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Two-dimensional linear partial differential equations in a convex polygon. (English) Zbl 0988.35129
From the author’s abstract: 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 repesentation in the complex \(k\)-plane of the solution \(q(x_1,x_2)\) in terms of a function \(\widehat q (k)\), and an integral representation in the \((x_1,x_2)\)-plane of \(\widehat q (k)\) in terms of the values of \(q\) and its derivatives on the boundary of the polygon. These boundary values are in general related, thus only some of them can be prescribed. (3) Given appropriate boundary conditions, express the part of \(\widehat q (k)\) 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 repesentations are obtained: (a) \(q(x,t)\) for a general dispersive evolution equation of order \(n\) in a domain bounded by a linearly moving boundary; (b) \(q(x,t)\) for the Laplace, modified Helmholtz and Helmholtz equations in a convex polygon.

35Q15 Riemann-Hilbert problems in context of PDEs
35C15 Integral representations of solutions to PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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