Convex integration for Lipschitz mappings and counterexamples to regularity

Müller, Stefan; Šverák, Vladimír

doi:10.4007/annals.2003.157.715

Cited by 227 publications

(282 citation statements)

References 20 publications

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“…This question is important because CI solutions are important, many counter examples to natural conjectures in PDE have been achieved via CI [13,19,31,32]. Minimising functional I is the simplest problem that constrains oscillation in some slight way where we can hope to see the effect of the existence of exact minimisers of (1.1).…”

Section: Background and Statement Of Main Resultsmentioning

confidence: 99%

The regularisation of theN-well problem by finite elements and by singular perturbation are scaling equivalent in two dimensions

Lorent¹

2008

ESAIM: COCV

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Abstract. Let K := SO (2) A1 ∪ SO (2) A2 . . . SO (2) AN where A1, A2, . . . , AN are matrices of nonzero determinant. We establish a sharp relation between the following two minimisation problems in two dimensions. Firstly the N -well problem with surface energy. Let p ∈ [1, 2], Ω ⊂ R 2 be a convex polytopal region. Defineand let AF denote the subspace of functions in W 2,2 (Ω) that satisfy the affine boundary condition Du = F on ∂Ω (in the sense of trace), where F ∈ K. We consider the scaling (with respect to ) of

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Section: Background and Statement Of Main Resultsmentioning

confidence: 99%

The regularisation of theN-well problem by finite elements and by singular perturbation are scaling equivalent in two dimensions

Lorent¹

2008

ESAIM: COCV

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“…Further drawing on an example of [17] it was shown that for N = n = 2 there exists a Lipschitz W 1 BMO-local minimiser of I satisfying the hypotheses (H1)-(H3), but which is non-differentiable on any open set. From this it is clear that a regularising condition like (1.4) is necessary for partial regularity for Lipschitz W 1 BMO-local minimisers.…”

Section: Significance Of the Regularity Resultsmentioning

confidence: 99%

Ana prioriCampanato type regularity condition for local minimisers in the calculus of variations

Dodd

2008

ESAIM: COCV

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Abstract. An a priori Campanato type regularity condition is established for a class of W 1 X local minimisers u of the general variational integralwhere Ω ⊂ R n is an open bounded domain, F is of class C 2 , F is strongly quasi-convex and satisfies the growth conditionfor a p > 1 and where the corresponding Banach spaces X are the Morrey-Campanato spaceand the space of bounded mean oscillation BMO(Ω, R N×n ). The admissible maps u : Ω → R N are of Sobolev class W 1,p , satisfying a Dirichlet boundary condition, and to help clarify the significance of the above result the sufficiency condition for W 1 BMO local minimisers is extended from Lipschitz maps to this admissible class.

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“…(iv) Dropping the assumption that u is C 1 is problematic for the reasons pointed out in [22]. See [14,Section 7] for examples of nowhere C 1 weak local minimizers based on the construction of [17]. The assumption K(v) < ∞ would appear to limit possible oscillations of ∇u in the direction tangent to ∂Ω, say, but there is still room for bad behaviour in the directions normal to ∂Ω.…”

Section: Uniqueness Of Cmentioning

confidence: 99%

Extending the Knops-Stuart-Taheri technique to $C^{1}$ weak local minimizers in nonlinear elasticity

Bevan

2011

Proc. Amer. Math. Soc.

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Abstract. We prove that any C 1 weak local minimizer of a certain class of elastic stored-energy functionals I(u) = Ω f (∇u) dx subject to a linear boundary displacement u 0 (x) = ξx on a star-shaped domain Ω with C 1 boundary is necessarily affine provided f is strictly quasiconvex at ξ. This is done without assuming that the local minimizer satisfies the Euler-Lagrange equations, and therefore extends in a certain sense the results of Knops and Stuart, and those of Taheri, to a class of functionals whose integrands take the value +∞ in an essential way.

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Convex integration for Lipschitz mappings and counterexamples to regularity

Cited by 227 publications

References 20 publications

The regularisation of theN-well problem by finite elements and by singular perturbation are scaling equivalent in two dimensions

The regularisation of theN-well problem by finite elements and by singular perturbation are scaling equivalent in two dimensions

Ana prioriCampanato type regularity condition for local minimisers in the calculus of variations

Extending the Knops-Stuart-Taheri technique to $C^{1}$ weak local minimizers in nonlinear elasticity

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