Nonlinear Bound States on Weakly Homogeneous Spaces

We prove the existence of ground state solutions for a class of nonlinear elliptic equations, arising in the production of standing wave solutions to an associated family of nonlinear Schrödinger equations. We examine two constrained minimization problems, which give rise to such solutions. One yields what we call F _λ-minimizers, the other energy minimizers. We produce such ground state solutions on a class of Riemannian manifolds called weakly homogeneous spaces, and establish smoothness, positivity, and decay properties. We also identify classes of Riemannian manifolds with no such minimizers, and classes for which essential uniqueness of positive solutions to the associated elliptic PDE fails.     

Young measure solutions and instability of the one-dimensional Perona-Malik equation

We apply a variational approach to the one-dimensional version of the widely used Perona-Malik equation in image processing. We rephrase the problem into the one related to the quasiconvex hull of a graph in the space of 2×2 matrices M2×2. We then use the solutions of some heat equations as the centre of the mass for the Young measure-valued solutions to construct the approximate solutions by using simple laminates. The approximate solutions can be viewed as solutions of a perturbation problem by W−1,p (or W−1,∞) functions. The sequences of the approximate solutions generates Young measure-valued solutions. Our results also show that the solutions of the one-dimensional Perona-Malik equation are unstable under small W−1,∞ perturbations.     

On Burgers fluids

We investigate several qualitative aspects concerning the solutions to the flow of a Burgers fluid, a model that has been used to describe a variety of materials: polymeric liquids, asphalt and asphalt mixtures, and the earth's mantle. Continuous dependence of the solutions with respect to initial data and supply terms, uniqueness of solutions and the impossibility of localization of the solutions are proved. Exponential stability and structural stability are analysed. We also consider uniaxial shear flows. We prove instability of solutions whenever the constitutive parameters do not satisfy certain relations. Also we study the spatial behaviour of the solutions. Copyright © 2006 John Wiley & Sons, Ltd.     

Solutions with peaks for a coagulation–fragmentation equation. Part I: stability of the tails

Abstract

The aim of this two-part paper is to investigate the stability properties of a special class of solutions to a coagulation–fragmentation equation. We assume that the coagulation kernel is close to the diagonal kernel, and that the fragmentation kernel is diagonal. We construct a two-parameter family of stationary solutions concentrated in Dirac masses. We carefully study the asymptotic decay of the tails of these solutions, showing that this behavior is stable. In a companion paper, we prove that for initial data which are sufficiently concentrated, the corresponding solutions approach one of these stationary solutions for large times.     

Stationary layered solutions in {\Bbb R}^2 for an Allen–Cahn system with multiple well potential

We study entire solutions on of the elliptic system where is a multiple-well potential. We seek solutions which are “heteroclinic,” in two senses: for each fixed they connect (at ) a pair of constant global minima of , and they connect a pair of distinct one dimensional stationary wave solutions when . These solutions describe the local structure of solutions to a reaction-diffusion system near a smooth phase boundary curve. The existence of these heteroclinic solutions demonstrates an unexpected difference between the scalar and vector valued Allen–Cahn equations, namely that in the vectorial case the transition profiles may vary tangentially along the interface. We also consider entire stationary solutions with a “saddle” geometry, which describe the structure of solutions near a crossing point of smooth interfaces. Received April 15, 1996 / Accepted: November 11, 1996     

Decay estimates for a thermoelastic system in waveguides

We investigate the linear system of thermoelasticity, consisting of an elasticity equation and a heat conduction equation, in a waveguide Ω=(0,1)×Rn−1, with certain boundary conditions. We consider the cases of homogeneous and inhomogeneous systems and prove decay estimates of the solutions, which are a key ingredient to showing the global existence of solutions to non-linear thermoelasticity, after having decomposed the solutions into various parts. We also give a simplified proof to the representation of the solutions to the Cauchy problem of thermoelasticity.     

The porous medium equation with measure data

We study the existence of solutions to the porous medium equation with a nonnegative, finite Radon measure on the right-hand side. We show that such problems have solutions in a wide class of supersolutions. These supersolutions are defined as lower semicontinuous functions obeying a parabolic comparison principle with respect to continuous solutions. We also consider the question of how the integrability of the gradient of solutions is affected if the measure is given by a function in L ^s , for a small exponent s > 1.     

Stationary points of the Yang-Mills action

We examine the structure of a recently discovered set of non-self-dual solutions of the Yang-Mills equations. These solutions have a symmetry that reduces the YM equations to a set of ODE's. The distinct solutions are indexed by two postive odd integers. We develop a scheme to approximate on a computer the solutions for small values of the indexing integers and present some numerical results. We then analyze the asymptotic behavior of the solutions as the indexing integers become large.     

Perturbations of Dynamic Equations

We give conditions on the coefficient matrix for certain perturbed linear dynamic equations on time scales ensuring that there exists a bounded solution (which is explicitly given) to which all other solutions converge, and similarly conditions ensuring a bounded solution from which all other solutions diverge. We also consider periodic time scales and corresponding linear dynamic equations with periodic coefficients and prove similar statements about periodic solutions to which all other solutions converge or from which all other solutions diverge.     

Minimal Zero Norm Solutions of Linear Complementarity Problems

In this paper, we study minimal zero norm solutions of the linear complementarity problems, defined as the solutions with smallest cardinality. Minimal zero norm solutions are often desired in some real applications such as bimatrix game and portfolio selection. We first show the uniqueness of the minimal zero norm solution for Z-matrix linear complementarity problems. To find minimal zero norm solutions is equivalent to solve a difficult zero norm minimization problem with linear complementarity constraints. We then propose a p norm regularized minimization model with p in the open interval from zero to one, and show that it can approximate minimal zero norm solutions very well by sequentially decreasing the regularization parameter. We establish a threshold lower bound for any nonzero entry in its local minimizers, that can be used to identify zero entries precisely in computed solutions. We also consider the choice of regularization parameter to get desired sparsity. Based on the theoretical results, we design a sequential smoothing gradient method to solve the model. Numerical results demonstrate that the sequential smoothing gradient method can effectively solve the regularized model and get minimal zero norm solutions of linear complementarity problems.     

