Accession Number : AD0002201


Title :   CONVEXITY OF DOMAIN FUNCTIONALS


Corporate Author : STANFORD UNIV CA APPLIED MATHEMATICS AND STATISTICS LABS


Personal Author(s) : GARABEDIAN, P R ; SCHIFFER, M


Full Text : https://apps.dtic.mil/dtic/tr/fulltext/u2/002201.pdf


Report Date : 04 Feb 1953


Pagination or Media Count : 110


Abstract : A rigorous theory is developed for variation of domain functions in a space of 3 dimensions as well as in the plane. The classical Hadamard variational formulas in space are discussed, and the so-called interior variational method is generalized to 3 dimensions. Interior variations of a 3- dimensional domain D are defined by means of differential mappings of D which depend on a small parameter Epsilon. The first-order shifts in terms of Epsilon of the Green's function, Neumann's function, and eigen-values, which result from this variation of D, are calculated rigorously by referring all varied quantities back to the original D through the infinitesimal mappings. Since proof is possible that the varied domain functions can be expanded in powers of Epsilon, the perturbation method is employed to calculate the second variations of these functions. Second variation expressions are obtained for the capacity, virtual mass, and eigenvalues corresponding to various particular ways in which D can be shifted. The variational theory is specialized to the case of 2 independent variables to show the existence of vortex sheets in axially symmetric, irrotational flow of an incompressible fluid. An external characterization of vortex sheets in 3-dimensional space without symmetry of any kind is sketched heuristically. In a study of the eigen functions and eigenvalues of the vibrating membrane, the second variation is used to show that under certain conformal mappings of a domain, which depend on a suitable parameter, the inverse square of the principal frequency of the domain becomes a convex function of the parameter.


Descriptors :   *VARIATIONAL METHODS , VIBRATION , PARAMETERS , QUANTITY , THEORY , EIGENVECTORS , EIGENVALUES , INTERNAL , SHIFTING , MEMBRANES , PERTURBATIONS , FLUIDS , INVERSION , INCOMPRESSIBILITY , CONVEX BODIES


Subject Categories : Numerical Mathematics


Distribution Statement : APPROVED FOR PUBLIC RELEASE