An index formula for elliptic systems in the plane

An index formula is proved for elliptic systems of P.D.E.'s with boundary values in a simply connected region in the plane. Let denote the elliptic operator and the boundary operator. In an earlier paper by the author, the algebraic condition for the Fredholm property, i.e. the Lopatinskii condition, was reformulated as follows. On the boundary, a square matrix function defined on the unit cotangent bundle of was constructed from the principal symbols of the coefficients of the boundary operator and a spectral pair for the family of matrix polynomials associated with the principal symbol of the elliptic operator. The Lopatinskii condition is equivalent to the condition that the function have invertible values. In the present paper, the index of is expressed in terms of the winding number of the determinant of .

     

An index formula for the product of linear relations

Let , and be linear spaces and let A and B be linear relations from to and from to , respectively. The main result of this note is a formula which relates the nullities and the defects of the relations A and B with those of the product relation BA.     

On the index instability for some nonlocal elliptic problems

The index of unbounded operators defined on generalized solutions of nonlocal elliptic problems in plane bounded domains is investigated. It is known that nonlocal terms with smooth coefficients having zero of a certain order at the conjugation points do not affect the index of the unbounded operator. In this paper, we construct examples showing that the index may change under nonlocal perturbations with coefficients not vanishing at the points of conjugation of boundary-value conditions. __________ Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 26, pp. 178–193, 2007.     

Asymptotically linear elliptic problems at resonance

Summary In this paper we study the existence of solutions for some semilinear, elliptic equation with homogeneous Dirichelet boundary conditions and a non linear term which is asymptotically linear «at resonance».In memoria di Paolo Bartolo     

A generalized Green's formula for elliptic problems in domains with edges

The usual Green's formula connected with the operator of a boundary-value problem fails when both of the solutions u and v that occur in it have singularities that are too strong at a conic point or at an edge on the boundary of the domain. We deduce a generalized Green's formula that acquires an additional bilinear form in u and v and is determined by the coefficients in the expansion of solutions near singularities of the boundary. We obtain improved asymptotic representations of solutions in a neighborhood of an edge of positive dimension, which together with the generalized Green's formula makes it possible, for example, to describe the infinite-dimensional kernel of the operator of an elliptic problem in a domain with edge. Bibliography: 14 titles. Translated fromProblemy Matematicheskogo Analiza, No. 13, 1992, pp. 106–147.     

Multiplicity results for asymptotically linear elliptic problems at resonance

In this paper several new multiplicity results for asymptotically linear elliptic problem at resonance are obtained via Morse theory and minimax methods. Some new observations on the critical groups of a local linking-type critical point are used to deal with the resonance case at 0.     

Quasilinear elliptic problems under strong resonance conditions

In this paper we establish the existence and multiplicity of solutions for a quasilinear elliptic problem under strong resonance conditions at infinity. In order to control the resonance we consider a new hypothesis on the nonlinear term. In all results we use Variational Methods, Critical Groups and the Morse Theory.     

An extended Galerkin analysis for elliptic problems

A general analysis framework is presented in this paper for many different types of finite element methods(including various discontinuous Galerkin methods). For the second-order elliptic equation-div(α▽u) = f, this framework employs four different discretization variables, uh, p_h, uhand p_h, where uh and phare for approximation of u and p =-α▽u inside each element, and uhand phare for approximation of the residual of u and p·n on the boundary of each element. The resulting 4-field discretization is proved to satisfy two types of inf-sup conditions that are uniform with respect to all discretization and penalization parameters. As a result, many existing finite element and discontinuous Galerkin methods can be analyzed using this general framework by making appropriate choices of discretization spaces and penalization parameters.     

Semilinear degenerate elliptic boundary value problems at resonance

The purpose of this paper is to study a class of semilinear degenerate elliptic boundary value problems at resonance which include as particular cases the Dirichlet and Robin problems. The approach here is based on the global inversion theorems between Banach spaces, and is distinguished by the extensive use of the ideas and techniques characteristic of the recent developments in the theory of partial differential equations. By making use of the Lyapunov–Schmidt procedure and the global inversion theorem, we prove existence and uniqueness theorems for our problem. The results here extend an earlier theorem due to Landesman and Lazer to the degenerate case.     

An expanded mixed covolume method for elliptic problems

We consider the mixed covolume method combining with the expanded mixed element for a system of first‐order partial differential equations resulting from the mixed formulation of a general self‐adjoint elliptic problem with a full diffusion tensor. The system can be used to model the transport of a contaminant carried by a flow in porous media. We use the lowest order Raviart‐Thomas mixed element space. We show the first‐order error estimate for the approximate solution in L² norm. We show the superconvergence both for pressure and velocity in certain discrete norms. We also get a finite difference scheme by using proper approximate integration formulas. Finally we give some numerical examples. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005