Boundary Problems For Elliptic Equations Research Articles

Generalized interior shock layer solution of nonlinear singularly perturbed elliptic equation for higher order

A class of singular perturbation boundary problem of elliptic equation for higher order is studied. Firstly, under suitable conditions, the generalized outer solution to reduced problems is considered. Then the generalized interior shock asymptotic solution to original problem is constructed by using the theory of generalized functions. Finally, using the fixed point theorem of functional analytic, the uniformly validity of the generalized asymptotic solution with interior shock and boundary layers is proved.

A singular perturbation free boundary problem for elliptic equations in divergence form

In this paper we study the free boundary problem arising as a limit as ɛ → 0 of the singular perturbation problem \({\textrm{div}(A(x)\nabla u) = \Gamma(x) \beta_\varepsilon(u)}\) , where A = A(x) is Holder continuous, βɛ converges to the Dirac delta δ0. By studying some suitable level sets of uɛ, uniform geometric properties are obtained and show to hold for the free boundary of the limit function. A detailed analysis of the free boundary condition is also done. At last, using very recent results of Salsa and Ferrari, we prove that if A and Γ are Lipschitz continuous, the free boundary is a C1,γ surface around \({\mathcal{H}^{N-1}}\) a.e. point on the free boundary.

Estimates for Solutions to Model Problems Connected with Quasistationary Approximation for the Stefan Problem

Model problems for elliptic equations are considered. The time-derivative of a solution can be contained in the boundary condition or in the conjugation condition. Suich problems appear, for example, in the study of free boundary problems for elliptic equations that can be considered as quasistationary approximations to free boundary problems for parabolic equations. Estimates for solutions to the model problems are obtained. Bibliography: 11 titles.

A free boundary problem of elliptic equation arising in supersonic flow past a conical body

We study free boundary value problems of elliptic equation caused by a supersonic flow past a non-symmetric conical body. The flow is described by the potential flow equation. In the self-similar coordinate system the problem can be reduced to a boundary value problem of second order nonlinear elliptic equation with a free boundary. Applying the partial hodograph transformation and the method of nonlinear alternative iteration we proved the existence of solution to this boundary value problem. Consequently, we also proved the conclusion that for the problem of supersonic flow past a conical body, if the conical body is slightly different from a circular cone with its vertex angle less than a given value determined by the parameters of the coming flow, then there exists a weak entropy solution with an attached conical shock.

INVESTIGATION OF POTENTIALS ASSOCIATED WITH BOUNDARY PROBLEMS FOR ELLIPTIC EQUATIONS

We consider potentials whose kernels are generalizations of the well-known functions of Levi for general boundary problems. We ascertain in which Holder spaces the operators act which are generated by these potentials. The results obtained are applied to the construction of Green's functions.

ON ELLIPTIC EQUATIONS

This paper, like the note on integral geometry in the last number of the “Uspekhi” , is an addendum to my paper [1]. The main idea of the paper is contained in § 2, where we pose the problem of describing linear elliptic equations and their boundary problems in topological terms. The most important of the properties in the large of the solutions of these equations and problems are preserved under small deformations of the problem and must therefore be, in some sense, homotopy invariants. The discovery and study of these invariants is the right way to sort out the whole multiplicity of boundary problems for elliptic equations and to classify these problems. § 1 is introductory, and the reader may skim through it lightly if he wishes. In this section we introduce the basic definitions and notations and, in preparation for the discussion of general boundary problems for elliptic equations in § 2, we advance some general considerations about the nature of these problems. In the preparation of this paper M. S. Agranovich and Z. Ya. Shapiro have given me considerable help, and I take this opportunity of expressing my thanks to them.