PreprintCC BY-NC-SA 4.0arXiv
Duality Conditions for Morrey Measures via Parabolic Capacity
Abstract
We establish Morrey-type conditions ensuring that a finite signed Radon measure belongs to the dual of the energy space of solutions to nonlinear parabolic equations of p-Laplacian type. More precisely, for the parabolic cylinders Qr,rp(z):=Br(x)×(t−rp,t+rp), we prove that |μ|(Qr,rp(z)∩(Ω× (0,T))) ≤ Mrn+p−ϑ, ϑ<p, implies duality, and this threshold is sharp in general. More generally, for cylinders with time length proportional to rq, q>1, the sufficient threshold is ϑ<min{p,q}. Finally, we discuss the resulting variational, energy, and renormalized solution theories for nonlinear parabolic equations with measure data, relating our conclusions to the existing literature.
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BibTeX
@misc{braglia2026duality,
title = {{Duality Conditions for Morrey Measures via Parabolic Capacity}},
author = {Lorenzo Braglia},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.32646},
eprint = {2609.32646},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.32646},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Duality Conditions for Morrey Measures via Parabolic Capacity
AU - Lorenzo Braglia
PY - 2026
DA - 2026/09/26
PB - arXiv
DO - 10.48550/arxiv.2609.32646
UR - https://arxiv.org/abs/2609.32646
AB - We establish Morrey-type conditions ensuring that a finite signed Radon measure belongs to the dual of the energy space of solutions to nonlinear parabolic equations of p-Laplacian type. More precisely, for the parabolic cylinders Q_r,r^p(z):=B_r(x)×(t−r^p,t+r^p), we prove that |μ|(Q_r,r^p(z)∩(Ω× (0,T))) ≤ Mr^n+p−ϑ, ϑ<p, implies duality, and this threshold is sharp in general. More generally, for cylinders with time length proportional to r^q, q>1, the sufficient threshold is ϑ<min{p,q}. Finally, we discuss the resulting variational, energy, and renormalized solution theories for nonlinear parabolic equations with measure data, relating our conclusions to the existing literature.
ER -