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This is a Quantitative Study study.

October 5, 2025Open Access

The inhomogeneous fractional Dirichlet problem

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The study reveals new boundary regularity results for the inhomogeneous Dirichlet problem involving fractional operators.
Explicit counterexamples demonstrate the sharpness of the derived results, particularly for the fractional Laplacian.
The analysis focuses on boundary conditions in generalized Hölder spaces, a key factor in addressing the Dirichlet problem.
Findings challenge prior methodologies by directly addressing the inhomogeneous problem without modifying exterior data.

Implication

This research demonstrates boundary regularity for inhomogeneous Dirichlet problems, suggesting new insights into fractional Laplacians and generalized Hölder spaces.

Abstract

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.

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Cite This Study

Florian Grube (2025) studied this question.

synapsesocial.com/papers/68e2537cd6d66a53c24743e1 https://doi.org/10.48550/arxiv.2510.01055

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Implication

This research demonstrates boundary regularity for inhomogeneous Dirichlet problems, suggesting new insights into fractional Laplacians and generalized Hölder spaces.

The inhomogeneous fractional Dirichlet problem

Key Points

Implication

Abstract

Discussion

Cite This Study

The inhomogeneous fractional Dirichlet problem

Key Points

Implication

Abstract

Discussion

Cite This Study

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