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.     

Guaranteed error bounds for finite element approximations of noncoercive elliptic problems and their applications

In this paper, we discuss with guaranteed a priori and a posteriori error estimates of finite element approximations for not necessarily coercive linear second order Dirichlet problems. Here, ‘guaranteed’ means we can get the error bounds in which all constants included are explicitly given or represented as a numerically computable form. Using the invertibility condition of concerning elliptic operator, guaranteed a priori and a posteriori error estimates are formulated. This kind of estimates plays essential and important roles in the numerical verification of solutions for nonlinear elliptic problems. Several numerical examples that confirm the actual effectiveness of the method are presented.     

The Neumann problem for the Helmholtz equation in a starlike planar domain

The interior and exterior Neumann problems for the Helmholtz equation in starlike planar domains are addressed by using a suitable Fourier-like technique. Attention is in particular focused on normal-polar domains whose boundaries are defined by the so called “superformula” introduced by Gielis. A dedicated numerical procedure based on a computer algebra system is developed in order to validate the proposed approach. In this way, highly accurate approximations of the solution, featuring properties similar to classical ones, are obtained. Computed results are found to be in good agreement with theoretical findings on Fourier series expansion presented by Carleson.     

Permanence under strong aggressions is possible

This paper analyzes the limiting behavior of the positive solutions of a general class of sublinear elliptic weighted mixed boundary value problems as the amplitude of the positive part of the lower order terms of the differential operator blows up to infinity. The main result establishes that the positive solutions approximate zero within the support of the positive part of the potential, whereas they stabilize to the positive solution of a certain elliptic mixed boundary value problem on its complement. Further, we use this result for deriving some general principles in competing species dynamics. Precisely, we shall show that in the presence of a refuge region two competing species must coexist if their reproduction rates are sufficiently large, independently of the strength of the competition. It should be emphasized that the abstract theory developed here allows measuring how large the reproduction rates should be for being permanent, providing us, simultaneously, with the limiting behavior of each of the species separately. Basically, when the pressure from the competitor grows the tackled species concentrates within its refuge. The results of this paper are substantial extensions of some pioneer results found by one of the authors in [16, Section 4]. The main ingredients in deriving the main results of this paper are the continuous dependence of the principal eigenvalue with respect to a general class of perturbations of the domain around its Dirichlet boundary – very recent result coming from [6] – and the continuous dependence of the positive solutions of the sublinear problem – coming from [7].     

Well-posed elliptic Neumann problems involving irregular data and domains

Non-linear elliptic Neumann problems, possibly in irregular domains and with data affected by low integrability properties, are taken into account. Existence, uniqueness and continuous dependence on the data of generalized solutions are established under a suitable balance between the integrability of the datum and the (ir)regularity of the domain. The latter is described in terms of isocapacitary inequalities. Applications to various classes of domains are also presented.     

Continuous dependence results for non-linear Neumann type boundary value problems

We obtain estimates on the continuous dependence on the coefficient for second-order non-linear degenerate Neumann type boundary value problems. Our results extend previous work of Cockburn et al., Jakobsen and Karlsen, and Gripenberg to problems with more general boundary conditions and domains. A new feature here is that we account for the dependence on the boundary conditions. As one application of our continuous dependence results, we derive for the first time the rate of convergence for the vanishing viscosity method for such problems. We also derive new explicit continuous dependence on the coefficients results for problems involving Bellman-Isaacs equations and certain quasilinear equation.     

Some remarks on Sobolev type inequalities

We extend two Sobolev type inequalities for balls to arbitrary smooth bounded domains. In the case of balls, one inequality is due to Brezis and Lieb and another is due to Escobar. The extension has been achieved by analyzing the asymptotic behaviour of solutions of certain semilinear Neumann problems.     

The Schur algorithm and its applications

The Schur algorithm and its time-domain counterpart, the fast Cholseky recursions, are some efficient signal processing algorithms which are well adapted to the study of inverse scattering problems. These algorithms use a layer stripping approach to reconstruct a lossless scattering medium described by symmetric two-component wave equations which model the interaction of right and left propagating waves. In this paper, the Schur and fast Chokesky recursions are presented and are used to study several inverse problems such as the reconstruction of nonuniform lossless transmission lines, the inverse problem for a layered acoustic medium, and the linear least-squares estimation of stationary stochastic processes. The inverse scattering problem for asymmetric two-component wave equations corresponding to lossy media is also examined and solved by using two coupled sets of Schur recursions. This procedure is then applied to the inverse problem for lossy transmission lines.The work of this author was supported by the Exxon Education FoundationThe work of this author was supported by the Air Force Office of Scientific Research under Grant AFOSR-82-0135A.     

On general boundary-value problems for nonlinear elliptic equations of second order in a multiply connected plane domain

This paper deals with some general irregular oblique derivative problems for nonlinear uniformly elliptic equations of second order in a multiply connected plane domain. Firstly, we state the well-posedness of a new set of modified boundary conditions. Secondly, we verify the existence of solutions of the modified boundary-value problem for harmonic functions, and then prove the solvability of the modified problem for nonlinear elliptic equations, which includes the original boundary-value problem (i.e. boundary conditions without involving undertermined functions data). Here, mainly, the location of the zeros of analytic functions, a priori estimates for solutions and the continuity method are used in deriving all these results. Furthermore, the present approach and setting seems to be new and different from what has been employed before.The research was partially supported by a UPGC Grant of Hong Kong.     

