Multiple Solutions for Nonlinear Coercive Problems with a Nonhomogeneous Differential Operator and a Nonsmooth Potential

We consider a nonlinear elliptic problem driven by a nonlinear nonhomogeneous differential operator and a nonsmooth potential. We prove two multiplicity theorems for problems with coercive energy functional. In both theorems we produce three nontrivial smooth solutions. In the second multiplicity theorem, we provide precise sign information for all three solutions (the first positive, the second negative and the third nodal). Out approach is variational, based on the nonsmooth critical point theory. We also prove an auxiliary result relating smooth and Sobolev local minimizer for a large class of locally Lipschitz functionals.     

On the existence of multiple solutions for a nonlocal BVP with vector-valued response

The existence of positive solutions for a nonlocal boundary-value problem with vector-valued response is investigated. We develop duality and variational principles for this problem. Our variational approach enables us to approximate solutions and give a measure of a duality gap between the primal and dual functional for minimizing sequences.     

Estimates of deviations from exact solutions of variational problems with linear growth functionals

Estimates of deviations from exact solutions of variational problems with linear growth functionals are derived. Since the original variational problem may have no minimizer in a reflexive Banach space, the estimates are presented in terms of the dual problem. The consistency of these estimates are proved, and their computationally convenient forms are obtained. Bibliography: 36 titles.     

Optimal Solutions to Relaxation in Multiple Control Problems of Sobolev Type with Nonlocal Nonlinear Fractional Differential Equations

We introduce the optimality question to the relaxation in multiple control problems described by Sobolev-type nonlinear fractional differential equations with nonlocal control conditions in Banach spaces. Moreover, we consider the minimization problem of multi-integral functionals, with integrands that are not convex in the controls, of control systems with mixed nonconvex constraints on the controls. We prove, under appropriate conditions, that the relaxation problem admits optimal solutions. Furthermore, we show that those optimal solutions are in fact limits of minimizing sequences of systems with respect to the trajectory, multicontrols, and the functional in suitable topologies.     

On a variational statement of a nonlocal boundary value problem for a fourth-order ordinary differential equation

We study a nonlocal boundary value problem for a fourth-order ordinary differential equation. We give a variational statement of the problem by constructing the corresponding functional. The minimization of this functional provides a solution of the problem.     

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.     

Sharp Threshold of Global Existence for a Nonlocal Nonlinear Schrödinger System in ℝ3

In this paper, the authors investigate the sharp threshold of a three-dimensional nonlocal nonlinear Schr\"{o}dinger system. It is a coupled system which provides the mathematical modeling of the spontaneous generation of a magnetic field in a cold plasma under the subsonic limit. The main difficulty of the proof lies in exploring the inner structure of the system due to the fact that the nonlocal effect may bring some hinderance for establishing the conservation quantities of the mass and of the energy, constructing the corresponding variational structure, and deriving the key estimates to gain the expected result. To overcome this, the authors must establish local well-posedness theory, and set up suitable variational structure depending crucially on the inner structure of the system under study, which leads to define proper functionals and a constrained variational problem. By building up two invariant manifolds and then making a priori estimates for these nonlocal terms, the authors figure out a sharp threshold of global existence for the system under consideration.     

Existence of uncertainty minimizers for the continuous wavelet transform

Continuous wavelet design is the endeavor to construct mother wavelets with desirable properties for the continuous wavelet transform (CWT). One class of methods for choosing a mother wavelet involves minimizing a functional, called the wavelet uncertainty functional. Recently, two new wavelet uncertainty functionals were derived from theoretical foundations. In both approaches, the uncertainty of a mother wavelet describes its concentration, or accuracy, as a time-scale probe. While an uncertainty minimizing mother wavelet can be proven to have desirable localization properties, the existence of such a minimizer was never studied. In this paper, we prove the existence of minimizers for the two uncertainty functionals.     

A constrained variational problem arising in attractive Bose–Einstein condensate with ellipse-shaped potential

We consider a minimization problem for the variational functional associated with a Gross–Pitaevskii equation arising in the study of an attractive Bose–Einstein condensate. Under an ellipse-shaped trapping potential, that is, the bottom of the trapping potential is an ellipse, we prove that any minimizer of the minimization problem blows up at one of the endpoints of the major axis of the ellipse if the parameter associated to the attractive interaction strength approaches a critical value.     

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.     

