THE SINGULARLY PERTURBED NONLINEAR ELLIPTIC SYSTEMS IN UNBOUNDED DOMAINS

Abstract. The singularly perturbed problems for elliptic systems in unbounded domains are considered. Under suitable conditions and by using the comparison theorem the existence and asymptotic behavior of solution for the boundary value problems studied,     

Schauder estimates for a class of second order elliptic operators on a cube

We consider a class of second order elliptic operators on a d-dimensional cube S_d. We prove that if the coefficients are of class C^k+δ(S_d), with k=0,1 and δ∈(0,1), then the corresponding elliptic problem admits a unique solution u belonging to C^k+2+δ(S_d) and satisfying non-standard boundary conditions involving only second order derivatives.     

Lipschitz continuity of Cheeger-harmonic functions in metric measure spaces

We use the heat equation to establish the Lipschitz continuity of Cheeger-harmonic functions in certain metric spaces. The metric spaces under consideration are those that are endowed with a doubling measure supporting a (1,2)-Poincaré inequality and in addition supporting a corresponding Sobolev-Poincaré-type inequality for the modification of the measure obtained via the heat kernel. Examples are given to illustrate the necessity of our assumptions on these spaces. We also provide an example to show that in the general setting the best possible regularity for the Cheeger-harmonic functions is Lipschitz continuity.     

A regularity result of solution to the compressible Stokes equations on a convex polygon

A compressible Stokes problem is analyzed in a convex polygon D. The goal of this paper is to sort out a singularity of the pressure function at the corner and to establish the corresponding regularity result of the resulted remainder part of the solution. For this a solution formula is derived and the singular function of the Stokes problem is considered. It is seen that the lowest order of the regularity of the system is the same as that of the (incompressible) Stokes one.     

The problem of Dirichlet for anisotropic quasilinear degenerate elliptic equations

We consider the Dirichlet problem for a class of anisotropic degenerate elliptic equations. New a priori estimates for solutions and for the gradient of solutions are established. Based on these estimates sufficient conditions guaranteeing the solvability of the problem are formulated. The results are new even in the semilinear case when the principal part is the Laplace operator.     

Spherically symmetric internal layers for activator-inhibitor systems—I. Existence by a Lyapunov-Schmidt reduction

Reaction-diffusion systems of activator-inhibitor type are studied on an N-dimensional ball with the homogeneous Neumann boundary conditions. Under the condition that the activator diffuses slowly, reacts rapidly and the inhibitor diffuses rapidly, reacts moderately, we show that the system admits a family of spherically symmetric internal transition layer equilibria. The method of proof consists of rigorous asymptotic expansions and a Lyapunov-Schmidt reduction.     

Spherically symmetric internal layers for activator-inhibitor systems II. Stability and symmetry breaking bifurcations

A reaction-diffusion system of activator-inhibitor type is studied on an N-dimensional ball with the homogeneous Neumann boundary conditions. We analyze the stability property of the spherically symmetric solutions and their symmetry-breaking bifurcations into layer solutions which are not spherically symmetric.     

G-convergence,Dirichlet to Neumann maps and invisibility

We establish optimal conditions under which the G-convergence of linear elliptic operators implies the convergence of the corresponding Dirichlet to Neumann maps. As an application we show that the approximate cloaking isotropic materials from [19] are independent of the source.     

Quasi-neutral limit to the drift-diffusion models for semiconductors with physical contact-insulating boundary conditions

In this paper the limit of vanishing Debye length in a bipolar drift-diffusion model for semiconductors with physical contact-insulating boundary conditions is studied in one-dimensional case. The quasi-neutral limit (zero-Debye-length limit) is proved by using the asymptotic expansion methods of singular perturbation theory and the classical energy methods. Our results imply that one kind of the new and interesting phenomena in semiconductor physics occurs.     

Invariant regions for quasilinear reaction-diffusion systems and applications to a two population model

We prove that some conditions are sufficient for regions to be invariant with respect to strongly coupled quasilinear parabolic systems indivergence form. This result can be applied to certain two population systems where we can compute the boundaries of the invariant regions by solving ordinary differential equations. Under simple conditions on the parameters we get bounded invariant regions.     

Uniform binary geometries

In a geometric lattice every interval can be mapped isomorphically into an upper interval (containing 1) by a strong map. A natural question thus arises as to what extent certain assumptions on the upper interval structure determine the whole lattice. We consider conditions of the following sort: that above a certain levelm any two upper intervals of the same length be isomorphic. This property, called uniformity, is studied for binary geometries. The geometries satisfying the strongest uniformity condition (m = 1) are determined (except for one open case). As is to be expected the corresponding problem for lower intervals is easier and is solved completely.     

