Pathological solutions to elliptic problems in divergence form with continuous coefficients

Jin, Tianling;Maz'ya, Vladimir;Van Schaftingen, Jean
(2009) Comptes rendus - Mathématique — Vol. 347, n° 13-14, p. 773-778 (2009)

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Abstract
We construct a function u is an element of W-loc(1.1) (B(0, 1)) which is a solution to div(A del u) = 0 in the sense of distributions, where A is continuous and u is not an element of W-loc(1.p) (B(0, 1)) for p > 1. We also give a function u is an element of W-loc(1.1)(B(0, 1)) such that u is an element of W-loc(1.p) (B(0, 1)) for every p < infinity, u satisfies div(A del u) = 0 with A continuous but u is not an element of W-loc(1.infinity) (B(0, 1)). This answers questions raised by H. Brezis (On a conjecture of J. Serrin, Rend. Lincei Mat. Appl. 19 (2008) 335-338). To cite this article: T Jin et al., C. R. Acad. Sci. Paris., Ser. 1 347 (2009). (C) 2009 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.
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Jin, T., Maz’ya, V., & Van Schaftingen, J. (2009). Pathological solutions to elliptic problems in divergence form with continuous coefficients. Comptes rendus - Mathématique, 347(13-14), 773-778. https://doi.org/10.1016/j.crma.2009.05.008 (Original work published 2009)