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1.
A Volume Constrained Variational Problem with Lower-Order Terms 总被引:1,自引:0,他引:1
We study a one-dimensional variational problem with two or more level set constraints. The existence of global and local
minimizers turns out to be dependent on the regularity of the energy density. A complete characterization of local minimizers
and the underlying energy landscape is provided. The Γ -limit when the phases exhaust the whole domain is computed. 相似文献
2.
Summary. The ABC lamellar phase of a triblock copolymer in the strong segregation region is studied on periodic and bounded intervals. In
the periodic case we find a family of local minimizers of the free energy functional all with a fine lamellar structure. Among
these local minimizers we identify the one most favored by the free energy, and hence determine the thickness of lamellar
microdomains. In the bounded interval case we show that perfect lamellar structure does not exist due to the boundary effect.
We view the strong segregation limit as a Γ -limit of the free energy by a proper choice of the material sample size. The key step is the spectral analysis of a large
matrix resulting from the second derivative of the Γ -limit. 相似文献
3.
J.M. Ball A. Taheri M. Winter 《Calculus of Variations and Partial Differential Equations》2002,14(1):1-27
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 相似文献
4.
Ian Tice 《Journal d'Analyse Mathématique》2008,106(1):129-190
In this paper, we continue the study of Lorentz space estimates for the Ginzburg-Landau energy started in [15]. We focus on
getting estimates for the Ginzburg-Landau energy with external magnetic field h
ex
in certain interesting regimes of h
ex
. This allows us to show that for configurations close to minimizers or local minimizers of the energy, the vorticity mass
of the configuration (u, A) is comparable to the L
2, ∞ Lorentz space norm of ∇
A
u. We also establish convergence of the gauge-invariant Jacobians (vorticity measures) in the dual of a function space defined
in terms of Lorentz spaces.
Supported by an NSF Graduate Research Fellowship. 相似文献
5.
A. J. Tromba 《Journal of Fixed Point Theory and Applications》2011,10(2):253-277
This is the second in a series of two papers discussing the elementary but beautiful and fundamental question (open for some
eighty years) of whether or not a minimal surface spanning a sufficiently smooth curve, which is a local minimizer, is immersed
up to and including the boundary. We show that C
k
minimizers of energy or area cannot have nonexceptional boundary branch points. 相似文献
6.
A capture-convergence rate theorem is proved for variable metric gradient projection processes near nonsingular local minimizers in convex feasible sets defined by nonlinear inequalities in a real Hilbert spaceX. The minimizers in question satisfy standard Kuhn-Tucker second-order sufficient conditions for local optimality.Investigation partially supported by National Science Foundation Research Grant No. DMS-85-03746. 相似文献
7.
We investigate existence, uniqueness and positivity of minimizers or critical points for an energy functional which contains only p-homogeneous and linear terms, 1
p-homogeneous part of the energy functional is that it be given by the p-th power of an equivalent, uniformly convex norm on the underlying Sobolev space. Finally, continuous dependence of minimizers on the energy functional is established. 相似文献
8.
Hamdi Zorgati 《Comptes Rendus Mathematique》2005,340(1):81-86
We consider a thin curved ferromagnetic film not submitted to an external magnetic field. The behavior of the film is described by an energy depending on the magnetization of the film verifying the saturation constraint. The energy is composed of an induced magnetostatic energy and an energy term with density including the exchange energy and the anisotropic energy. We study the behavior of this energy when the thickness of the curved film goes to zero. We show with Γ-convergence arguments that the minimizers of the free energy converge to the minimizers of a local energy depending on a two-dimensional magnetization. To cite this article: H. Zorgati, C. R. Acad. Sci. Paris, Ser. I 340 (2005). 相似文献
9.
Derivation of lower critical magnetic field for anisotropic Ginzburg-Landau model in superconductivity 总被引:1,自引:0,他引:1
In this paper, the authors discuss the vortex structure of an anisotropic Ginzburg-Landau model for superconducting thin film
proposed by Du. We obtain the estimate for the lower critical magnetic field $
H_{C_1 }
$
H_{C_1 }
which is the first critical value of h
ex
corresponding to the first phase transition in which vortices appear in the superconductor. We also find local minimizers
of the anisotropic superconducting thin film with a large parameter κ, and for the applied magnetic field near the critical field we discuss the asymptotic behavior of the local minimizers. 相似文献
10.
In this paper we prove the local boundedness of minimizers of integral functionals with non-standard growth conditions. 相似文献
11.
David Cruz-Uribe Patrizia Di Gironimo Luigi D’Onofrio 《Czechoslovak Mathematical Journal》2012,62(1):111-116
In this paper we establish a continuity result for local minimizers of some quasilinear functionals that satisfy degenerate
elliptic bounds. The non-negative function which measures the degree of degeneracy is assumed to be exponentially integrable.
