APPLICATION OF THE BUBNOV-GALERKIN METHOD TO NONLINEAR STATIONARY MAGNETIC FIELD PROBLEMS WITH INHOMOGENEOUS BOUNDARY CONDITIONS

A numerical procedure based on the Bubnov-Galerkin method is presented for the approximate solution of a nonlinear stationary magnetic field problem with inhomogeneous boundary conditions. Neumann-type boundary conditions are included in the operator equation describing the boundary value problem. Their incorporation is presented both with the aid of a special functional Hilbert space and by the use of generalized functions. The Dirichlet- type boundary conditions are reduced to ones of Neumann type

Diagonalization of Non-selfadjoint Analytic Semigroups and Application to the Shape Memory Alloys Operator

AbstractTo a densely defined, but not necessarily selfadjoint, operator A on a Hilbert space H we consider on R+×H the following abstract “elliptic” problem of Dirichlet type:[formula] Then, in this paper, we establish that for every t>0, the solution [formula] can be expanded into a series of generalized eigenvectors of the operator A provided that its resolvent belongs to Carleman class Cp for some p∈]0,12[. A similar result holds for t large enough if the inverse A−1 belongs to Carleman class Cp for every p>12. (See Theorem 3.1 and Theorem 3.2.) Furthermore, we apply these obtained results to the shape memory alloys non-selfadjoint operator [formula] and Dn=∂n/∂xn when acting on an appropriate Hilbert space E of functions on the interval [0,1], by establishing that the inverse C−1 belongs to the Carleman class Cp for every p>12, so that we get in this case more regularity in the sense that the operatorial series ∑∞k=1etCPk converges strongly in E to the analytic semigroup etC for every t>0 (the Pk are the projectors into the root subspaces of C). A similar result holds for [formula] provided that t is large enough. (See Theorem 4.1 and Theorem 4.2 in Section 4 for the precise result.

A remark on Schatten-von Neumann properties of resolvent differences of generalized Robin Laplacians on bounded domains

In this note we investigate the asymptotic behaviour of the $s$-numbers of the resolvent difference of two generalized self-adjoint, maximal dissipative or maximal accumulative Robin Laplacians on a bounded domain $\Omega$ with smooth boundary $\partial\Omega$. For this we apply the recently introduced abstract notion of quasi boundary triples and Weyl functions from extension theory of symmetric operators together with Krein type resolvent formulae and well-known eigenvalue asymptotics of the Laplace-Beltrami operator on $\partial\Omega$. It will be shown that the resolvent difference of two generalized Robin Laplacians belongs to the Schatten-von Neumann class of any order $p$ for which $p>(dim\Omega-1)/3$. Moreover, we also give a simple sufficient condition for the resolvent difference of two generalized Robin Laplacians to belong to a Schatten-von Neumann class of arbitrary small order. Our results extend and complement classical theorems due to M.Sh.Birman on Schatten-von Neumann properties of the resolvent differences of Dirichlet, Neumann and self-adjoint Robin Laplacians