The boundary-contact problem of elasticity for homogeneous anisotropic media with a contact on some part of the boundaries

The existence and uniqueness of solutions of the boundary-contact problem of elasticity for homogeneous anisotropic media with a contact on some part of their boundaries are investigated in the Besov and Bessel potential classes using the methods of the potential theory and the theory of pseudodifferential equations on manifolds with boundary. The smoothness of the solutions obtained is studied.     

Two-dimensional steady-state oscillation problems of anisotropic elasticity

The paper deals with the two-dimensional exterior boundary value problems of the steady-state oscillation theory for anisotropic elastic bodies. By means of the limiting absorption principle the fundamental matrix of the oscillation equations is constructed and the generalized radiation conditions of Sommerfeld-Kupradze type are established. Uniqueness theorems of the basic and mixed type boundary value problems are proved.     

Viscosity solutions for elliptic-parabolic problems

We study an elliptic-parabolic problem appearing in the theory of partially saturated flows in the framework of viscosity solutions. This is part of current investigation to understand the theory of viscosity solutions for PDE problems involving free boundaries. We prove that the problem is well posed in the viscosity setting and compare the results with the weak theory. Dirichlet or Neumann boundary conditions are considered.     

Traces of functions in Hardy and Besov spaces on Lipschitz domains with applications to compensated compactness and the theory of Hardy and Bergman type spaces

In this paper we study conditions guaranteeing that functions defined on a Lipschitz domain Ω have boundary traces in Hardy and Besov spaces on ∂Ω. In turn these results are used to develop a new approach to the theory of compensated compactness and the theory of non-locally convex Hardy and Bergman type spaces.     

Borel summability of divergent solutions for singular first-order partial differential equations with variable coefficients. Part I

This article part I and the forthcoming part II are concerned with the study of the Borel summability of divergent power series solutions for singular first-order linear partial differential equations of nilpotent type. Under one restriction on equations, we can divide them into two classes. In this part I, we deal with the one class and obtain the conditions under which divergent solutions are Borel summable. (The other class will be studied in part II.) In order to assure the Borel summability of divergent solutions, global analytic continuation properties for coefficients are required despite of the fact that the domain of the Borel sum is local.     

Basic boundary value problems of thermoelasticity for anisotropic bodies with cuts. II

In the first part [1] of the paper the basic boundary value problems of the mathematical theory of elasticity for three-dimensional anisotropic bodies with cuts were formulated. It is assumed that the two-dimensional surface of a cut is a smooth manifold of an arbitrary configuration with a smooth boundary. The existence and uniqueness theorems for boundary value problems were formulated in the Besov and Bessel-potential ( _p ^s ) spaces. In the present part we give the proofs of the main results (Theorems 7 and 8) using the classical potential theory and the nonclassical theory of pseudodifferential equations on manifolds with a boundary.     

On elliptic operator pencils with general boundary conditions

In this paper parameter-dependent partial differential operators are investigated which satisfy the condition of N-ellipticity with parameter, an ellipticity condition formulated with the use of the Newton polygon. For boundary value problems with general boundary operators we define N-ellipticity including an analogue of the Shapiro-Lopatinskii condition. It is show that the boundary value problem is N-elliptic if and only if an a priori estimate with respect to certain parameter-dependent norms holds. These results are closely connected with singular perturbation theory and lead to uniform estimates, for problems of Vishik-Lyusternik type containing a small parameter.Supported in part by the Deutsche Forschungsgemeinschaft and by Russian Foundation of Fundamental Research, Grant 00-01-00387.     

On a system of partial differential equations of Monge-Kantorovich type

We consider a system of PDEs of Monge-Kantorovich type arising from models in granular matter theory and in electrodynamics of hard superconductors. The existence of a solution of such system (in a regular open domain Ω⊂Rn), whose construction is based on an asymmetric Minkowski distance from the boundary of Ω, was already established in [G. Crasta, A. Malusa, The distance function from the boundary in a Minkowski space, Trans. Amer. Math. Soc., submitted for publication]. In this paper we prove that this solution is essentially unique. A fundamental tool in our analysis is a new regularity result for an elliptic nonlinear equation in divergence form, which is of some interest by itself.     

