Partial regularity of strong local minimizers of quasiconvex integrals with (<i>p, q</i>)-growth

Schemm, Sabine; Schmidt, Thomas

doi:10.1017/s0308210507001278

Cited by 6 publications

(6 citation statements)

References 55 publications

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“…The regularity of strong W 1,p -minimisers was first considered by Kristensen & Taheri [44], who established a partial regularity theorem in the case of superquadratic (p ≥ 2) growth. These results have been extended for instance in [13,60,10,11].…”

Section: Extension To Strong Local Minimisersmentioning

confidence: 82%

Partial regularity for minima of higher-order quasiconvex integrands with natural Orlicz growth

Irving¹

2021

Preprint

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A partial regularity theorem is presented for minimisers of k thorder functionals subject to a quasiconvexity and general growth condition. We will assume a natural growth condition governed by an N -function satisfying the ∆ 2 and ∇ 2 conditions, assuming no quantitative estimates on the second derivative of the integrand; this is new even in the k = 1 case. These results will also be extended to the case of strong local minimisers.

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Section: Extension To Strong Local Minimisersmentioning

confidence: 82%

Partial regularity for minima of higher-order quasiconvex integrands with natural Orlicz growth

Irving¹

2021

Preprint

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“…In [35], the obstacle of not being able to iterate the Caccioppoli inequality carrying the term in (1.2) is overcome for W 1,q -local minimizers by means of a measure-theoretical argument, thanks to which strong convergence can be established bypassing the need of the usual estimates that give higher integrability. Other interesting regularity results concerning local minimizers can be found in [5,6,9,12,15,39].…”

Section: Introductionmentioning

confidence: 90%

Partial regularity for local minimizers of variational integrals with lower order terms

Cordero

2021

Preprint

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We consider functionals of the formwhere Ω ⊆ R n is open and bounded. The integrand F : Ω × R N × R N×n → R is assumed to satisfy the classical assumptions of a polynomial p-growth and strong quasiconvexity. In addition, F is Hölder continuous in its first two variables uniformly with respect to the third variable, and bounded below by a quasiconvex function depending only on z ∈ R N×n . We establish that, for every α ∈ (0, 1), strong local minimizers of F are of class C 1,α in a subset Ω0 ⊆ Ω with Ω0 of full n-dimensional measure. This extends the partial regularity result for strong local minimizers by Kristensen and Taheri (2003) to the case where the integrand depends also on u. Furthermore, we provide a direct strategy for this result, in contrast to the blow-up argument used for the case of homogeneous integrands.

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“…It is negative between these roots and positive otherwise. In particular, if ż ≥ 0 then , from (29), it follows that…”

Section: T(r)mentioning

confidence: 99%

“…Elastic situations subject to double (or M −) covering boundary data have been intensely discussed in [3,4,9,11,12]. Additional everywhere regularity results can be found in [6,15,16,[23][24][25] and for a list of partial regularity results regarding poly-or quasiconvex integrands see [1,13,14,18,19,21,23,26,27,29].…”

Section: Introductionmentioning

confidence: 99%

On the assembly of $C^1-$stationary points of a polyconvex functional and finite BOP-theory

Dengler¹

2022

Preprint

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In this work the following energy is consideredwhere B ⊂ R 2 denotes the unit ball, u ∈ W 1,2 (B, R 2 ) and ρ : R → R + 0 smooth and convex with ρ(s) = 0 for all s ≤ 0 and ρ becomes affine when s exceeds some value s0 > 0. Additionally, we may impose (M ∈ N \ {0})−covering maps as boundary conditions in a suitable fashion. For such situations we then construct radially symmetric M −covering stationary points of the energy, which are at least C 1 (in some circumstances even C ∞ ), and verify more refined properties, which these stationary points need to satisfy. We do so by following the strategy first and foremost developed by P. Bauman, N. C. Owen, and D. Phillips (BOP) confirming and generalising that the method remains valid beyond the M = 2−case for an arbitrary M. Furthermore, as far as we know, this is the first treatise of BOP-theory in finite elasticity. The finiteness, not imposing such strict conditions, allows for a richer class of possible behaviours of the stationary points, making it more difficult to completely determine them.

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Partial regularity of strong local minimizers of quasiconvex integrals with (p, q)-growth

Cited by 6 publications

References 55 publications

Partial regularity for minima of higher-order quasiconvex integrands with natural Orlicz growth

Partial regularity for minima of higher-order quasiconvex integrands with natural Orlicz growth

Partial regularity for local minimizers of variational integrals with lower order terms

On the assembly of $C^1-$stationary points of a polyconvex functional and finite BOP-theory

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