Local minimizers and rearrangements

We are concerned witha priori estimates for functionsu which locally minimize, in the topology ofL , functionals of the Calculus of Variations. Sharp pointwise upper bounds for the spherically symmetric rearrangement ofu are proved. Such result enables us to get conditions for the boundedness ofu and estimates for ess sup¦u¦.     

Local minimizers in micromagnetics and related problems

Let be a smooth bounded domain and consider the energy functional Here is a small parameter and the admissible function m lies in the Sobolev space of vector-valued functions and satisfies the pointwise constraint for a.e. . The induced magnetic field is related to m via Maxwell's equations and the function is assumed to be a sufficiently smooth, non-negative energy density with a multi-well structure. Finally is a constant vector. The energy functional arises from the continuum model for ferromagnetic materials known as micromagnetics developed by W.F. Brown [9]. In this paper we aim to construct local energy minimizers for this functional. Our approach is based on studying the corresponding Euler-Lagrange equation and proving a local existence result for this equation around a fixed constant solution. Our main device for doing so is a suitable version of the implicit function theorem. We then show that these solutions are local minimizers of in appropriate topologies by use of certain sufficiency theorems for local minimizers. Our analysis is applicable to a much broader class of functionals than the ones introduced above and on the way to proving our main results we reflect on some related problems. Received: 20 November 2000 / Accepted: 4 December 2000 / Published online: 4 May 2001     

Local boundedness of minimizers in a limit case

We prove the local boundedness of minimizers of a functional with anisotropic polynomial growth. The result here obtained is optimal if compared with previously know counterexamples. This work has been performed as a part of a National Research Project, supported by MPI (40%, 1987).     

Local minimizers of the Ginzburg-Landau functional with prescribed degrees

We consider, in a smooth bounded multiply connected domain D⊂R², the Ginzburg-Landau energy subject to prescribed degree conditions on each component of ∂D. In general, minimal energy maps do not exist [L. Berlyand, P. Mironescu, Ginzburg-Landau minimizers in perforated domains with prescribed degrees, preprint, 2004]. When D has a single hole, Berlyand and Rybalko [L. Berlyand, V. Rybalko, Solution with vortices of a semi-stiff boundary value problem for the Ginzburg-Landau equation, J. Eur. Math. Soc. (JEMS), in press, 2008, http://www.math.psu.edu/berlyand/publications/publications.html] proved that for small ε local minimizers do exist. We extend the result in [L. Berlyand, V. Rybalko, Solution with vortices of a semi-stiff boundary value problem for the Ginzburg-Landau equation, J. Eur. Math. Soc. (JEMS), in press, 2008, http://www.math.psu.edu/berlyand/publications/publications.html]: Eε(u) has, in domains D with 2,3,… holes and for small ε, local minimizers. Our approach is very similar to the one in [L. Berlyand, V. Rybalko, Solution with vortices of a semi-stiff boundary value problem for the Ginzburg-Landau equation, J. Eur. Math. Soc. (JEMS), in press, 2008, http://www.math.psu.edu/berlyand/publications/publications.html]; the main difference stems in the construction of test functions with energy control.     

Local minimizers of one-dimensional variational problems and obstacle problems

In this Note we suggest a direct approach to study local minimizers of one-dimensional variational problems. To cite this article: M.A. Sychev, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Strong versus weak local minimizers for the perturbed Dirichlet functional

Let be a bounded domain and . In this paper we consider functionals of the form where the admissible function belongs to the Sobolev space of vector-valued functions and is such that the integral on the right is well defined. We state and prove a sufficiency theorem for local minimizers of where . The exponent is shown to depend on the dimension and the growth condition of and an exact expression is presented for this dependence. We discuss some examples and applications of this theorem. Received: 20 July 2000 / Accepted: 7 June 2001 / Published online: 19 October 2001     

Local minimizers with vortex pinning to Ginzburg-Landau functional in three dimensions

We consider the pinning effect for full Ginzburg-Landau functional. The existence of local minimizers with vortices locating in the pinning regions is obtained.     

Local boundedness of minimizers of integrals of the calculus of variations

In this paper we prove the local boundedness of minimizers of integral functionals with non-standard growth conditions.     

Local minimizers and planar interfaces in a phase-transition model with interfacial energy

Interfacial energy is often incorporated into variational solid-solid phase transition models via a perturbation of the elastic energy functional involving second gradients of the deformation. We study consequences of such higher-gradient terms for local minimizers and for interfaces. First it is shown that at slightly sub-critical temperatures, a phase which globally minimizes the elastic energy density at super-critical temperatures is an L ¹-local minimizer of the functional including interfacial energy, whereas it is typically only a W ^1,??-local minimizer of the purely elastic functional. The second part deals with the existence and uniqueness of smooth interfaces between different wells of the multi-well elastic energy density. Attention is focussed on so-called planar interfaces, for which the deformation depends on a single direction x · N and the deformation gradient then satisfies a rank-one ansatz of the form ${Dy(x) = A + u(x \cdot N) \otimes N}$ , where A and ${B=A+a \otimes N}$ are the gradients connected by the interface.     

