}(\mathbb{R}^d)$.
We study weak solutions to nonlocal equations governed by integrodifferential operators. Solutions are defined with the help of symmetric nonlocal bilinear forms. Throughout this work, our main emphasis is on operators with general, possibly singular, measurable kernels. We obtain regularity results which are robust with respect to the differentiability order of the equation. Furthermore, we provide a general tool for the derivation of Hölder a-priori estimates from the weak Harnack inequality. This tool is applicable for several local and nonlocal, linear and nonlinear problems on metric spaces. Another aim of this work is to provide comparability results for nonlocal quadratic forms.
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.
One aim of this note is to give an overview of some developments in the area of Dirichlet forms. A second aim is to review the new book "Semi-{D}irichlet forms and {M}arkov processes" by Yoichi Oshima. The book appeared last year, but first versions were written as lecture notes 25 years ago. We first give a rather short and light introduction into the field of Dirichlet forms with a special emphasis on the subjects presented in the book under consideration. After a small account on the history of Dirichlet forms we comment on the book by Oshima against the background of related works.
We study growth lemmas and questions of regularity for generators of Markov processes. The generators are allowed to have an arbitrary order of differentiability less than 2. In general, this order is represented by a function and not by a number. The approach enables a careful study of regularity issues up to the phase boundary between integro-differential (positive order of differentiability) and integral operators (nonnegative order of differentiability). The proof is based on intrinsic scaling properties of the underlying operators and stochastic processes.
In this note we set up the elliptic and the parabolic Dirichlet problem for linear nonlocal operators. As opposed to the classical case of second order differential operators, here the "boundary data" are prescribed on the complement of a given bounded set. We formulate the problem in the classical framework of Hilbert spaces and prove unique solvability using standard techniques like the Fredholm alternative.
We survey recent regularity results for parabolic equations involving nonlocal operators like the fractional Laplacian. We extend the results of Felsinger-Kassmann (2013) and obtain regularity estimates for nonlocal operators with kernels not being absolutely continuous with respect to the Lebesgue measure.
We prove Hölder regularity results for a class of nonlinear elliptic integro-differential operators with integration kernels whose ellipticity bounds are strongly directionally dependent. These results extend those in [9] and are also uniform as the order of operators approaches 2.
Weak solutions to parabolic integro-differential operators of order $α\in (α_0, 2)$ are studied. Local a priori estimates of Hölder norms and a weak Harnack inequality are proved. These results are robust with respect to $α\nearrow 2$. In this sense, the presentation is an extension of Moser's result in 1971.
The aim of this note is to show that Poincaré inequalities imply corresponding weighted versions in a quite general setting. Fractional Poincaré inequalities are considered, too. The proof is short and does not involve covering arguments.
We prove regularity estimates for functions which are harmonic with respect to certain jump processes. The aim of this article is to extend the method of Bass-Levin[BL02] and Bogdan-Sztonyk[BS05] to more general processes. Furthermore, we establish a new version of the Harnack inequality that implies regularity estimates for corresponding harmonic functions.