On the existence of positive solutions and their continuous dependence on functional parameters for some class of elliptic problems

We discuss the existence and the dependence on functional parameters of solutions of the Dirichlet problem for a kind of the generalization of the balance of a membrane equation. Since we shall propose an approach based on variational methods, we treat our equation as the Euler-Lagrange equation for a certain integral functional J. We will not impose either convexity or coercivity of the functional. We develop a duality theory which relates the infimum on a special set X of the energy functional associated with the problem, to the infimum of the dual functional on a corresponding set X^d. The links between minimizers of both functionals give a variational principle and, in consequence, their relation to our boundary value problem. We also present the numerical version of the variational principle. It enables the numerical characterization of approximate solutions and gives a measure of a duality gap between primal and dual functional for approximate solutions of our problem.     

Projection-iterative version of the pointwise relaxation method

We propose a projection-iterative modification of the pointwise relaxation method for the solution of variational problems of mechanics of deformable solids. For the general case of the problem of conventional minimization of functionals in Hilbert spaces, convergence is proved. For the problem of minimization of quadratic functional, a theoretical justification is performed by the finite-element method.     

Variational properties of the solutions of singular second-order differential equations and systems

This paper is devoted to the existence and variational characterization of the weak solutions of the Dirichlet boundary value problem for singular second-order ordinary differential equations and systems. The solution appears as a minimizer of the energy functional associated with the equation, and in the case of systems, as a Nash-type equilibrium of the set of energy functionals. The results are connected with the recent abstract fixed point theory due to the second author and with its application, given by the first author, to semilinear operator problems of Michlin type.     

On duality for a generalized Monge-Kantorovich problem

The existence of a minimizer for a special version of the Monge-Kantorovich problem is solved using classical tools of functional analysis, and it is shown that the minimizer is not unique. Our variational problem has a connection to the Michell problem in optimal design.     

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.     

Weighted energy-dissipation functionals for doubly nonlinear evolution

This paper is concerned with the Weighted Energy-Dissipation (WED) functional approach to doubly nonlinear evolutionary problems. This approach consists in minimizing (WED) functionals defined over entire trajectories. We present the features of the WED variational formalism and analyze the related Euler-Lagrange problems. Moreover, we check that minimizers of the WED functionals converge to the corresponding limiting doubly nonlinear evolution. Finally, we present a discussion on the functional convergence of sequences of WED functionals and present some application of the abstract theory to nonlinear PDEs.     

Finding the Strictly Local and $\epsilon $-Global Minimizers of Concave Minimization with Linear Constraints

This paper considers the concave minimization problem with linear constraints, proposes a technique which may avoid the unsuitable Karush-Kuhn-Tucker points, then combines this technique with Frank-Wolfe method and simplex method to form a pivoting method which can determine a strictly local minimizer of the problem in a finite number of iterations. Based on strictly local minimizers, a new cutting plane method is proposed. Under some mild conditions, the new cutting plane method is proved to be finitely terminated at an $\epsilon $-global minimizer of the problem.     

When only global optimization matters

In this short note, the objective of which is essentially pedagogical, we show that in the well-known problem which consists of minimizing the rank of a matrix, every admissible point is a local minimizer. Hence, in this problem like in various other ones, only global minimization matters.     

The parameterized Steiner problem and the singular Plateau problem via energy

The Steiner problem is the problem of finding the shortest network connecting a given set of points. By the singular Plateau Problem, we will mean the problem of finding an area-minimizing surface (or a set of surfaces adjoined so that it is homeomorphic to a 2-complex) spanning a graph. In this paper, we study the parametric versions of the Steiner problem and the singular Plateau problem by a variational method using a modified energy functional for maps. The main results are that the solutions of our one- and two-dimensional variational problems yield length and area minimizing maps respectively, i.e. we provide new methods to solve the Steiner and singular Plateau problems by the use of energy functionals. Furthermore, we show that these solutions satisfy a natural balancing condition along its singular sets. The key issue involved in the two-dimensional problem is the understanding of the moduli space of conformal structures on a 2-complex.

     

On the uniqueness and stability of solutions of extremal problems for the stationary Navier-Stokes equations

We consider inverse extremal problems for the stationary Navier-Stokes equations. In these problems, one seeks an unknown vector function occurring in the Dirichlet boundary condition for the velocity and the solution of the considered boundary value problem on the basis of the minimization of some performance functional. We derive new a priori estimates for the solutions of the considered extremal problems and use them to prove theorems of the local uniqueness and stability of solutions for specific performance functionals.