Spherical Means for PDEs

Abstract

Part 1 Scalar second-order PDEs: spherical mean-value relations for the Laplace equation and integral formulation of the Dirichlet problem direct spherical mean-value relation converse mean-value theorem integral equation equivalent to the Dirichlet problem Poisson-Jensen formula the diffusion and Helmholtz equations diffusion equation Helmholtz equation generalized second-order elliptic equations parabolic equations heat equation parabolic equation with variable co-efficients expansion of the parabolic means. Part 2 High-order elliptic equations: Balayage operator the biharmonic equation direct spherical mean-value relation the generalized Poisson formula rigid fixing of the boundary non-homogeneous biharmonic equation fourth-order equation governing the bending of a plate on an elastic base surface metaharmonic equations polyharmonic equation general case. Part 3 Triangular systems of elliptic equations: a one-component diffusion system a two-component diffusion system a coupled biharmonic-harmonic equation. Part 4 Systems of elasticity theory: the Lame equation direct spherical mean-value theorem converse spherical mean-value theorem pseudo-vibration elastic equation thermo-elastic equation. Part 5 The generalized Poisson formula for the Lame equation: plane elasticity generalized spatial Poisson formula for the Lame equation an alternative derivation of the Poisson formula. Part 6 Spherical means for the stress and strain tensors. Part 7 Applications to the random-walk-on-spheres method. (Part contents)