Uniqueness Theorem of the Regularizable Redial Ginzburg-Landau Type Minimizers

The author proves the uniqueness of the regularizable radial minimizers of a Ginzburg-Landau type functional in the case n - 1 < p < n,and the location of the zeros of the regularizable radial minimizers of this functional is discussed.     

Limiting properties of the second order Ginzburg–Landau minimizers

The author studies the weak convergence for the gradient of the minimizers for a second order energy functional when the parameter tends to 0. And this paper is also concerned with the location of the zeros and the blow-up points of the gradient of the minimizers of this functional. Finally, the strong convergence of the gradient of the radial minimizers is obtained.     

一类Ginzburg-Landau型泛函的径向极小元的渐近性

The behavior of radial minimizers for a Ginzburg-Landau type functional is considered. The weak convergence of minimizers in W1,n is improved to the strong convergence in W1,n. Some estimates of the rate of the convergence for the module of minimizers are presented.     

Mixed Type Duality for Set-valued Optimization Problems via Higher-order Radial Epiderivatives

In this article, we introduce a notion of higher-order radial epiderivative for set-valued maps and study its properties. A generalized concept of higher-order strict minimizers in set-valued optimization is proposed as well. By virtue of the radial epiderivative, we establish a mixed dual problem, and then weak, strong, and converse duality theorems are obtained in dealing with generalized strict minimizers.     

Symmetry and uniqueness of minimizers of Hartree type equations with external Coulomb potential

In the present article we study the radial symmetry and uniqueness of minimizers of the energy functional, corresponding to the repulsive Hartree equation in external Coulomb potential. To overcome the difficulties, resulting from the “bad” sign of the nonlocal term, we modify the reflection method and obtain symmetry and uniqueness results.     

Newton's problem of the body of minimal resistance under a single-impact assumption

We consider the problem of the body of minimal resistance as formulated in [2], Sect. 5: minimize , where is the unit disc of , in the class of radial functions satisfying a geometrical property (1), corresponding to a single-impact assumption ( is a given parameter). We prove the existence of a critical value of M. For , there exist a unique local minimizer of the functional. For , the set of local minimizers is not compact in , though they all achieve the same value of the functional. Received February 15, 2000 / Accepted May 2, 2000 / Published online September 14, 2000     

Existence of minimizers for Newton's problem of the body of minimal resistance under a single impact assumption

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].     

On an Isoperimetric Problem with a Competing Nonlocal Term I: The Planar Case

This paper is concerned with a study of the classical isoperimetric problem modified by an addition of a nonlocal repulsive term. We characterize existence, nonexistence, and radial symmetry of the minimizers as a function of mass in the situation where the nonlocal term is generated by a kernel given by an inverse power of the distance. We prove that minimizers of this problem exist for sufficiently small masses and are given by disks with prescribed mass below a certain threshold when the interfacial term in the energy is dominant. At the same time, we prove that minimizers fail to exist for sufficiently large masses due to the tendency of the low‐energy configuration to split into smaller pieces when the nonlocal term in the energy is dominant. In the latter regime, we also establish linear scaling of energy with mass, suggesting that for large masses low‐energy configurations consist of many roughly equal‐size pieces far apart. In the case of slowly decaying kernels, we give a complete characterization of the minimizers. © 2012 Wiley Periodicals, Inc.     

Existence of planar curves minimizing length and curvature

We consider the problem of reconstructing a curve that is partially hidden or corrupted by minimizing the functional $ \smallint \sqrt {1 + K_\gamma ^2 ds} $ \smallint \sqrt {1 + K_\gamma ^2 ds} , depending both on the length and curvature K. We fix starting and ending points as well as initial and final directions. For this functional we discuss the problem of existence of minimizers on various functional spaces. We find nonexistence of minimizers in cases in which initial and final directions are considered with orientation. In this case, minimizing sequences of trajectories may converge to curves with angles. We instead prove the existence of minimizers for the “time-reparametrized” functional $ \smallint ||\dot \gamma (t)||\sqrt {1 + K_\gamma ^2 dt} $ \smallint ||\dot \gamma (t)||\sqrt {1 + K_\gamma ^2 dt} for all boundary conditions if the initial and final directions are considered regardless of orientation. In this case, minimizers may present cusps (at most two) but not angles.     

Polyconvex energies and cavitation

We study the existence of singular minimizers in the class of radial deformations for polyconvex energies that grow linearly with respect to the Jacobian.     

