A Riemann-Hilbert approach to the Laplace equation (Q1593773)

Let \(q(x,y)\) be a solution of the Laplace equation in an arbitrary convex polygon and let \(Q(z)= q_x- iq_y\) \((z= x+ iy)\) which thus satisfies the analyticity condition \(\partial Q/\partial\overline z= 0\). In a preceding paper [``On a transform method for the Laplace equation in a polygon'', Preprint (2000)], the authors have introduced a transform method for solving the Laplace equation in an arbitrary convex polygon. This method is based on the fact that a function \(Q(z)\), holomorphic in the interior of a convex polygon, admits a simple integral representation; cf. the main result, Theorem 2.1, in the present paper (which is explicitly proved here). This integral representation yields \(q(x,y)\) in terms of a certain function \(\rho(k)\) (``spectral function''), which is expressed as an integral over the boundary of the polygon involving \(q_x\) and \(q_y\). The determination of \(\rho(k)\) implies the difficulty that some of the boundary values appearing in \(\rho(k)\) are in general unknown if a well-posed boundary value problem is prescribed. In the present paper, the derivation of the integral representation is performed by undertaking the spectral analysis of the equation \(\mu_z- ik\mu= Q\) (\(=q_x- iq_y\)). This gives rise to a Riemann-Hilbert problem in the complex \(k\)-plane which can be solved in closed form. The paper contains several examples and useful remarks. Besides, a further (well-known) direct method is introduced which also involves a Riemann-Hilbert problem. The applicability of both, this alternative and the above new, methods is discussed. Furthermore, the feasibility of extensions of the new method to more complicated domains (also non-convex domains) and to other linear partial differential equations (e.g., the Helmholtz equation or the biharmonic equation) is emphasized; cf. forthcoming papers by the first author and the co-authors R. Craster and D. Crowdy, respectively.