Research Projects

Let u be an eigenfunction of the Laplacian with Dirichlet boundary conditions on a planar doubly connected domain D. In 1960, Weinberger proved that there exists a closed curve (called "effectless cut''}), consisting of a finite number of analytic curves along which the normal derivative of u vanishes. Moreover, this curve separates the inner and the outer boundaries of D. The eigenvalues of Laplacian on these subdomains (separated by this curve) with mixed boundary conditions are the same and coincide with the first Dirichlet eigenvalue. We would like to explore the existence of such curves in the higher dimensional domains. Recently, several authors studied similar problems that deal with the identification of more such curves (Neuman lines) and some special domains (Neumann domains) for the Morse functions. These special domains and curves are identified using the stable and unstable manifolds of critical points of ??.