publications

#Published Research Papers

#The Dirichlet Problem for Lévy-stable operators with \(L^2\)-data

Grube, F., Hensiek, T. & Schefer, W. The Dirichlet problem for Lévy-stable operators with \(L^2\)-data. Calc. Var. 63, 74 (2024). https://doi.org/10.1007/s00526-024-02679-8

#Robust nonlocal trace and extension theorems

Grube, F. and Hensiek, T. “Robust nonlocal trace spaces and Neumann problems”. In: Nonlinear Analysis 241 (2024), p. 113481. issn: 0362-546X. doi: https://doi.org/10.1016/j.na.2023.113481.

#Maximum principle for stable operators

Grube, F. and Hensiek, T., Maximum principle for stable operators, Math. Nachr. 296 (2023), 5684–5702. https://doi.org/10.1002/mana.202200354

#Preprints

#The Dirichlet Problem for Lévy-stable operators with \(L^2\)-data

We prove Sobolev regularity for distributional solutions to the Dirichlet problem for generators of \(2s\)-stable processes and exterior data, inhomogeneity in weighted \(L^2\)-spaces. This class of operators includes the fractional Laplacian. For these rough exterior data the theory of weak variational solutions is not applicable. Our regularity estimate is robust in the limit \(s\to1−\) which allows us to recover the local theory. T.Hensiek, W.Schefer, FG (2023) arxiv

#Robust nonlocal trace and extension theorems

We prove trace and extension results for Sobolev-type function spaces that are well suited for nonlocal Dirichlet and Neumann problems including those for the fractional p-Laplacian. Our results are robust with respect to the order of differentiability. In this sense they are in align with the classical trace and extension theorems. M. Kassmann, FG (2023) arxiv

#Robust nonlocal trace spaces and Neumann problems

We prove trace and extension results for fractional Sobolev spaces of order \(s\in(0,1)\). These spaces are used in the study of nonlocal Dirichlet and Neumann problems on bounded domains. The results are robust in the sense that the continuity of the trace and extension operators is uniform as s approaches 1 and our trace spaces converge to \(H^{1/2}(\partial \Omega)\). We apply these results in order to study the convergence of solutions of nonlocal Neumann problems as the integro-differential operators localize to a symmetric, second order operator in divergence form. T. Hensiek, FG (2022) arXiv

#Maximum principle for stable operators

We prove a weak maximum principle for nonlocal symmetric stable operators. This includes the fractional Laplacian. The main focus of this work is the regularity of the considered function. T. Hensiek, FG (2022) arXiv

#Norm inflation for the Zakharov system

The Cauchy problem for the classical Zakharov system is shown to be ill-posed in the sense of norm inflation in a range of Sobolev spaces for all dimensions d. This proves several results on well-posedness, which includes existence of solutions, uniqueness and continuous dependence on the initial data, to be sharp up to endpoints. FG (2022) arXiv