Basic boundary value problems of thermoelasticity for anisotropic bodies with cuts. I

The three-dimensional problems of the mathematical theory of thermoelasticity are considered for homogeneous anisotropic bodies with cuts. It is assumed that the two-dimensional surface of a cut is a smooth manifold of an arbitrary configuration with a smooth boundary. The existence and uniqueness theorems for boundary value problems of statics and pseudo-oscillations are proved in the Besov ( ) and Bessel-potential ( ) spaces by means of the classical potential methods and the theory of pseudodifferential equations on manifolds with boundary. Using the embedding theorems, it is proved that the solutions of the considered problems are Hölder continuous. It is shown that the displacement vector and the temperature distribution function areC -regular with any exponent <1/2.This paper consists of two parts. In this part all the principal results are formulated. The forthcoming second part will deal with the auxiliary results and proofs.     

Generalized dirac operators on nonsmooth manifolds and Maxwell's equations

We develop a function theory associated with Dirac type operators on Lipschitz subdomains of Riemannian manifolds. The main emphasis is on Hardy spaces and boundary value problems, and our aim is to identify the geometric and analytic assumptions guaranteeing the validity of basic results from complex function theory in this general setting. For example, we study Plemelj-Calderón-Seeley-Bojarski type splittings of Cauchy boundary data into traces of ‘inner’ and ‘outer’ monogenics and show that this problem has finite index. We also consider Szegö projections and the corresponding L^p-decompositions. Our approach relies on an extension of the classical Calderón-Zygmund theory of singular integral operators which allow one to consider Cauchy type operators with variable kernels on Lipschitz graphs. In the second part, where we explore connections with Maxwell's equations, the main novelty is the treatment of the corresponding electro-magnetic boundary value problem by recasting it as a ‘half’ Dirichlet problem for a suitable Dirac operator.     

Asymptotic stability of viscous contact discontinuity to an inflow problem for compressible Navier-Stokes equations

This paper is concerned with an initial-boundary value problem for one-dimensional full compressible Navier-Stokes equations with inflow boundary conditions in the half space R₊=(0,+∞). The asymptotic stability of viscous contact discontinuity is established under the conditions that the initial perturbations and the strength of contact discontinuity are suitably small. Compared with the free-boundary and the initial value problems, the inflow problem is more complicated due to the additional boundary effects and the different structure of viscous contact discontinuity. The proofs are given by the elementary energy method.     

Sturm–Liouville problems with parameter dependent potential and boundary conditions

This paper deals with the computation of the eigenvalues of Sturm–Liouville problems with parameter dependent potential and boundary conditions. We shall extend the domain of application of the method based on sampling theory to the case where the classical Whittaker–Shannon–Kotel’nikov theorem is not applicable. A few numerical examples will be presented.     

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.     

Stability of hyperbolic–parabolic mixed type equations with partial boundary condition

In this paper, we are concerned with the hyperbolic–parabolic mixed type equations with the non-homogeneous boundary condition. If it is degenerate on the boundary, the part of the boundary whose boundary value should be imposed, is determined by the entropy condition from the convection term. If there is no convection term in the equation, we show that the stability of solutions can be proved without any boundary condition. If the equation is completely degenerate, we show that the stability of solutions can be established just based on the partial boundary condition.     

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.     

Reduction and quotient equations for differential equations with symmetries

The method of reduction previously known in the theory of Hamiltonian systems with symmetries is developed in order to obtain exact group-invariant solutions of systems of partial differential equations. This method leads to representations of quotient equations which are very convenient for the systematic analysis of invariant solutions of boundary value problems. In the case of partially invariant solutions, necessary and sufficient conditions of their invariance with respect to subalgebras of symmetry algebras are given. The concept of partial symmetries of differential equations is considered.     

Boundary value problems for rotationally symmetric mean curvature flows

The motion of surfaces by their mean curvature has been studied by several authors from different points of view. K. A. Brake studied this problem from the geometric measure theory point of view, the parametric problem was studied by G. Huisken [5]. Nonparametric mean curavture flow with boundary conditions was studied in [6] and [7]. Rotationally symmetric mean curvature flows have been treated by G. Dziuk, B. Kawohl [3], but also by S. Altschuler, S. B. Angenent and Y. Giga [2]. In this paper we consider the case in which the initial surface has rotational symmetry and we shall generalize the results in [3] in the sense that we shall give more general boundary conditions which enforce the formation of a singularity in finite time. The proofs rely entirely on parabolic maximum principles. Received: 6 September 2006     

Variational formulations of nonlinear constrained boundary value problems

Variational formulations of nonlinear constrained boundary value problems in reflexive Banach spaces are discussed from a compositional duality approach. The mixed variational compatibility conditions of the theory correspond to the surjectivity of the primal coupling boundary and interior operators.