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.     

Dirichlet problems on varying domains

The aim of the paper is to characterise sequences of domains for which solutions to an elliptic equation with Dirichlet boundary conditions converge to a solution of the corresponding problem on a limit domain. Necessary and sufficient conditions are discussed for strong and uniform convergence for the corresponding resolvent operators. Examples are given to illustrate that most results are optimal.     

Initial boundary value problem for a nonconservative system in elastodynamics

This article is concerned with the initial boundary value problem for a nonconservative system of hyperbolic equation appearing in elastodynamics in the space time domain x > 0, t > 0. The number of boundary conditions, to be prescribed at the boundary x = 0, depends on the number of characteristics entering the domain. Because our system is nonlinear, the characteristic speeds depends on the unknown and the direction of the characteristics curves are known apriori. As it is well known, the boundary condition has to be understood in a generalised way. One of the standard way is using vanishing viscosity method. We use this method to construct solution for a particular class of initial and boundary data, namely the initial and boundary datas that lie on the level sets of one of the Riemann invariants.     

Analytical methods for an elliptic singular perturbation problem in a circle

We consider an elliptic perturbation problem in a circle by using the analytical solution that is given by a Fourier series with coefficients in terms of modified Bessel functions. By using saddle point methods we construct asymptotic approximations with respect to a small parameter. In particular we consider approximations that hold uniformly in the boundary layer, which is located along a certain part of the boundary of the domain.     

Some uniqueness and continuous dependence theorems for nonlinear elastodynamics in exterior domains

AMS (MOS) Nos. 73C50, 73C15; 35B30, 35M05

Uniqueness and Höolder continuous dependence upon the initial data of the null solution to the initial boundary value problem of nonlinear hyperelasticity are proved for exterior domains subject to mild asymptotic behaviour on the displacement, velocity and stress components. The strain-energy is not required to be locally sign-definite although at sufficiently large spatial distances it must be non-negative. Other limitations imposed on the strain-energy become identically satisfied upon reduction to the linear theory.     

Global solvability for the Kirchhoff equations in exterior domains of dimension three

We consider the initial (boundary) value problem for the Kirchhoff equations in exterior domains or in the whole space of dimension three, and show that these problems admit time-global solutions, provided the norms of the initial data in the usual Sobolev spaces of appropriate order are sufficiently small. We obtain uniform estimates of the L¹(R) norms with respect to time variable at each point in the domain, of solutions of initial (boundary) value problem for the linear wave equations. We then show that the estimates above yield the unique global solvability for the Kirchhoff equations.     

The behavior of the best Sobolev trace constant and extremals in thin domains

In this paper, we study the asymptotic behavior of the best Sobolev trace constant and extremals for the immersion W1,p(Ω)?Lq(∂Ω) in a bounded smooth domain when it is contracted in one direction. We find that the limit problem, when rescaled in a suitable way, is a Sobolev-type immersion in weighted spaces over a projection of Ω, W1,p(P(Ω),α)?Lq(P(Ω),β).For the special case p=q, this problem leads to an eigenvalue problem with a nonlinear boundary condition. We also study the convergence of the eigenvalues and eigenvectors in this case.     

Eigenvalue asymptotics for an elliptic boundary problem involving an indefinite weight

We are concerned here with the eigenvalue asymptotics for a non-selfadjoint elliptic boundary problem involving an indefinite weight function which vanishes on a set of positive measure. The asymptotic behaviour of the eigenvalues is well known for the case of second order operators. However for higher order operators, results have only been established under the restriction that the order of the operator exceeds the dimension of the underlying Euclidean space in which the problem is set. In this paper we establish the eigenvalue asymptotics for the case of higher order operators without any such restriction.Supported in part by the John Knopfmacher Centre for Applicable Analysis and Number Theory, University of the Witwatersrand.     

Superlinear elliptic equations with singular coefficients on the boundary

In this paper, a superlinear elliptic equation whose coefficient diverges on the boundary is studied in any bounded domain Ω under the zero Dirichlet boundary condition. Although the equation has a singularity on the boundary, a solution is smooth on the closure of the domain. Indeed, it is proved that the problem has a positive solution and infinitely many solutions without positivity, which belong to or . Moreover, it is proved that a positive solution has a higher order regularity up to .     

Rearrangements and minimization of the principal eigenvalue of a nonlinear Steklov problem

This paper, motivated by Del Pezzo et al. (2006) [1], discusses the minimization of the principal eigenvalue of a nonlinear boundary value problem. In the literature, this type of problem is called Steklov eigenvalue problem. The minimization is implemented with respect to a weight function. The admissible set is a class of rearrangements generated by a bounded function. We merely assume the generator is non-negative in contrast to [1], where the authors consider weights which are positively away from zero, in addition to being two-valued. Under this generality, more physical situations can be modeled. Finally, using rearrangement theory developed by Geoffrey Burton, we are able to prove uniqueness of the optimal solution when the domain of interest is a ball.     

On a Darboux type multidimensional problem for second-order hyperbolic systems

The correct formulation of a Darboux type multidimensional problem for second-order hyperbolic systems is investigated. The correct formulation of such a problem in the Sobolev space is proved for temporal type surfaces on which the boundary conditions of a Darboux type problem are given.     

