Title:

Some problems involving elliptic partial differential equations with mixed boundary conditions

A function phi satisfies the equation phixx+ phigammagamma+R2phi= 0 in a region R bounded by a closed curve C on which mixed boundary conditions are specified, for example phi=0 on a part A of the boundary and, dphi/m=0 on a part B, where = A + B. It is required to find the values of k for which the equation possesses solutions satifying the mixed boundary conditions. Two variational principles are given for these eigenvalues, and conditions are obtained under which these two principles would give upper and lower bounds for the lowest eigenvalue. Transcendental equations, obtained for the determination of the lowest eigenvalue, are shown to be functions of an unknown function which is, for example, the value of phi on the part B of the boundary, or of dphi/dn on part A. If a first order approximation to this function is made, it appears that the resulting approximation to the eigenvalue is of second order. The general theory, obtained for a simple closed curve, is extended to investigate a curve enclosing a certain type of composite region, and it is shown that the conditions under which the two variational principles would give upper and lower bounds are similar to those of the simpler problem. Several problems are worked out in detail and some numerical results are obtained for comparisonwith results obtained by other authors by other methods. It is shown that in all of the chosen problems for which the two variational principles can be given, the conditions that they give upper and lower bounds are satisfied. An alternative method of solution of these problems is given, using conformal transformations in a slightly modified form of Schwinger's "equivalent static problem" method. The paper concludes with a brief note on the application of variational principles to mixed boundary problems in potential theory.
