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Schauder estimates for ratios of elliptic solutions under zero set inclusion
Abstract
Let u,v be two nontrivial functions that solve div(A∇ u)=div(A∇ v)=0 in Br⊂ℝn, n≥2, r>0, with A symmetric, uniformly elliptic and Lipschitz continuous, and assume the zero set inclusion Z(u)⊆ Z(v). We prove that the ratio w=v/u continuously extends, together with its gradient, across Z(u)={u=0}. In particular, w∈ C1,αloc(Br) for any α∈(0,1). Furthermore, given any integer k≥2 and α∈(0,1), if A∈ Ck−1,α(Br), then w∈ Ck,αloc(Br). The estimates are uniform in classes of coefficients A with prescribed ellipticity and regularity bounds, and in normalized classes of denominators u with uniformly bounded doubling constant.
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BibTeX
@misc{vita2026schauder,
title = {{Schauder estimates for ratios of elliptic solutions under zero set inclusion}},
author = {Stefano Vita},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.28077},
eprint = {2610.28077},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.28077},
}
RIS
TY - UNPB
TI - Schauder estimates for ratios of elliptic solutions under zero set inclusion
AU - Stefano Vita
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.28077
UR - https://arxiv.org/abs/2610.28077
AB - Let u,v be two nontrivial functions that solve div(A∇ u)=div(A∇ v)=0 in B_r⊂ℝ^n, n≥2, r>0, with A symmetric, uniformly elliptic and Lipschitz continuous, and assume the zero set inclusion Z(u)⊆ Z(v). We prove that the ratio w=v/u continuously extends, together with its gradient, across Z(u)={u=0}. In particular, w∈ C^1,α_loc(B_r) for any α∈(0,1). Furthermore, given any integer k≥2 and α∈(0,1), if A∈ C^k−1,α(B_r), then w∈ C^k,α_loc(B_r). The estimates are uniform in classes of coefficients A with prescribed ellipticity and regularity bounds, and in normalized classes of denominators u with uniformly bounded doubling constant.
ER -