Accession Number : AD0019095


Title :   CONVESITY OF FUNCTIONALS BY TRANSPLANTATION


Corporate Author : STANFORD UNIV CA APPLIED MATHEMATICS AND STATISTICS LABS


Personal Author(s) : Polya, G ; Schiffer, M ; Helfenstein, Heinz


Full Text : https://apps.dtic.mil/dtic/tr/fulltext/u2/019095.pdf


Report Date : 19 Oct 1953


Pagination or Media Count : 125


Abstract : The dependence of various functionals on their domain of definition is discussed. The functionals are defined by certain extremum problems. The methods of transplanting extremum functions and of variation are applied to the problem of utilizing the knowledge of the functional for a few special domains to obtain knowledge about the same functional in the general case. Functionals such as torsional rigidity, virtual mass, outer conformal radius, and electrostatic capacity are treated. A discussion is given of a theorem of Poincare which permits an easy simultaneous estimation of the N first eigenvalues of a general type of eigenvalue problem. The convexity of various combinations of eigenvalues is studied for the case in which the domain of definition is deformed by stretching or by conformal transformation. The usefulness of the fact that the initial domain D(1) has symmetry properties is indicated. The invariance of the class of harmonic functions in D under conformal mapping can be used to derive convexity statements for some functionals connected with the Green's function for Laplace's equation. A numerical application is given for the torsional rigidity of isosceles triangles and rectangles.


Descriptors :   *CONFORMAL MAPPING , HARMONICS , NUMERICAL ANALYSIS , EIGENVALUES , ESTIMATES , SYMMETRY , EQUATIONS , GREENS FUNCTIONS , INVARIANCE , RIGIDITY , ELECTROSTATICS , RADIUS(MEASURE) , TORSION , TRANSFORMATIONS , CONVEX SETS , LAPLACE TRANSFORMATION , CONFORMAL STRUCTURES , FUNCTIONS , CAPACITY(QUANTITY)


Subject Categories : Theoretical Mathematics


Distribution Statement : APPROVED FOR PUBLIC RELEASE