Finite energy solutions of self-adjoint elliptic mixed boundary value problems

Abstract

This paper describes existence, uniqueness and special eigenfunction representations of H 1 -solutions of second order, self-adjoint, elliptic equations with both interior and boundary source terms. The equations are posed on bounded regions with Dirichlet conditions on part of the boundary and Neumann conditions on the complement. The system is decomposed into separate problems defined on orthogonal subspaces of H 1 ( ). One problem involves the equation with the interior source term and the Neumann data. The other problem just involves the homogeneous equation with Dirichlet data. Spectral representations of the solution operators for each of these problems are found. The solutions are described using bases that are, respectively, eigenfunctions of the differential operator with mixed null boundary conditions, and certain mixed Steklov eigenfunctions. These series converge strongly in H 1 ( ). Necessary and sufficient conditions for the Dirichlet part of the boundary data to have a finite energy extension are described. The solutions for a problem that models a cylindrical capacitor is found explicitly.

Spectral representations, and approximations, of divergence-free vector fields

“…These come from the expansion of the solution in terms of the basis of harmonic Σ−Steklov eigenfunctions. Similar results for elliptic boundary value problems have been described by the first author in [4] and [7].…”

Spatial‐temporal security stability analysis and regulation against different types of attacks on industrial control systems

“…Here , $$$ \nabla y $$$ is the gradient of $$$ y $$$ with $$$ \nabla y=\partial y/\partial {s}_i $$$. The constant $$$ {C}_F $$$ (also called Friedrich constant) is the least positive eigenvalue $$$ {\lambda}_{\mathrm{min}} $$$ of the following eigenproblem $$$$ -\Delta x=\Lambda \kern0.3em x\kern0.60em \mathrm{in}\kern0.3em \Omega, \kern1.20em \left(\nabla x\right)\cdotp \nu +x=0\kern0.60em \mathrm{on}\kern0.3em \mathrm{\partial \Omega }, $$$$ where $$$ \Delta x $$$ is the Laplacian operator acting on the function $$$ x $$$, and $$$ \nu $$$ is the unit out normal vector 28 …”

“…where Δx is the Laplacian operator acting on the function x, and 𝜈 is the unit out normal vector. 28 Lemma 2 (Poincaré inequality). The following inequality holds…”