Existence of extremal solutions by approximation to a first-order initial dynamic equation with Carathéodory's conditions and discontinuous non-linearities*

This paper is devoted to the study of the existence of extremal solutions to a first-order initial value problem on an interval of an arbitrary time scale. We prove the existence of extremal solutions for problems satisfying Carathéodory's conditions. Moreover, they are approximated uniformly by a sequence of lower and upper solutions to this problem, respectively.

We also can warrant the existence and approximation of extremal solutions for the problem by relaxing their continuity properties.     

Generalized motion by mean curvature with Neumann conditions and the Allen-Cahn model for phase transitions

We study asharpinterface model for phase transitions which incorporates the interaction of the phase boundaries with the walls of a container Ω. In this model, the interfaces move by their mean curvature and are normal to δΩ. We first establish local-in-time existence and uniqueness of smooth solutions for the mean curvature equation with a normal contact angle condition. We then discuss global solutions by interpreting the equation and the boundary condition in a weak (viscosity) sense. Finally, we investigate the relation of the aforementioned model with atransitionlayer model. We prove that if Ω isconvex, the transition-layer solutions converge to the sharp-interface solutions as the thickness of the layer tends to zero. We conclude with a discussion of the difficulties that arise in establishing this result in nonconvex domains. Communicated by David Kinderlehrer     

Solutions to the Landau–Lifshitz system with nonhomogenous Neumann boundary conditions arising from surface anisotropy and super-exchange interactions in a ferromagnetic media

We study the solutions to the Landau–Lifshitz system in a bilayered ferromagnetic body when super-exchange and surface anisotropy interactions are present at the interface between the layers. We prove the existence of weak solutions in infinite time and strong solutions in finite time.     

Computational investigations of the Prouhet-Tarry-Escott Problem

We describe a method for searching for ideal symmetric solutions to the Prouhet-Tarry-Escott Problem. We report results of extensive searches for solutions of sizes up to 12. We found two solutions of size 10 that are smaller by two orders of magnitude than the solution found by A. Letac in the 1940s, which was the smallest size 10 solution known before our search.

     

Exponential decay for the solutions of nonlinear elliptic systems posed in unbounded cylinders

We study the asymptotic behavior at infinity of the solutions of a nonlinear elliptic system posed in a cylinder of infinite length. The problem is written in a variational formulation, where we ask the derivative of the solutions to be in Lp. We show that an exponential decay at infinity for the second member implies exponential decay for the derivative of the solutions. We also give an application of this result to the study of boundary layers problems.     

Singular solutions of Hessian fully nonlinear elliptic equations

We study Hessian fully nonlinear uniformly elliptic equations and show that the second derivatives of viscosity solutions of those equations (in 12 or more dimensions) can blow up in an interior point of the domain. We prove that the optimal interior regularity of such solutions is no more than C1+?, showing the optimality of the known interior regularity result. The same is proven for Isaacs equations. We prove the existence of non-smooth solutions to fully nonlinear Hessian uniformly elliptic equations in 11 dimensions. We study also the possible singularity of solutions of Hessian equations defined in a neighborhood of a point and prove that a homogeneous order 0<α<1 solution of a Hessian uniformly elliptic equation in a punctured ball should be radial.     

Hölder continuity of solutions to a degenerate elliptic equation in the presence of the Lavrentiev phenomenon

We study a second-order elliptic equation for which the Dirichlet problem can be posed in a nonunique way due to the so-called Lavrentiev phenomenon. In the corresponding weighted Sobolev space smooth functions are not dense, which leads to the existence of W – solutions and H – solutions. For H - solutions, we establish the Hölder continuity. We also discuss this question for W – solutions, for which the situation is more complicated.     

Poisson Total Boundedness of Solutions of Systems of Differential Equations and Lyapunov Vector Functions

We introduce the notions of Poisson total boundedness of solutions, partial Poisson total boundedness of solutions, and partial Poisson total boundedness of solutions with partly controlled initial conditions. We use the Lyapunov vector function method to obtain sufficient conditions for the Poisson total boundedness of solutions, the partial Poisson total boundedness of solutions, and the partial Poisson total boundedness of solutions with partly controlled initial conditions. As a consequence, we obtain sufficient conditions for the above-mentioned kinds of Poisson total boundedness of solutions based on the Lyapunov function method.

     

Population monotonic solutions on convex games

The Dutta-Ray solution and the Shapley value are two well-known examples of population-monotonic solutions on the domain of convex games. We provide a new formula for the Dutta-Ray solution from which population-monotonicity immediately follows. Then we define a new family of population-monotonic solutions, which we refer to as “sequential Dutta-Ray solutions.” We also show that it is possible to construct several symmetric and population-monotonic solutions by using the solutions in this family. Received September 1998/Revised version: December 1999     

A Free Boundary Problem with Curvature

Abstract

In this paper we are interested in a free boundary problem with a motion law involving the mean curvature term of the free boundary. Viscosity solutions are introduced as a notion of global-time solutions past singularities. We show the comparison principle for viscosity solutions, which yields the existence of minimal and maximal solutions for given initial data. We also prove uniqueness of the solution for several classes of initial data and discuss the possibility of nonunique solutions.