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Florian Grube

Publications and source records attributed to Florian Grube.

10 recordsLinked to original sources
arXiv2026

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 L\'evy 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 L\'evy 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.

math.AP↗
arXiv2026

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 $p$-Laplacian $(-\Delta_p)^s$ for a range of $s$ and $p$. The result extends to strong solutions of the fractional nonlinear Schr\"odinger equation. We adapt the recent proofs of the weak UCP by Berger, Schilling and Prasad.

math.AP↗
arXiv2025

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.

math.AP↗
arXiv2024

Boundary regularity and Hopf lemma for nondegenerate stable operators

We prove sharp boundary H{ö}lder 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.

math.AP↗
arXiv2023

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 $s\to 1-$ which allows us to recover the local theory.

math.AP↗
arXiv2023

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.

math.AP↗
arXiv2022

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 $H^{1/2}(\partial Ω)$. 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.

math.AP↗
arXiv2022

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.

math.AP↗
arXiv2022

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 $H^s(\mathbb{R}^d)\times H^l(\mathbb{R}^d)$ 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.

math.AP↗