A short proof of the existence of the solution to elliptic boundary problem

<p>There are several methods for proving the existence of the solution to the elliptic boundary problem \(Lu=f \text{in} D,\quad u|_S=0,\quad (*)\). Here <em>L</em> is an elliptic operator of second order, f is a given function, and uniqueness of the solution to problem (*) is assumed. The known methods for proving the existence of the solution to (*) include variational methods, integral equation methods, method of upper and lower solutions. In this paper a method based on functional analysis is proposed. This method is conceptually simple. It requires some a priori estimates and a continuation in a parameter method, which is well-known.</p>

Eigenvalue asymptotics for an elliptic boundary problem

We consider an elliptic boundary problem in a bounded region Ω ⊂ ℝn wherein the spectral parameter is multiplied by a real-valued weight function with the property that it, together with its reciprocal, is essentially bounded in Ω. The problem is considered under limited smoothness assumptions and under an ellipticity with parameter condition. Then, fixing our attention upon the operator induced on L2(Ω) by the boundary problem under null boundary conditions, we establish results pertaining to the asymptotic behaviour of the eigenvalues of this operator under weaker smoothness assumptions than have hitherto been supposed.

An elliptic boundary problem involving a semi-definite weight

We consider an elliptic boundary problem defined in a bounded region Ω ⊂ Rn and where the spectral parameter is multiplied by a weight function ω(x). We suppose that ω(x) ≠ 0 for x ∈ Ω, but vanishes in a specified manner on the boundary of Ω. Under limited smoothness assumptions, we derive results pertaining to existence and uniqueness of and a priori estimates for solutions of the boundary problem. If S(λ) denotes the operator pencil induced in L2(Ω) by the boundary problem with zero boundary conditions, then results are also derived pertaining to the spectral properties of S(λ).

An elliptic boundary problem involving an indefinite weight

The spectral theory for non-self-adjoint elliptic boundary problems involving an indefinite weight function has only been established for the case of higher-order operators under the assumption that the reciprocal of the weight function is essentially bounded. In this paper we are concerned with the spectral theory for a case where the weight function vanishes on a set of positive measure.