Stability of supersonic contact discontinuity for two-dimensional steady compressible Euler flows in a finite nozzle

In this paper, we study the stability of supersonic contact discontinuity for the two-dimensional steady compressible Euler flows in a finitely long nozzle of varying cross-sections. We formulate the problem as an initial–boundary value problem with the contact discontinuity as a free boundary. To deal with the free boundary value problem, we employ the Lagrangian transformation to straighten the contact discontinuity and then the free boundary value problem becomes a fixed boundary value problem. We develop an iteration scheme and establish some novel estimates of solutions for the first order of hyperbolic equations on a cornered domain. Finally, by using the inverse Lagrangian transformation and under the assumption that the incoming flows and the nozzle walls are smooth perturbations of the background state, we prove that the original free boundary problem admits a unique weak solution which is a small perturbation of the background state and the solution consists of two smooth supersonic flows separated by a smooth contact discontinuity.     

Concentrating solutions for a planar elliptic problem involving nonlinearities with large exponent

We consider the boundary value problem Δu+up=0 in a bounded, smooth domain Ω in R² with homogeneous Dirichlet boundary condition and p a large exponent. We find topological conditions on Ω which ensure the existence of a positive solution up concentrating at exactly m points as p→∞. In particular, for a nonsimply connected domain such a solution exists for any given m?1.     

Existence and instability of spike layer solutions to singular perturbation problems

An abstract framework is given to establish the existence and compute the Morse index of spike layer solutions of singularly perturbed semilinear elliptic equations. A nonlinear Lyapunov-Schmidt scheme is used to reduce the problem to one on a normally hyperbolic manifold, and the related linearized problem is also analyzed using this reduction. As an application, we show the existence of a multi-peak spike layer solution with peaks on the boundary of the domain, and we also obtain precise estimates of the small eigenvalues of the operator obtained by linearizing at a spike layer solution.     

Rayleigh-type isoperimetric inequality with a homogeneous magnetic field

We prove that the two dimensional free magnetic Schrödinger operator, with a fixed constant magnetic field and Dirichlet boundary conditions on a planar domain with a given area, attains its smallest possible eigenvalue if the domain is a disk. We also give some rough bounds on the lowest magnetic eigenvalue of the disk.This article was processed by the author using the style filepljour1m from Springer-Verlag.     

On the solvability of a Darboux type non-characteristic spatial problem for the wave equation

The question of the correct formulation of a Darboux type non-characteristic spatial problem for the wave equation is investigated. The correct solvability of the problem is proved in the Sobolev space for surfaces of the temporal type on which Darboux type boundary conditions are given.     

Energy minimizing harmonic maps with an obstacle at the free boundary

We consider partial regularity for energy minimizing maps satisfying a partially free boundary condition. This condition takes the form of the requirement that a relatively open subset of the boundary of the domain manifold be mapped into a closed submanifold with non-empty boundary, contained in the target manifold. We obtain an optimal estimate on the Hausdorff dimension of the singular set of such a map. Our result can be interpreted as regularity result for a vector-valued Signorini, or thin-obstacle, problem.     

On chordal and bilateral SLE in multiply connected domains

We discuss the possible candidates for conformally invariant random non-self-crossing curves which begin and end on the boundary of a multiply connected planar domain, and which satisfy a Markovian-type property. We consider both, the case when the curve connects a boundary component to itself (chordal), and the case when the curve connects two different boundary components (bilateral). We establish appropriate extensions of Loewner’s equation to multiply connected domains for the two cases. We show that a curve in the domain induces a motion on the boundary and that this motion is enough to first recover the motion of the moduli of the domain and then, second, the curve in the interior. For random curves in the interior we show that the induced random motion on the boundary is not Markov if the domain is multiply connected, but that the random motion on the boundary together with the random motion of the moduli forms a Markov process. In the chordal case, we show that this Markov process satisfies Brownian scaling and discuss how this limits the possible conformally invariant random non-self-crossing curves. We show that the possible candidates are labeled by two functions, one homogeneous of degree zero, the other homogeneous of degree minus one, which describes the interaction of the random curve with the boundary. We show that the random curve has the locality property for appropriate choices of the interaction term. The research of the first author was supported by NSA grant H98230-04-1-0039. The research of the second author was supported by a grant from the Max-Planck-Gesellschaft.     

Optimal control for a parabolic problem in a domain with highly oscillating boundary

In this article we study the homogenization of an optimal control problem for a parabolic equation in a domain with highly oscillating boundary. We identify the limit problem, which is an optimal control problem for the homogenized equation and with a different cost functional.     

Large-Time Behaviour of Solutions of the Exterior-Domain Cauchy–Dirichlet Problem for the Porous Media Equation with Homogeneous Boundary Data

The title of this paper states precisely what the subject is. The first part of the paper concerns the radially-symmetric problem in the exterior of the unit ball. It is shown that in time the solution of the problem converges to one of two specific self-similar solutions of the porous media equation, dependent upon the dimensionality of the problem. Moreover, the free boundary of the solution converges to that of the self-similar solution. The critical space dimension is two, for which there is no distinction between the self-similar solutions, and the form of the convergence is exceptional. The technique used is a comparison principle involving a variable that is a weighted integral of the solution. The second part of the paper is devoted to the problem in an arbitrary spatial domain with no conditions of symmetry. A special invariance principle and the results obtained for the radially-symmetric case are used to determine the large-time behaviour of solutions and their free boundaries. This behaviour is decidedly different from when the boundary data are fixed and not homogeneous.