Some distribution solutions of integral equations and their application to partial differential equations

Abstract

We consider two integral operators, L and L ^k defined by

Let L ²(p)(L ₂(q)) be the space of functions defined on [−1, 1] and integrable with respect to the weight function (1−x ²)^−½((1−x ²)½) . Let W₂ ¹(q) be the space of functions, f, absolutely continuous on [−1,1] with f ∈ L ₂(q) and W ₂ ⁻¹(q) be its dual. It has previously been shown that L and L _k are one to one, continuous maps of L ₂(q) onto W ₂ ^l(q). Here we show that these mappings can be extended to mappings L and L _k which are one-to-one continuous maps of W ₂ ⁻¹(q) onto L₂(p). These results are applied to the problem of solving the two dimensional Laplace and Helmholtz equations with boundary data given on the interval [−1,1] of the x axis.

^†University of Maryland, College Park, Md. 20742. This research was supported in part by the National Science Foundation under Grant GP 12838 with the University of Maryland.

^†University of Maryland, College Park, Md. 20742. This research was supported in part by the National Science Foundation under Grant GP 12838 with the University of Maryland.

Notes

^†University of Maryland, College Park, Md. 20742. This research was supported in part by the National Science Foundation under Grant GP 12838 with the University of Maryland.