The Fujita exponent for the Cauchy problem in the hyperbolic space

It is well known that the heat kernel in the hyperbolic space has a different behavior for large times than the one in the Euclidean space. The main purpose of this paper is to study its effect on the positive solutions of Cauchy problems with power nonlinearities. Existence and non-existence results for local solutions are derived. Emphasis is put on their long time behavior and on Fujita?s phenomenon. To have the same situation as for the Cauchy problem in RN, namely finite time blow up for all solutions if the exponent is smaller than a critical value and existence of global solutions only for powers above the critical exponent, we must introduce a weight depending exponentially on the time. In this respect the situation is similar to problems in bounded domains with Dirichlet boundary conditions. Important tools are estimates for the heat kernel in the hyperbolic space and comparison principles.     

Approximate controllability of evolution systems with nonlocal conditions

We study the approximate controllability for the abstract evolution equations with nonlocal conditions in Hilbert spaces. Assuming the approximate controllability of the corresponding linearized equation we obtain sufficient conditions for the approximate controllability of the semilinear evolution equation. The results we obtained are a generalization and continuation of the recent results on this issue. At the end, an example is given to show the application of our result.     

Concentration of solutions for some singularly perturbed mixed problems: Asymptotics of minimal energy solutions

In this paper we carry on the study of asymptotic behavior of some solutions to a singularly perturbed problem with mixed Dirichlet and Neumann boundary conditions, started in the first paper [J. Garcia Azorero, A. Malchiodi, L. Montoro, I. Peral, Concentration of solutions for some singularly perturbed mixed problems: Existence results, Arch. Ration. Mech. Anal., in press]. Here we are mainly interested in the analysis of the location and shape of least energy solutions when the singular perturbation parameter tends to zero. We show that in many cases they coincide with the new solutions produced in [J. Garcia Azorero, A. Malchiodi, L. Montoro, I. Peral, Concentration of solutions for some singularly perturbed mixed problems: Existence results, Arch. Ration. Mech. Anal., in press].     

Bergman-Toeplitz operators: Radial component influence

We analyze the influence of the radial component of a symbol to spectral, compactness, and Fredholm properties of Toeplitz operators, acting on the Bergman space. We show that there existcompact Toeplitz operators whose (radial) symbols areunbounded near the unit circle . Studying this question we give several sufficient, and necessary conditions, as well as the corresponding examples. The essential spectra of Toeplitz operators with pure radial symbols have sufficiently rich structure, and even can be massive.TheC ^*-algebras generated by Toeplitz operators with radial symbols are commutative, but the semicommutators[T _a, T_b)=T_a·T_b–T_a·b are not compact in general. Moreover for bounded operatorsT _a andT _b the operatorT _a·b may not be bounded at all.This work was partially supported by CONACYT Project 27934-E, México.The first author acknowledges the RFFI Grant 98-01-01023, Russia.     

Stabilities in multi-valued dynamical systems

For the study of the various stabilities in multi-valued dynamical systems, we obtain some necessary and sufficient conditions of the notions of stability and Lyapunov stability, and investigate the connection between the concepts of attractions and the suitable versions of limit sets. Also we consider the notion of characteristic 0⁺ in multi-valued dynamical systems and obtain necessary and sufficient conditions for the concept of characteristic 0⁺.     

Boundary behaviour for solutions of elliptic singular equations with a gradient term

We investigate the homogeneous Dirichlet boundary value problem for a class of second-order nonlinear elliptic partial differential equations with a gradient term and singular data. Under general conditions on the data, we study the behaviour of the solution near the boundary of the domain. Under suitable additional conditions we also investigate the second-order term in the asymptotic expansion of the solution in terms of the distance from the boundary.     

Continuity of attractors for a reaction-diffusion problem with nonlinear boundary conditions with respect to variations of the domain

In this work we show, for a class of dissipative semilinear parabolic problems, that the global compact attractor varies continuously with respect to parameters in the equations. Applications to a parabolic problem with nonlinear boundary conditions are also obtained.     

Quasineutral limit of a time-dependent drift-diffusion-Poisson model for p-n junction semiconductor devices

In this paper the vanishing Debye length limit of the bipolar time-dependent drift-diffusion-Poisson equations modelling insulated semiconductor devices with p-n junctions (i.e., with a fixed bipolar background charge) is studied. For sign-changing and smooth doping profile with ‘good’ boundary conditions the quasineutral limit (zero-Debye-length limit) is performed rigorously by using the multiple scaling asymptotic expansions of a singular perturbation analysis and the carefully performed classical energy methods. The key point in the proof is to introduce a ‘density’ transform and two λ-weighted Liapunov-type functionals first, and then to establish the entropy production integration inequality, which yields the uniform estimate with respect to the scaled Debye length. The basic point of the idea involved here is to control strong nonlinear oscillation by the interaction between the entropy and the entropy dissipation.     

On boundary-value problems for the laplacian in bounded and in unbounded domains with perforated boundaries

In this paper we consider boundary-value problems in domains with perforated boundaries. We use the classification of homogenized (limit) problems depending on the ratio of small parameters, which characterize the diameter of the holes and the distance between them. We study the analogue of the Helmholtz resonator for domains with a perforated boundary.