Abstract
In this article we derive an (almost) optimal scaling law for a singular perturbation problem associated with the Tartar square. As in Winter (Eur J Appl Math 8(2):185–207, 1997), Chipot (Numer Math 83(3):325–352, 1999), our upper bound quantifies the well-known construction which is used in the literature to prove the flexibility of the Tartar square in the sense of the flexibility of approximate solutions to the differential inclusion. The main novelty of our article is the derivation of an (up to logarithmic powers matching) ansatz free lower bound which relies on a bootstrap argument in Fourier space and is related to a quantification of the interaction of a nonlinearity and a negative Sobolev space in the form of “a chain rule in a negative Sobolev space”. Both the lower and the upper bound arguments give evidence of the involved “infinite order of lamination”.
Highlights
In this article we study a singularly perturbed variational problem for a differential inclusion associated with the Tartar square
Tartar square—which was introduced in several places in the literature [1,7,51,59, 64]—is the following set K ⊂ R2×2: K := A1, A2, A3, A4 with A1 = −01 −03, A2 = −3 0, A3 = − A1, A4 = − A2
It is known that arbitrarily small perturbations of the Tartar square enjoy even stronger flexibility in the sense that if Kδ ⊂ R2×2 is an arbitrarily small, open neighbourhood of K in R2×2, there are infinitely many, non-affine solutions to the differential inclusion ∇u ∈ Kδ which can be obtained by the method of convex integration [48]
Summary
In this article we study a singularly perturbed variational problem for a differential inclusion associated with the Tartar square. The Tartar square, T4, and more generally its siblings the TN -structures, are well-known sets in matrix space with important ramifications in the calculus of variations and the theoretical study of differential inclusions [12,28,32,48–50,62], the theory of partial differential equations, in particular as building blocks for convex integration schemes, ranging from elliptic and parabolic equations [46,47,63] to equations of fluid dynamics [21,24,61], and with various consequences for applications, for instance for the analysis of certain phase transformations [10,60] and related differential inclusions [43,52]. For further applications and implications we refer to the lecture notes and survey articles [32,33,45,53]
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