Convex integration for Lipschitz mappings and counterexamples to regularity

By Stefan Müller, Vladimír Sverák. 2004

302 citations recorded by the source corpus.

We study Lispchitz solutions of partial differential relations $\nabla u\in K$, where $u$ is a vector-valued function in an open subset of $R^n$. In some cases the set of solutions turns out to be surprisingly large. The general theory is then used to construct counter-examples to regularity of solutions of Euler-Lagrange systems satisfying classical ellipticity conditions.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/4990144699b1f434cf8b2d2113034a8e4202858d

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In Applied Mathematics

In Computational Theory and Mathematics

Citations in the corpus

The Singular Set of Lipschitzian Minima of Multiple Integrals (2007, background): “This was recently shown by Müller & Šverák in [33], where they demonstrated even the absence of partial regularity for critical, non-minimizing points of uniformly strictly quasiconvex integrals v 7→ ∫”
Convex integration and phenomenologies in turbulence (2019, methodology): “These surveys discuss in detail the Nash-Kuiper theorem [152, 118], Gromov’s h-principle [92], convex integration constructions for flexible differential inclusions inspired by the work of Müller and Šverák [151], the Scheffer [173]- Shnirelman [179] constructions, leading to the constructions of”
Regularity of minima: An invitation to the dark side of the calculus of variations (2006, background): “The example of Müller & ˇ Sverák is completely different from the ones considered in Section 3, and is based on a delicate construction resting on a the use of the so-called Tartar's " T 4 -configuration " , and of Gromov's convex integration theory [168]; for more on the issue see also their paper ”
Lack of Uniqueness for Weak Solutions of the Incompressible Porous Media Equation (2009, methodology): “This is an obstruction for the available versions of convex integration, the ones based on Baire category [5, 10, 15] and direct constructions [9, 11, 14].”
Relaxation of the incompressible porous media equation (2011, methodology): “The advantage of the method used in [1] is that it is rather robust, and can be used in situations where an explicit calculation of KΛ is out of reach due to the high complexity (see also [10, 12]).”
The Regularity of Critical Points of Polyconvex Functionals (2004, methodology): “In [ Ma03 ], Muller & Sverak constructed an example of a strongly quasiconvex F so that the corresponding 2 × 2 system (1) admits weak solutions that are Lipschitz but not C 1 in any open subset of � . Their method is based on a modification”
On Admissibility Criteria for Weak Solutions of the Euler Equations (2007, background): “In a recent paper [7], we have shown how the general framework of convex integration [5,12,17,23] combined with Tartar’s programme on oscillation phenomena in conservation laws [24] (see also [13] for an overview) can be applied to (1).”
Boundary Regularity in Variational Problems (2010, background): “By now standard references on partial regularity include [25, 23, 20, 2, 18, 31, 45, 46, 50, 51]; an up-dated presentation is in [42].”
Partial Regularity of Strong Local Minimizers in the Multi-Dimensional Calculus of Variations (2003, methodology): “2) (and (ii)) is due to Müller & Šverák [32].”
The Euler equations as a differential inclusion (2007, background): “Therefore the wave cone corresponding to (8) is given by Λ = { (v, u, q) ∈ Rn × Sn0 × R : det ( u+ qIn v v 0 ) = 0 } .”