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Unbounded variation solutions for uniformly elliptic equations in nondivergence form in dimension three
Abstract
For each nonnegative integer m, we construct smooth symmetric 3× 3 coefficient matrices Am satisfying the fixed ellipticity bound I≤ Am≤ 281I for which the smooth solutions of uniformly elliptic equations in nondivergence form tr(Am(x)D2 um)=Am(x):D2um=0 in B2⊂ ℝ3 have common Dirichlet data, satisfy |um|L∞(B2)≤1, but limm→ ∞|Dum|L1(B1)=∞. Thus, there is no interior W1,1 estimate depending only on ellipticity in dimension three, and consequently no such W1,p estimate for any p≥1. This resolves in the negative an open question raised by Nadirashvili, Tkachev, and Vlăduţ. The construction also gives a uniformly convergent limit u∉ BVloc(B1) for a measurable uniformly elliptic coefficient matrix obtained as an L1 limit of the Am.
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BibTeX
@misc{le2026unbounded,
title = {{Unbounded variation solutions for uniformly elliptic equations in nondivergence form in dimension three}},
author = {Nam Q. Le and Qi Sun and Hung V. Tran},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.13380},
eprint = {2608.13380},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.13380},
}
RIS
TY - UNPB
TI - Unbounded variation solutions for uniformly elliptic equations in nondivergence form in dimension three
AU - Nam Q. Le
AU - Qi Sun
AU - Hung V. Tran
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2608.13380
UR - https://arxiv.org/abs/2608.13380
AB - For each nonnegative integer m, we construct smooth symmetric 3× 3 coefficient matrices A_m satisfying the fixed ellipticity bound I≤ A_m≤ 2^81I for which the smooth solutions of uniformly elliptic equations in nondivergence form tr(A_m(x)D^2 u_m)=A_m(x):D^2u_m=0 in B_2⊂ ℝ^3 have common Dirichlet data, satisfy |u_m|_L^∞(B_2)≤1, but lim_m→ ∞|Du_m|_L^1(B_1)=∞. Thus, there is no interior W^1,1 estimate depending only on ellipticity in dimension three, and consequently no such W^1,p estimate for any p≥1. This resolves in the negative an open question raised by Nadirashvili, Tkachev, and Vlăduţ. The construction also gives a uniformly convergent limit u∉ BV_loc(B_1) for a measurable uniformly elliptic coefficient matrix obtained as an L^1 limit of the A_m.
ER -