Convex integration and the Lp theory of elliptic equations

Abstract

This paper deals with the L p theory of linear elliptic partial differential equations with bounded measurable coefficients. We construct in two dimensions examples of weak and so-called very weak solutions, with critical integrability properties, both to isotropic equations and to equations in non-divergence form. These examples show that the general L p theory, developed in [1,24] and [2], cannot be extended under any restriction on the essential range of the coefficients. Our constructions are based on the method of convex integration, as used by S. Müller and V.Šverák in [30] for the construction of counterexamples to regularity in elliptic systems, combined with the staircase type laminates introduced in [15].

Orientation-Preserving Young Measures

“…Also, since the p-growth condition turns out to be crucial, one could speak of "p-convex integration" for this variant and, further, our convergence principle is different from the one usually employed in convex integration as it involves different generalized convex hulls, see Section 7 for details. This gives rise to an application of convex integration which distinguishes between different degrees of integrability (rather than differentiability), a phenomenon that has, to the authors' knowledge, previously appeared only in [AFS08] and work of Yan [Yan96,Yan01a,Yan03] where convex integration techniques are employed for unbounded sets using laminar constructions resembling those of the present work. Indeed, convex integration typically shows flexibility below a certain threshold regularity, whereas at higher regularity the situation is rigid.…”

Improved Hölder regularity for strongly elliptic PDEs

“…Whether this exponent is the optimal one is a question we leave open, though Theorem 16 may indicate that the optimal exponent is larger than p K . Nevertheless, there are previous studies that show that solutions to elliptic equations may admit improved Hölder regularity but no improved Sobolev regularity -see for example [4], [7] and [20].…”

Magnetic helicity, weak solutions and relaxation of ideal MHD

“…As B • E is a compensated compactness quantity, our result is inspired by construction of gradients of homeomorphisms with vanishing Jacobian determinant. In fact, those were inspired by the construction of very weak solutions to elliptic equations [F04,AFS08]. Notice that such constructions can only exist in function spaces where the corresponding compensated compactness quantity is no longer weakly continuous.…”

“…The proof, presented in Section 2, relies on an anisotropic version of convex integration in L p through staircase laminates from [F03,AFS08] which might be of independent interest.…”