Characterization of energy minimizers in micromagnetics

We present a new characterization of minimizing sequences and possible minimizers (all called the minimizing magnetizations) for a nonlocal micromagnetic-like energy (without the exchange energy). Our method is to replace the nonlocal energy functional and its relaxation with certain local integral functionals on divergence-free fields obtained by a two-step minimization of some auxiliary augmented functionals. Through this procedure, the minimization problem becomes equivalent to the minimization of a new local variational functional, called the dual variational functional, which has a unique minimizer. We then precisely characterize the minimizing magnetizations of original nonlocal functionals in terms of the unique minimizer of the dual variational functional. Finally, we give some remarks and ideas on solving the dual minimization problem.     

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).     

Local minimizers versus X-local minimizers

An important property known, among other cases, for W ^1,p (Ω) versus ${C^1(\overline{\Omega})}$ -local minimizers of certain functions is extended to the general situation of local minimizers of a functional I on a Banach space Y versus X-local minimizers of I provided X is a Banach space continuously and densely embedded in Y.     

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¦.     

Existence of minimizers for spectral problems

In this paper we show that any increasing functional of the first k   eigenvalues of the Dirichlet Laplacian admits a (quasi-)open minimizer among the subsets of R^N${\mathbb{R}}^N$ of unit measure. In particular, there exists such a minimizer which is bounded, where the bound depends on k and N, but not on the functional.     

Existence and uniqueness of minimizers for L2-constrained problems related to fractional Kirchhoff equation

In this paper, we discuss the existence and uniqueness of solutions of the constrained variational problem with respect to the fractional Kirchhoff equation. For the exponent p<p_*(s,N), a complete classification with respect to p for the existence of solutions of the fractional Kirchhoff functional on the L²-normalized manifold was given. Furthermore, all these solutions are unique up to translations, and our methods depend only on some simple energy estimates.     

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).     

Gradient bounds for minimizers of free discontinuity problems related to cohesive zone models in fracture mechanics

In this note we consider a free discontinuity problem for a scalar function, whose energy depends also on the size of the jump. We prove that the gradient of every smooth local minimizer never exceeds a constant, determined only by the data of the problem.     

Symmetry of minimizers of some fractional problems

We establish the existence and symmetry of all minimizers of a constrained variational problem involving the fractional gradient. This problem is closely connected to some fractional kinetic equations.     

Stability of minimizers of set optimization problems

We study the asymptotic behavior of sequences of minimization problems in set optimization. More precisely, considering a sequence of set optimization problems \((P_n)\) converging in some sense to a set optimization problem (P) we investigate the upper and lower convergences of the sets of minimizers of the problems \((P_n)\) to the set of minimizers of the problem (P).     

Limiting domain wall energy for a problem related to micromagnetics

We study the asymptotic limit of a family of functionals related to the theory of micromagnetics in two dimensions. We prove a compactness result for families of uniformly bounded energy. After studying the corresponding one‐dimensional profiles, we exhibit the Γ‐limit (“wall energy”), which is a variational problem on the folding of solutions of the eikonal equation |∇g| = 1. We prove that the minimal wall energy is twice the perimeter. © 2001 John Wiley & Sons, Inc.     

Symmetry of minimizers for some nonlocal variational problems

We present a new approach to study the symmetry of minimizers for a large class of nonlocal variational problems. This approach which generalizes the Reflection method is based on the existence of some integral identities. We study the identities that lead to symmetry results, the functionals that can be considered and the function spaces that can be used. Then we use our method to prove the symmetry of minimizers for a class of variational problems involving the fractional powers of Laplacian, for the generalized Choquard functional and for the standing waves of the Davey-Stewartson equation.     

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 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.     

Structure of minimizers in linear-quadratic Bolza problems of optimal control

In this paper, we study intersections of extremals in a linear-quadratic Bolza problem of optimal control. The structure of the inter-sections is described. We show that this structure implies the semipositive definiteness of the quadratic cost functional. In addition, we derive necessary and sufficient conditions for the existence of minimizers.     

Regularity for minimizers of some variational problems in plasticity theory

A variational problem for a functional depending on the symmetric part of the gradient of the unknown vectorvalued function is considered. We assume that the integrand of the problem has power growth with exponent less than two. We prove the existence of summable second derivatives near a flat piece of the boundary. In the two-dimensional case, Hölder continuity up to the boundary of the strain and stress tensors is established. Bibliography: 6 titles.     

Boundary regularity for almost minimizers of quasiconvex variational problems

We consider boundary regularity for almost minimizers of quasiconvex variational integrals with polynomial growth of order p ≥ 2, and obtain a general criterion for an almost minimizer to be regular in the neighbourhood of a given boundary point. Combined with existing results on interior partial regularity, the proof yields directly the optimal regularity for an almost minimizer in this neighbourhood.