Two-dimensional linear partial differential equations in a convex polygon

摘要

A method is introduced for solving boundary-value problems for linear partial differential equations (PDEs) in convex polygons. It consists of three algorithmic steps. (1) Given a PDE, construct two compatible eigenvalue equations. (2) Given a polygon, perform the simultaneous spectral analysis of these two equations. This yields an integral representation in the complex k-plane of the solution q(x(1),x(2)) in terms of a function (q) over cap (k), and an integral representation in the (x(1),x(2))-plane of (q) over cap (k) in terms of the values of q and of its derivatives on the boundary of the polygon. These boundary values are in general related, thus only some of them carl be prescribed. (3) Given appropriate boundary conditions, express the part of (q) over cap (k) involving the unknown boundary values in terms of the boundary conditions. This is based on the existence of a simple global relation formulated in the complex k-plane, and on the invariant properties of this relation. As an illustration, the following integral representations are obtained: (a) q(x, t) for a general dispersive evolution equation of order n in a domain bounded by a linearly morling boundary; (b) q(x, y) for the Laplace, modified Helmholtz and Helmholtz equations in a convex polygon. These general formulae and the analysis of the associated global relations are used to discuss typical boundary-value problems for evolution equations and for elliptic equations. [References: 22]