The minimizers are shown to have a modulus of continuity controlled by log log(1/|x|)−1. Our proof adapts ideas developed for solutions of degenerate elliptic equations by J. Onninen, X. Zhong: Continuity of solutions of linear, degenerate elliptic equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 6 (2007), 103–116. 相似文献
12.
We set up a formula for the Fréchet and ε-Fréchet subdifferentials of the difference of two convex functions. We even extend it to the difference of two approximately
starshaped functions. As a consequence of this formula, we give necessary and sufficient conditions for local optimality in
nonconvex optimization. Our analysis relies on the notion of gap continuity of multivalued maps and involves concepts of independent
interest such as the notions of blunt and sharp minimizers and the notion of equi-subdifferentiability.
相似文献
13.
Gonzalo Contreras Jorge Delgado Renato Iturriaga 《Bulletin of the Brazilian Mathematical Society》1997,28(2):155-196
Define the critical levelc(L) of a convex superlinear LagragianL as the infimum of thek such that the LagragianL+k has minimizers with fixed endpoints and free time interval. We provide proofs for Mañé's statements [7] characterizingc(L) in termos of minimizing measures ofL, and also giving graph, recurrence covering and cohomology properties for minimizers ofL+c(L). It is also proven thatc(L) is the infimum of the energy levelsk such that the following for of Tonelli's theorem holds:There exists minimizers of the L+k-action joining any two points in the projection of E=k among curves with energy k.To the memory of Ricardo Mañé.partially supported by CNPq-Brazil.partially supported by Conacyt-Mexico, grant 3398E9307. 相似文献
14.
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. 相似文献
15.
We study the stable configurations of a thin three-dimensional weakly prestrained rod subject to a terminal load as the thickness of the section vanishes. By \(\Gamma \)-convergence we derive a one-dimensional limit theory and show that isolated local minimizers of the limit model can be approached by local minimizers of the three-dimensional model. In the case of isotropic materials and for two-layers prestrained three-dimensional models the limit energy further simplifies to that of a Kirchhoff rod-model of an intrinsically curved beam. In this case we study the limit theory and investigate global and/or local stability of straight and helical configurations. 相似文献
16.
We prove that the infimum of Newton's functional of minimal resistanceF(u):=∫Ω
dx/(1+|▽u(x)|2), where Ω ⊂R
2 is a strictly convex domain, is not attained in a wide class of functions satisfying a single-impact assumption, proposed
in [1]. On the other hand, we prove that the infimum is attained in the subclass of radial functions; hence the minimizers
are the local minimizers already described in [3]. 相似文献
17.
In this work, we study a nonsmooth optimization problem with generalized inequality constraints and an arbitrary set constraint.
We present necessary conditions for a point to be a strict local minimizer of order k in terms of higher-order (upper and lower) Studniarski derivatives and the contingent cone to the constraint set. In the
same line, when the initial space is finite dimensional, we develop sufficient optimality conditions. We also provide sufficient
conditions for minimizers of order k using the lower Studniarski derivative of the Lagrangian function. Particular interest is put for minimizers of order two,
using now a special second order derivative which leads to the Fréchet derivative in the differentiable case. 相似文献
18.
Discrete global descent method for discrete global optimization and nonlinear integer programming 总被引:2,自引:0,他引:2
A novel method, entitled the discrete global descent method, is developed in this paper to solve discrete global optimization
problems and nonlinear integer programming problems. This method moves from one discrete minimizer of the objective function
f to another better one at each iteration with the help of an auxiliary function, entitled the discrete global descent function.
The discrete global descent function guarantees that its discrete minimizers coincide with the better discrete minimizers
of f under some standard assumptions. This property also ensures that a better discrete minimizer of f can be found by some classical local search methods. Numerical experiments on several test problems with up to 100 integer
variables and up to 1.38 × 10104 feasible points have demonstrated the applicability and efficiency of the proposed method. 相似文献
19.
We study the regularity of vector-valued local minimizers in $ W^{1,p}, p > 1 $, of the integral functional
where is an open set in $ \mathbb{R}^N $ and f is a continuous function,
convex with respect to the last variable, such that $ 0 \leq f(x,u,t)\leq C(1+t^p) $.We prove that if f = f(x, t),
or f = f(x, u, t) and
$ p \leq N $, then local minimizers are locally Hölder continuous for any exponent less than 1.
If f = f(x, u, t) and
p < N then
local minimizers are Höolder continuous for every exponent less than 1 in an open set
$ \Omega_0 $ such that the Hausdorff dimension of $ \Omega \backslash \Omega_0 $ is less than
N–p.AMS Subject Classification: 49N60. 相似文献
20.
We study the behaviour of thin elastic bodies of fixed cross-section and of height h, with h → 0. We show that critical points of the energy functional of nonlinear three-dimensional elasticity converge to critical points of the von Kármán functional, provided that their energy per unit height is bounded by Ch 4 (and that the stored energy density function satisfies a technical growth condition). This extends recent convergence results for absolute minimizers. 相似文献