Non-classical solutions of the $p$-Laplace equation

Abstract

In this paper we answer Iwaniec and Sbordone's conjecture [19] concerning very weak solutions to the p-Laplace equation. Namely, on one hand we show that distributional solutions of the p-Laplace equation in W 1,r for p = 2 and r > max{1, p − 1} are classical weak solutions if their weak derivatives belong to certain cones. On the other hand, we construct via convex integration non-energetic distributional solutions if this cone condition is not met, thus answering negatively Iwaniec and Sbordone's conjecture in general.

Rigidity of Euclidean product structure: breakdown for low Sobolev exponents

“…Staircase laminates and differential inclusions. Since the original application of Faraco, staircase laminates have been applied in several situations where one can expect endpoint weak L p bounds [20,6,7,1,3,14,13,8]. Although staircase laminates are a very versatile tool, we were unable to find a general treatment of staircase laminates analogous to the case of bounded laminates described in Section 2.2 and in particular a corresponding generalization of Lemma 2.2.…”

“…Example 4. Our fourth example is from [8] and arises in the theory of the p-harmonic operator. For any p ∈ (1, ∞), let…”

“…A closely related notion was introduced by M. Sychev [29] for bounded sets K. It turns out that the concept of reduction is both simpler and more powerful for unbounded sets K; see the discussion after Definition 1.1. As an illustration we show in the Appendix how a result on optimal L p regularity for elliptic systems with bounded measurable coefficients [1] as well as recent results on irregular solutions of the p-Laplace equation [8] can easily be obtained by this method, once one can perform a certain algebraic construction in matrix space leading to a staircase laminate (see Definition 3.1, Proposition 3.1, and Theorem 4.3).…”

“…(1.13) Moreover, as is typical in problems of this type, we seek solutions which in addition satisfy affine boundary conditions. Differential inclusions of this type have a long history, both in the Lipschitz setting for compact K [15,22,10,25,29,23,20,17,24,5,30] as well as the Sobolev setting for unbounded K [12,20,6,7,1,3,21,26,14,13,8]. In most of these works the differential inclusion (1.13) is solved using convex integration, an iterative construction of highly oscillatory approximate solutions which are locally almost one-dimensional.…”

“…In the original proof in [14], the laminates for different m are not considered separately, but combined in a complicated and careful bookkeeping strategy. (b) Very weak solutions to isotropic second-order elliptic equations with measurable coefficients, following [1] (c) Recent examples of very weak solutions of the p-Laplace equation [8] Although these examples indicate the wide applicability of our framework, as a word of warning, we point out that our approach so far depends on the fact that one ultimately can reduce to the full space R d×m . For instance, this restriction means that we are not able to directly apply the general results in this paper to the case of higher integrability of W 1,2 weak solutions of second-order elliptic equations in [1].…”