Pathological solutions to elliptic problems in divergence form with continuous coefficients

By Tianling Jin, Vladimir G. Maz'ya, Jean Van Schaftingen. 2009

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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.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/f45a8ec6a380d23a30dd6fd081505bc6f7c5287e

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Citations in the corpus

A nonlinear Stein theorem (2014, background): “In fact, already in the case of linear and homogeneous elliptic equations div (A(x)Du) = 0 (4) the gradient of solutions is in general unbounded for continuos but not Dinicontinuous matrices A(·), as shown in [13].”
New perturbation methods for nonlinear parabolic problems (2012, background): “Counterexamples (see [14]) valid already in the case of linear elliptic equations of the type div(A(x)Du) = 0 show that when coefficients A(·) (i.”
Riesz Potentials and Nonlinear Parabolic Equations (2013, background): “Merely continuous coefficients are not sufficient to ensure that the gradient belongs even to BMO, see [14].”
Gradient estimates via linear and nonlinear potentials (2010, background): “As already mentioned in the case p < 2, we find that the same optimal conditions valid for linear equations actually works in the general degenerate case p = 2; see again [14].”
On C1, C2, and weak type-(1,1) estimates for linear elliptic operators (2016, background): “See also [11] and [7] for the corresponding counterexamples.”
Regularity for quasilinear equations on degenerate singular sets (2012, background): “Also, it has been shown, see [10], that continuity of aij is not enough to assure gradient bounds for solutions.”
Nonlinear Measure Data Problems (2011, background): “10) solutions are not Lipschitz if the elliptic matrix A(x) has just continuousbut-not-Dini-continuous entries [51].”