Weighted and nonlocal Poisson equations on domains with capacity density conditions
Abstract
We establish necessary and sufficient conditions for global Hölder solvability of weighted Poisson equations and related fractional problems. On domains satisfying a capacity density condition, a nonnegative measure with bounded support depth admits a bounded globally Hölder continuous zero-boundary solution if and only if it satisfies an interior Morrey-type growth condition. Conversely, uniform boundary decay estimates for solutions with localized nonnegative sources characterize the capacity density condition. Analogous characterizations are established for the restricted and spectral fractional Laplacians. Our approach is based on weighted strong barriers and also yields two-weight Hardy inequalities, compact trace embeddings, and global Hölder solvability for nonlinear Dirichlet-to-Neumann operators.
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BibTeX
@misc{hara2026weighted,
title = {{Weighted and nonlocal Poisson equations on domains with capacity density conditions}},
author = {Takanobu Hara},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.32070},
eprint = {2610.32070},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.32070},
}
RIS
TY - UNPB
TI - Weighted and nonlocal Poisson equations on domains with capacity density conditions
AU - Takanobu Hara
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.32070
UR - https://arxiv.org/abs/2610.32070
AB - We establish necessary and sufficient conditions for global Hölder solvability of weighted Poisson equations and related fractional problems. On domains satisfying a capacity density condition, a nonnegative measure with bounded support depth admits a bounded globally Hölder continuous zero-boundary solution if and only if it satisfies an interior Morrey-type growth condition. Conversely, uniform boundary decay estimates for solutions with localized nonnegative sources characterize the capacity density condition. Analogous characterizations are established for the restricted and spectral fractional Laplacians. Our approach is based on weighted strong barriers and also yields two-weight Hardy inequalities, compact trace embeddings, and global Hölder solvability for nonlinear Dirichlet-to-Neumann operators.
ER -