publications
(10) Boundary regularity for general elliptic operators of order \(2s\)
We establish optimal \(C^s\) boundary regularity for the most general class of (linear and translation invariant) nonlocal elliptic operator of order \(2s\). Namely, we consider Lévy operators that are symmetric and its Fourier symbol satisfies \(\mathcal{A}(\xi)\asymp |\xi|^{2s}\) in \(\mathbb{R}^d\). This was only known when the kernel of the operator (or Lévy measure) is either homogeneous or comparable to that of the fractional Laplacian, with different proofs in each case. Our new proofs extend both at the same time, and work in a very general class of domains, under a \(C^1\)-Dini-type condition. X. Ros-Oton, FG (2026) arxiv
(9) A strong-type unique continuation principle for the fractional $p$-Laplacian
We provide a simple and direct proof of a strong-type unique continuation principle for the fractional \((-\Delta_p)^s\)(-\Delta_p)^s$ for a range of $s$ and $p$. The result extends to strong solutions of the fractional nonlinear Schrödinger equation. We adapt the recent proofs of the weak UCP by Berger, Schilling and Prasad. FG (2026) arxiv
(8) The inhomogeneous fractional Dirichlet problem
We study boundary regularity for the inhomogeneous Dirichlet problem for \(2s\)-stable operators in generalized Hölder spaces. Moreover, we provide explicit counterexamples that showcase the sharpness of our results. Our approach directly addresses the inhomogeneous Dirichlet problem, rather than subtracting an appropriate extension of the exterior data. Even for the fractional Laplacian, our result is new. FG (2025) arxiv
(7) Pointwise estimates of the fundamental solution to the fractional Kolmogorov equation
We prove sharp two-sided estimates of the fundamental solution to the fractional Kolmogorov equation in \(\mathbb{R}\times\mathbb{R}\) using Fourier methods. Additionally, we provide an explicit form of the fundamental solution in case of the square root of the Laplacian. FG (2024) arxiv
(6) Boundary regularity and Hopf lemma for nondegenerate stable operators
We prove sharp boundary Hölder regularity for solutions to equations involving stable integro-differential operators in bounded open sets satisfying the exterior \(C^{1,\text{Dini}}\)-property. This result is new even for the fractional Laplacian. A Hopf-type boundary lemma is proven, too. An additional feature of this work is that the regularity estimate is robust as \(s\to 1-\) and we recover the classical results for second order equations. FG (2024) arxiv
(5) The Dirichlet Problem for Lévy-stable operators with \(L^2\)-data
Grube, F., Hensiek, T., and Schefer, W. The Dirichlet problem for Lévy-stable operators with \(L^2\)-data. Calc. Var. 63, 74 (2024). doi: https://doi.org/10.1007/s00526-024-02679-8
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
(4) Robust nonlocal trace and extension theorems
Grube, F. and Kassmann, M. Robust nonlocal trace and extension theorems. Analysis & PDE, Vol. 18 (2025), No. 10, pp. 2367–2414. doi: https://doi.org/10.2140/apde.2025.18.2367
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
(3) Robust nonlocal trace spaces and Neumann problems
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.
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
(2) Maximum principle for stable operators
Grube, F. and Hensiek, T., Maximum principle for stable operators, Math. Nachr. 296 (2023), 5684–5702. doi: https://doi.org/10.1002/mana.202200354
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
(1) Norm inflation for the Zakharov system
Florian Grube. Norm inflation for the Zakharov system. Communications on Pure and Applied Analysis (2025). doi: https://doi.org/10.3934/cpaa.2025060
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
Dissertation
Grube, F. "Integro-Differential Operators on Bounded Domains". Bielefeld: Universität Bielefeld. (2025) https://doi.org/10.4119/unibi/3005241