Front propagation in infinite cylinders. II. The sharp reaction zone limit

This paper applies the variational approach developed in part I of this work [22] to a singular limit of reaction–diffusion–advection equations which arise in combustion modeling. We first establish existence, uniqueness, monotonicity, asymptotic decay, and the associated free boundary problem for special traveling wave solutions which are minimizers of the considered variational problem in the singular limit. We then show that the speed of the minimizers of the approximating problems converges to the speed of the minimizer of the singular limit. Also, after an appropriate translation the minimizers of the approximating problems converge strongly on compacts to the minimizer of the singular limit. In addition, we obtain matching upper and lower bounds for the speed of the minimizers in the singular limit in terms of a certain area-type functional for small curvatures of the free boundary. The conclusions of the analysis are illustrated by a number of numerical examples.     

Existence and stability of solitary-wave solutions of equations of Benjamin-Bona-Mahony type

We consider solitary-wave solutions of equations of Benjamin-Bona-Mahony type. We show that for a large class of equations of BBM type, there do exist stable sets consisting of solitary-wave profile functions. In the case of generalized BBM equations, we found that there are profile functions of stable solitary waves that are not the minimizers of the associated variational problem. Such a phenomenon is not known to exist for equations of Korteweg-de Vries type.     

On a weighted total variation minimization problem

The present article is devoted to the study of a constrained weighted total variation minimization problem, which may be viewed as a relaxation of a generalized Cheeger problem and is motivated by landslide modeling. Using the fact that the set of minimizers is invariant by a wide class of monotone transformations, we prove that level sets of minimizers are generalized Cheeger sets and obtain qualitative properties of the minimizers: they are all bounded and all achieve their essential supremum on a set of positive measure.     

Existence and regularity of minimizers of a functional for unsupervised multiphase segmentation

We consider a variational model for image segmentation proposed in Sandberg et al. (2010) [12]. In such a model the image domain is partitioned into a finite collection of subsets denoted as phases. The segmentation is unsupervised, i.e., the model finds automatically an optimal number of phases, which are not required to be connected subsets. Unsupervised segmentation is obtained by minimizing a functional of the Mumford–Shah type (Mumford and Shah, 1989 [1]), but modifying the geometric part of the Mumford–Shah energy with the introduction of a suitable scale term. The results of computer experiments discussed in [12] show that the resulting variational model has several properties which are relevant for applications. In this paper we investigate the theoretical properties of the model. We study the existence of minimizers of the corresponding functional, first looking for a weak solution in a class of phases constituted by sets of finite perimeter. Then we find various regularity properties of such minimizers, particularly we study the structure of triple junctions by determining their optimal angles.     

Existence of minimizers of the Mumford-Shah functional with singular operators and unbounded data

We consider the regularization of linear inverse problems by means of the minimization of a functional formed by a term of discrepancy to data and a Mumford-Shah functional term. The discrepancy term penalizes the L ² distance between a datum and a version of the unknown function which is filtered by means of a non-invertible linear operator. Depending on the type of the involved operator, the resulting variational problem has had several applications: image deblurring, or inverse source problems in the case of compact operators, and image inpainting in the case of suitable local operators, as well as the modeling of propagation of fracture. We present counterexamples showing that, despite this regularization, the problem is actually in general ill-posed. We provide, however, existence results of minimizers in a reasonable class of smooth functions out of piecewise Lipschitz discontinuity sets in two dimensions. The compactness arguments we developed to derive the existence results stem from geometrical and regularity properties of domains, interpolation inequalities, and classical compactness arguments in Sobolev spaces.     

Variational problems with topological constraints

We introduce a decomposition that captures much of the topology of level sets for functions in certain Sobolev spaces, and allows the definition of an analog of the decreasing rearrangement of a function which respects the topology of level sets. In a variety of settings this decomposition is preserved under weak limits, and so is useful in establishing existence of minimizers of various variational problems with prescribed topological properties. These include variational problems in which ‘topological rearrangements’ are fixed, and ones in which the functional depends on derivatives of rearrangements. Received: December 16, 1998 / Accepted: July 16, 1999     

The regularity of minimizers of a radially symmetric free boundary problem

We treat a variational problem for a functional with a characteristic function term which causes the free boundary, and investigate the regularity of minimizers in the radially symmetric case. The regularity results depend upon the quantity of the coefficient of the term.     

Action minimizers under topological constraints in the planar equal-mass four-body problem

It is shown that in the planar equal-mass four-body problem, there exist two sets of action minimizers connecting two planar boundary configurations with fixed symmetry axes and specific order constraints: a double isosceles configuration and an isosceles trapezoid configuration, while order constraints are introduced on the boundary configurations. By applying the level estimate method, these minimizers are shown to be collision-free and they can be extended to two new sets of periodic or quasi-periodic orbits.     

A gradient bound for the Grad-Kruskal-Kulsrud functional

We establish the existence of minimizers with a bounded gradient for a variational problem arising in the Grad-Kruskal-Kulsrud model for the equilibrium of a confined plasma. The variational problem involves derivatives of the nondecreasing rearrangement of minimizers. Results are derived for bounded convex domains in R ². Our results answer in the negative (in the case of convex domains) a question raised by Grad concerning the possibility of singular behavior of the magnetic field at the point of maximum flux. The main approach is to use an approximating free boundary problem to handle the nonlinear nonlocal nature of the variational functional. Limiting minimizers of the variational problem are shown to have bounded gradient and to satisfy a weak equation that for one model problem takes the form where ψ*, μ_ψ are, respectively, the nondecreasing rearrangement and the distribution function of ψ. © 1996 John Wiley & Sons, Inc.     