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.     

Bounded-from-below solutions of the Hamilton-Jacobi equation for optimal control problems with exit times: vanishing lagrangians,eikonal equations,and shape-from-shading

We study the Hamilton-Jacobi equation for undiscounted exit time control problems with general nonnegative Lagrangians using the dynamic programming approach. We prove theorems characterizing the value function as the unique bounded-from-below viscosity solution of the Hamilton-Jacobi equation that is null on the target. The result applies to problems with the property that all trajectories satisfying a certain integral condition must stay in a bounded set. We allow problems for which the Lagrangian is not uniformly bounded below by positive constants, in which the hypotheses of the known uniqueness results for Hamilton-Jacobi equations are not satisfied. We apply our theorems to eikonal equations from geometric optics, shape-from-shading equations from image processing, and variants of the Fuller Problem.     

Parabolic equations with BMO coefficients in Lipschitz domains

In this paper, we are concerned with certain natural Sobolev-type estimates for weak solutions of inhomogeneous problems for second-order parabolic equations in divergence form. The geometric setting is that of time-independent cylinders having a space intersection assumed to be locally given by graphs with small Lipschitz coefficients, the constants of the operator being uniformly parabolic. We prove the relevant L^p estimates, assuming that the coefficients are in parabolic bounded mean oscillation (BMO) and that their parabolic BMO semi-norms are small enough.     

On elliptic problems satisfying Landesman-Laser conditions

In this paper we consider two elliptic problems. The first one is a Dirichlet problem while the second is Neumann. We extend all the known results concerning Landesman-Laser conditions by using the Mountain-Pass theorem with the Cerami (PS) condition.     

An approximation problem in inverse scattering theory

In [3] a new method was introduced for solving the inverse scattering problem for acoustic waves in an inhomogeneous medium. This method is based on the solution of a new class of boundary value problems for the reduced wave equation called interior transmission problems. In this paper it is shown that if there is absorption there exists at most one solution to the interior transmission problem and an approximate solution can be found such that the metaharmonic part is a Herglotz wave function. These results provide the necessary theoretical basis for the inverse scattering method introduced in [3]     

Exterior problems in the half-space for the Laplace operator in weighted Sobolev spaces

The purpose of this work is to solve exterior problems in the half-space for the Laplace operator. We give existence and unicity results in weighted Lp's theory with 1<p<∞. This paper extends the studies done in [C. Amrouche, V. Girault, J. Giroire, Dirichlet and Neumann exterior problems for the n-dimensional Laplace operator, an approach in weighted Sobolev spaces, J. Math. Pures Appl. 76 (1) (1997) 55-81] with Dirichlet and Neumann conditions.     

Stability results for Mountain Pass and Linking type solutions of semilinear problems involving Dirichlet forms

Some stability results for Mountain Pass and Linking type solutions of semilinear problems involving a very general class of Dirichlet forms are stated. The non linear terms are supposed to have a suitable superlinear growth and the family of Dirichlet forms is required to be dominated from below and from above by a fixed diffusion type form. Some concrete examples are also given.     

Corner singularities and boundary layers in a simple convection-diffusion problem

A singularly perturbed convection-diffusion problem posed on the unit square is considered. Its solution may have exponential and parabolic boundary layers, and corner singularities may also be present. Pointwise bounds on the solution and its derivatives are derived. The dependence of these bounds on the small diffusion coefficient, on the regularity of the data, and on the compatibility of the data at the corners of the domain are all made explicit. The bounds are derived by decomposing the solution into a sum of solutions of elliptic boundary-value problems posed on half-planes, then analyzing these simpler problems.     

Comparison principle for some classes of nonlinear elliptic equations

We prove a comparison principle for second order quasilinear elliptic operators in divergence form when a first order term appears. We deduce uniqueness results for weak solutions to Dirichlet problems when data belong to the natural dual space.     

On a semilinear elliptic problem without (PS) condition

An existence result for semilinear elliptic problems whose associated functionals do not satisfy a Palais-Smale condition is proved. The nonlinearity of our problem fits none of the conditions in Ambrosetti and Rabinowitz (J. Funct. Anal. 14 (1973) 349), de Figueiredo et al. (J. Math. Pures Appl. 61 (1982) 41) and Gidas and Spruck (Comm. Part. Diff. Eq. 6 (1981) 883). Some truncation happens to be essential, and in the argument some new results on Liouville-type theorems are established.     

Diffusive Logistic Equations with Degenerate Boundary Conditions

The purpose of this paper is to provide a careful and accessible exposition of static bifurcation theory for a class of degenerate boundary value problems for diffusive logistic equations with indefinite weights that model population dynamics in environments with spatial heterogeneity. We discuss the changes that occur in the structure of the positive solutions as a parameter varies near the first eigenvalue of the linearized problem, and prove that the most favorable situations will occur if there is a relatively large favorable region (with good resources and without crowding effects) located some distance away from the boundary of the environment.Dedicated to Professor Mitsuru Ikawa on the occasion of his 60th birthday