Gradient estimates for generalized double phase problems with two modulating coefficients
Abstract
We prove local Calderón-Zygmund estimates for distributional solutions to non-uniformly elliptic equations in divergence form modeled on the energy density a(x)G(|z|)+b(x)H(|z|), where G and H are Young functions and a(·), b(·) are nonnegative coefficients, Hölder continuous with exponents α and β. Either coefficient may vanish, and only their sum is bounded away from zero, so that each phase can dominate or disappear. Under a gap condition that reduces to q/p≤1+min{α,β}/n for G(t)=tp and H(t)=tq, including the borderline case of equality, we show that the energy density of the gradient inherits the Orlicz integrability of the energy density of the datum. The proof combines a fractional differentiability estimate, which gives higher integrability up to the borderline growth, with comparison arguments that freeze the two coefficients in the appropriate order.
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BibTeX
@misc{oh2026gradient,
title = {{Gradient estimates for generalized double phase problems with two modulating coefficients}},
author = {Jehan Oh and Ambesh Kumar Pandey},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.26543},
eprint = {2608.26543},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.26543},
}
RIS
TY - UNPB
TI - Gradient estimates for generalized double phase problems with two modulating coefficients
AU - Jehan Oh
AU - Ambesh Kumar Pandey
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2608.26543
UR - https://arxiv.org/abs/2608.26543
AB - We prove local Calderón-Zygmund estimates for distributional solutions to non-uniformly elliptic equations in divergence form modeled on the energy density a(x)G(|z|)+b(x)H(|z|), where G and H are Young functions and a(·), b(·) are nonnegative coefficients, Hölder continuous with exponents α and β. Either coefficient may vanish, and only their sum is bounded away from zero, so that each phase can dominate or disappear. Under a gap condition that reduces to q/p≤1+min{α,β}/n for G(t)=t^p and H(t)=t^q, including the borderline case of equality, we show that the energy density of the gradient inherits the Orlicz integrability of the energy density of the datum. The proof combines a fractional differentiability estimate, which gives higher integrability up to the borderline growth, with comparison arguments that freeze the two coefficients in the appropriate order.
ER -