Multiplicity results for interfaces of Ginzburg-Landau-Allen-Cahn equations in periodic media

The Ginzburg-Landau-Allen-Cahn equation is a variational model for phase coexistence and for other physical problems. It contains a term given by a kinetic part of elliptic type plus a double-well potential. We assume that the functional depends on the space variables in a periodic way.We show that given a plane with rational normal, there are minimal solutions, satisfying the following properties. These solutions are asymptotic to the pure phases and are separated by an interface. The convergence to the pure phases is exponentially fast. The interface lies at a finite distance M from the chosen plane, where M is a universal constant. Furthermore, these solutions satisfy some monotonicity properties with respect to integer translations (namely, integer translations are always comparable to the function).We then show that all the interfaces of the global periodic minimizers satisfy similar monotonicity and plane-like properties.We also consider the case of possibly irrationally oriented planes. We show that either there is a one parameter family of minimizers whose graphs provide a field of extremals or there are at least two solutions, one which is a minimizer and another one which is not. These solutions also have interfaces bounded by a universal constant, they enjoy monotonicity properties with respect to integer translations and the nonminimal solutions are trapped inside a gap of the lamination induced by the minimizers.     

Neumann p-Laplacian problems with a reaction term on metric spaces

We use a variational approach to study existence and regularity of solutions for a Neumann p-Laplacian problem with a reaction term on metric spaces equipped with a doubling measure and supporting a Poincaré inequality. Trace theorems for functions with bounded variation are applied in the definition of the variational functional and minimizers are shown to satisfy De Giorgi type conditions.

     

Global minimizers for the doubly-constrained Helfrich energy: the axisymmetric case

Since the pioneering work of Canham and Helfrich, variational formulations involving curvature-dependent functionals, like the classical Willmore functional, have proven useful for shape analysis of biomembranes. We address minimizers of the Canham–Helfrich functional defined over closed surfaces enclosing a fixed volume and having fixed surface area. By restricting attention to axisymmetric surfaces, we prove the existence of global minimizers.     

Variational approach to contact problems in nonlinear elasticity

We use variational methods to study problems in nonlinear 3-dimensional elasticity where the deformation of the elastic body is restricted by a rigid obstacle. For an assigned variational problem we first verify the existence of constrained minimizers whereby we extend previous results. Then we rigorously derive the Euler-Lagrange equation as necessary condition for minimizers, which was possible before only under strong smoothness assumptions on the solution. The Lagrange multiplier corresponding to the obstacle constraint provides structural information about the nature of frictionless contact. In the case of contact with, e.g., a corner of the obstacle, we derive a qualitatively new contact condition taking into account the deformed shape of the elastic body. By our analysis it is shown here for the first time rigorously that energy minimizers really solve the mechanical contact problem. Received: 20 October 2000 / Accepted: 7 June 2001 / Published online: 5 September 2002     

Variational problems with several volume constraints on the level sets

We consider functionals of the kind on , and we study the problem , where consists of those functions u whose level sets satisfy certain volume constraints , where denotes Lebesgue measure, and , are given numbers. Examples show that this problem may have no solution, even for simple smooth F. As a consequence, we relax the constraint to , i.e. , and we show that the minimizers over exist and are H?lder continuous. Then we prove several existence theorems for the original problem, showing that, under suitable assumptions on the integrand function F, every minimizer over actually belongs to . Received: 31 January 2001 / Accepted: 23 February 2001 / Published online: 23 July 2001     

The Structure of Minimizers of the Frame Potential on Fusion Frames

In this paper we study the fusion frame potential that is a generalization of the Benedetto-Fickus (vectorial) frame potential to the finite-dimensional fusion frame setting. We study the structure of local and global minimizers of this potential, when restricted to suitable sets of fusion frames. These minimizers are related to tight fusion frames as in the classical vector frame case. Still, tight fusion frames are not as frequent as tight frames; indeed we show that there are choices of parameters involved in fusion frames for which no tight fusion frame can exist. We exhibit necessary and sufficient conditions for the existence of tight fusion frames with prescribed parameters, involving the so-called Horn-Klyachko’s compatibility inequalities. The second part of the work is devoted to the study of the minimization of the fusion frame potential on a fixed sequence of subspaces, with a varying sequence of weights. We related this problem to the index of the Hadamard product by positive matrices and use it to give different characterizations of these minima.     

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.     

Stability of the Minimizers of Least Squares with a Non-Convex Regularization. Part I: Local Behavior

Many estimation problems amount to minimizing a piecewise C^m objective function, with m ≥ 2, composed of a quadratic data-fidelity term and a general regularization term. It is widely accepted that the minimizers obtained using non-convex and possibly non-smooth regularization terms are frequently good estimates. However, few facts are known on the ways to control properties of these minimizers. This work is dedicated to the stability of the minimizers of such objective functions with respect to variations of the data. It consists of two parts: first we consider all local minimizers, whereas in a second part we derive results on global minimizers. In this part we focus on data points such that every local minimizer is isolated and results from a C^m-1 local minimizer function, defined on some neighborhood. We demonstrate that all data points for which this fails form a set whose closure is negligible.     

Higher Critical Points in an Elliptic Free Boundary Problem

We study higher critical points of the variational functional associated with a free boundary problem related to plasma confinement. Existence and regularity of minimizers in elliptic free boundary problems have already been studied extensively. But because the functionals are not smooth, standard variational methods cannot be used directly to prove the existence of higher critical points. Here we find a nontrivial critical point of mountain pass type and prove many of the same estimates known for minimizers, including Lipschitz continuity and nondegeneracy. We then show that the free boundary is smooth in dimension 2 and prove partial regularity in higher dimensions.     

A minimizing property of homographic solutions

In this paper, we prove that the homographic solutions to the rhombus four body prob- lem are the variational minimizers of the Lagrangian action restricted on a holonomically constrained rhombus loop space.