Lower semicontinuity of surface energies

Fonseca, Irene

doi:10.1017/s0308210500015018

Cited by 38 publications

(37 citation statements)

References 28 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We can now use Fonseca's Varifold Theorem [16,Theorem 3.6] that says that there is a family of probability measures {λ x } x∈Ω on the unit sphere S in M and a non-negative measure π on Ω such that…”

Section: Representation Formulasmentioning

confidence: 99%

Sufficient conditions for strong local minima: The case of $C^{1}$ extremals

Grabovsky

Mengesha

2008

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

In this paper we settle a conjecture of Ball that uniform quasiconvexity and uniform positivity of the second variation are sufficient for a C 1 extremal to be a strong local minimizer. Our result holds for a class of variational functionals with a power law behavior at infinity. The proof is based on the decomposition of an arbitrary variation of the dependent variable into its purely strong and weak parts. We show that these two parts act on the functional independently. The action of the weak part can be described in terms of the second variation, whose uniform positivity prevents the weak part from decreasing the functional. The strong part "localizes", i.e. its action can be represented as a superposition of "Weierstrass needles", which cannot decrease the functional either, due to the uniform quasiconvexity conditions.

show abstract

Section: Representation Formulasmentioning

confidence: 99%

Sufficient conditions for strong local minima: The case of $C^{1}$ extremals

Grabovsky

Mengesha

2008

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…There are several tools how to deal with concentrations. They can be considered as generalization of Young measures, see for example DiPerna's and Majda's treatment of concentrations [9], Alibert's and Bouchitté's approach [2] or Fonseca's method described in [13]. An overview can be found in [36,40].…”

Section: Introductionmentioning

confidence: 99%

“…To our knowledge, the first attempt to characterize both oscillations and concentrations in sequences of gradients is due to Fonseca, Müller, and Pedregal [14]. They describe concentrations by means of a varifold while oscillations by gradient Young measures, following the works [3,4,13,35]. The authors give necessary and sufficient conditions on the varifold, so that they can fully describe effects of concentrations and oscillations on Keywords and phrases.…”

Section: Introductionmentioning

confidence: 99%

Oscillations and concentrations in sequences of gradients

Kałamajska

Kružík

2007

ESAIM: COCV

View full text Add to dashboard Cite

Abstract.We use DiPerna's and Majda's generalization of Young measures to describe oscillations and concentrations in sequences of gradients, {∇u k }, bounded in L p (Ω; R m×n ) if p > 1 and Ω ⊂ R n is a bounded domain with the extension property in W 1,p . Our main result is a characterization of those DiPerna-Majda measures which are generated by gradients of Sobolev maps satisfying the same fixed Dirichlet boundary condition. Cases where no boundary conditions nor regularity of Ω are required and links with lower semicontinuity results by Meyers and by Acerbi and Fusco are also discussed.Mathematics Subject Classification. 49J45, 35B05.

show abstract

“…If The other fundamental tool used in the proof of Theorem 2.10 is the notion of indicator measures (see FONSECA [7], RESHETNYAK [12]). They allow one to establish continuity and lower semicontinuity properties for energies of the type (1.1).…”

Section: Brunn-minkowski Theorem 212mentioning

confidence: 99%

“…This proof relies on the BrunnMinkowski Theorem and on the parametrized indicator measures (see FONSECA [7], RESHETNYAK [12]). These probability measures are very helpful to handle oscillating weakly converging sequences of surfaces and continuity and lower semicontinuity of functional of the type (1.1).…”

Section: Introductionmentioning

confidence: 99%

A uniqueness proof for the Wulff Theorem

Fonseca

Müller

1991

Proceedings of the Royal Society of Edinburgh: Section A Mathem

Self Cite

175

127

View full text Add to dashboard Cite

SynopsisThe Wulff problem is a generalisation of the isoperimetric problem and is relevant for the equilibrium of (small) elastic crystals. It consists in minimising the (generally anisotropic) surface energy among sets of given volume. A solution of this problem is given by a geometric construction due to Wulff. In the class of sets of finite perimeter this was first shown by J. E. Taylor who, using methods of geometric measure theory, also proved uniqueness. Here a more analytic uniqueness proof is presented. The main ingredient is a sharpened version of the Brunn–Minkowski inequality.

show abstract

Lower semicontinuity of surface energies

Abstract: SynopsisUsing the theory of indicator measures, a lower semicontinuity result for quasiconvex functions in W1,1 and assuming only L1 convergence is obtained.

Cited by 38 publications

References 28 publications

Sufficient conditions for strong local minima: The case of $C^{1}$ extremals

Sufficient conditions for strong local minima: The case of $C^{1}$ extremals

Oscillations and concentrations in sequences of gradients

A uniqueness proof for the Wulff Theorem

Contact Info

Product

Resources

About