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(Version 3) We study the problem of finding functions, defined within and onan ellipse, whose Laplacian is -1 and which satisfy a homogeneous Robinboundary condition on the ellipse. The parameter in the Robin condition isdenoted by beta. The general solution and various asymptotic approximations areobtained. The integral of the solution over the ellipse, denoted by Q, is aquantity of interest in some physical applications. The dependence of Q on betaand the ellipse geometry is found. Several methods are used.

Functions with constant Laplacian satisfying Robin boundary conditions on an ellipse