Régularité des solutions d'un problème variationnel sous contrainte de convexité

We study the minimizers, in the class of convex functions, of an elliptic functional with nonhomogeneous Dirichlet boundary conditions. We prove C¹ regularity of the minimizers under the assumption that the upper envelope of admissible functions is C¹. This condition is optimal at least when the functional depends only on the gradient [4]. We then extend this result to a problem without boundary conditions arising in an economic model introduced by Rochet and Choné in [5].     

Evolution by Non-Convex Functionals

We establish a semi-group solution concept for flows that are generated by generalized minimizers of non-convex energy functionals. We use relaxation and convexification to define these generalized minimizers. The main part of this work consists in exemplary validation of the solution concept for a non-convex energy functional. For rotationally invariant initial data it is compared with the solution of the mean curvature flow equation. The basic example relates the mean curvature flow equation with a sequence of iterative minimizers of a family of non-convex energy functionals. Together with the numerical evidence this corroborates the claim that the non-convex semi-group solution concept defines, in general, a solution of the mean curvature equation.     

Duality in reconstruction systems

We consider reconstruction systems (RS’s), which are G-frames in a finite dimensional setting, and that includes the fusion frames as projective RS’s. We describe the spectral picture of the set of RS operators for the projective systems with fixed weights. We also introduce a functional defined on dual pairs of RS’s, called the joint potential, and study the structure of the minimizers of this functional. In the case of irreducible RS’s the minimizers are characterize as the tight systems. In the general case we give spectral and geometric characterizations of the minimizers of the joint potential. At the end of the paper we show several examples that illustrate our results.     

An inverse problem for Hamiltonian systems

In a linear Hamiltonian system for which the Dirichlet principle is valid, solutions to boundary value problems can be identified as the unique minimizers of the quadratic functional associated with the system. The inverse problem, in which coefficient functions in the differential equations are identified as unique minimizers of a related functional, is discussed, together with conditions under which recovery can occur.     

Towards a Fully Nonlinear Sharp Sobolev Trace Inequality

We classify local minimizers of $\int\sigma_2+\oint H_2$ among all conformally flat metrics in the Euclidean $(n+1)$-ball, $n=4$ or $n=5$, for which the boundary has unit volume, subject to an ellipticity assumption. We also classify local minimizers of the analogous functional in the critical dimension $n+1=4$. If minimizers exist, this implies a fully nonlinear sharp Sobolev trace inequality. Our proof is an adaptation of the Frank-Lieb proof of the sharp Sobolev inequality, and in particular does not rely on symmetrization or Obata-type arguments.     

Masse fluide in rotazione

Problems concerning the existence of local minimizers of a functional which represents in the chance number 3 the positions of balance about a fluid surrounding solid sphere are considered. The existence of these minimizers is demonstrated when coefficient relative to centripetal force is quite big.     

Heteroclinic solutions of a van der Waals model with indefinite nonlocal interactions

We construct heteroclinic the global minimizers of a nonlocal free energy functional that van der Waals derived in 1893. We study the case where the nonlocality satisfies only a weakened type of ellipticity, which precludes the use of comparison methods. In the interesting case when the local part of the energy is nonconvex, we construct a classical the global minimizer by studying a relaxed functional corresponding to the convexification of the local part and exclude the possibility of minimizers of the relaxed functional having rapid oscillations. We also construct examples where the global minimizer is not monotonic.     

Anisotropic Chan–Vese segmentation

In this paper we study a variant to Chan–Vese (CV) segmentation model with rectilinear anisotropy. We show existence of minimizers in the 2-phases case and how they are related to the (anisotropic) Rudin–Osher–Fatemi (ROF) denoising model. Our analysis shows that in the natural case of a piecewise constant on rectangles image ($\mathrm{PCR}$ function in short), there exists a minimizer of the CV functional which is also piecewise constant on rectangles over the same grid that the one defined by the original image. In the multiphase case, we show that minimizers of the CV multiphase functional also share this property in the case that the initial image is a $\mathrm{PCR}$ function. We also investigate a multiphase and anisotropic version of the Truncated ROF algorithm, and we compare the solutions given by this algorithm with minimizers of the multiphase anisotropic CV functional.     

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     

关于双调和算子非线性基本方程正解的径向对称性

本文证明了双调和算子非线性基本方程的正解都是径向对称的，这些正解也是达到某类Ｓｏｂｏｌｅｖ最佳嵌入常数的采小元。