Local and global minimizers for a variational energy involving a fractional norm

We study existence, uniqueness, and other geometric properties of the minimizers of the energy functional $$ \|u\|^2_{H^s(\Omega)}+\int\limits_\Omega W(u)\,{d}x, $$ where ${\|u\|_{H^s(\Omega)}}$ denotes the total contribution from Ω in the H ^s norm of u and W is a double-well potential. We also deal with the solutions of the related fractional elliptic Allen-Cahn equation on the entire space ${\mathbb{R}^n}$ . The results collected here will also be useful for forthcoming papers, where the second and the third author will study the Γ-convergence and the density estimates for level sets of minimizers.     

Relative minimizers of energy are relative minimizers of area

Let Γ be a closed, regular Jordan curve in ${{\mathbb R}^3}$ which is of class C ^1,μ , 0 <  μ <  1, and denote by ${{\mathcal C}(\Gamma)}$ the class of the disk-type surfaces ${X : B \to {\mathbb R}^3}$ with continuous, monotonic boundary values, mapping ${\partial B}$ onto Γ. One easily sees that any minimal surface ${X \in {\mathcal C}(\Gamma)}$ is a relative minimizer of energy, i.e. of Dirichlet’s integral D, if it is a relative minimizer of the area functional A. Here we prove conversely: If an immersed ${X \in {\mathcal C}(\Gamma)}$ is a C ¹-relative minimizer of D in ${{\mathcal C}(\Gamma)}$ , then it also is a C ^1,μ -relative minimizer of A in ${{\mathcal C}(\Gamma)}$ .     

Local minimizers with vortices to the Ginzburg‐Landau system in three dimensions

We construct local minimizers to the Ginzburg‐Landau energy in certain three‐dimensional domains based on the asymptotic connection between the energy and the total length of vortices using the theory of weak Jacobians. Whenever there exists a collection of locally minimal line segments spanning the domain, we can find local minimizers with arbitrarily assigned degrees with respect to each segment. © 2003 Wiley Periodicals, Inc.     

The Ekeland variational principle for Henig proper minimizers and super minimizers

In this paper we consider, for the first time, approximate Henig proper minimizers and approximate super minimizers of a set-valued map F with values in a partially ordered vector space and formulate two versions of the Ekeland variational principle for these points involving coderivatives in the sense of Ioffe, Clarke and Mordukhovich. As applications we obtain sufficient conditions for F to have a Henig proper minimizer or a super minimizer under the Palais-Smale type conditions. The techniques are essentially based on the characterizations of Henig proper efficient points and super efficient points by mean of the Henig dilating cones and the Hiriart-Urruty signed distance function.     

W versus C local minimizers and multiplicity results for quasilinear elliptic equations

In this paper we show that the local minimizers of a class of functionals in the C¹-topology are still their local minimizers in . Using this fact, we study the multiplicity of solutions for a class of quasilinear elliptic equations via critical point theory.     

Planelike minimizers in periodic media

We show that given an elliptic integrand ?? in ?^d that is periodic under integer translations, and given any plane in ?^d, there is at least one minimizer of ?? that remains at a bounded distance from this plane. This distance can be bounded uniformly on the planes. We also show that, when folded back to ?^d/?^d, the minimizers we construct give rise to a lamination. One particular case of these results is minimal surfaces for metrics invariant under integer translations. The same results hold for other functionals that involve volume terms (small and average zero). In such a case the minimizers satisfy the prescribed mean curvature equation. A further generalization allows the formulation and proof of similar results in manifolds other than the torus provided that their fundamental group and universal cover satisfy some hypotheses. © 2001 John Wiley & Sons, Inc.     

All periodic minimizers are unstable

We consider the periodic action functional associated to some Lagrangian verifying the Legendre convexity condition and show that all minimizers are unstable. Supported by D.G.I. MTM2005-03483, Ministerio de Educación y Ciencia, Spain.     

Mumford-Shah minimizers on thin plates

We show that limits of Mumford-Shah minimizers in product domains Ω = Ω′× (0,t), t small, are Mumford-Shah minimizers in one less dimension. The main ingredient of the proof is a symmetry argument from Dal Maso, Morel, and Solimini. AMS classification 49K99, 49Q20