CONVESITY OF FUNCTIONALS BY TRANSPLANTATION
STANFORD UNIV CA APPLIED MATHEMATICS AND STATISTICS LABS
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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 D1 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 Greens function for Laplaces equation. A numerical application is given for the torsional rigidity of isosceles triangles and rectangles.
- Theoretical Mathematics