Degenerate elliptic operators: capacity, flux and separation

Let $S=\{S_t\}_{t\geq0}$ be the semigroup generated on L_2(\Ri^d) by a self-adjoint, second-order, divergence-form, elliptic operator $H$ with Lipschitz continuous coefficients. Further let $\Omega$ be an open subset of \Ri^d with Lipschitz continuous boundary $\partial\Omega$. We prove that $S$ leaves $L_2(\Omega)$ invariant if, and only if, the capacity of the boundary with respect to $H$ is zero or if, and only if, the energy flux across the boundary is zero. The global result is based on an analogous local result.Comment: 18 page

Degenerate elliptic operators in one dimension

Let $H$ be the symmetric second-order differential operator on L_2(\Ri) with domain C_c^\infty(\Ri) and action $H\varphi=-(c \varphi')'$ where c\in W^{1,2}_{\rm loc}(\Ri) is a real function which is strictly positive on \Ri\backslash\{0\} but with $c(0)=0$. We give a complete characterization of the self-adjoint extensions and the submarkovian extensions of $H$. In particular if $\nu=\nu_+\vee\nu_-$ where $\nu_\pm(x)=\pm\int^{\pm 1}_{\pm x} c^{-1}$ then $H$ has a unique self-adjoint extension if and only if $\nu\not\in L_2(0,1)$ and a unique submarkovian extension if and only if $\nu\not\in L_\infty(0,1)$. In both cases the corresponding semigroup leaves $L_2(0,\infty)$ and $L_2(-\infty,0)$ invariant. In addition we prove that for a general non-negative c\in W^{1,\infty}_{\rm loc}(\Ri) the corresponding operator $H$ has a unique submarkovian extension.Comment: 28 page

Markov uniqueness of degenerate elliptic operators

Let $\Omega$ be an open subset of \Ri^d and $H_\Omega=-\sum^d_{i,j=1}\partial_i c_{ij} \partial_j$ a second-order partial differential operator on $L_2(\Omega)$ with domain $C_c^\infty(\Omega)$ where the coefficients $c_{ij}\in W^{1,\infty}(\Omega)$ are real symmetric and $C=(c_{ij})$ is a strictly positive-definite matrix over $\Omega$. In particular, $H_\Omega$ is locally strongly elliptic. We analyze the submarkovian extensions of $H_\Omega$, i.e. the self-adjoint extensions which generate submarkovian semigroups. Our main result establishes that $H_\Omega$ is Markov unique, i.e. it has a unique submarkovian extension, if and only if \capp_\Omega(\partial\Omega)=0 where \capp_\Omega(\partial\Omega) is the capacity of the boundary of $\Omega$ measured with respect to $H_\Omega$. The second main result establishes that Markov uniqueness of $H_\Omega$ is equivalent to the semigroup generated by the Friedrichs extension of $H_\Omega$ being conservative.Comment: 24 page

Development of a space qualified high reliability rotary actuator for spaceflight use. Volume 2: Appendices to technical report

A space-qualified, high reliability, 150 ft-lb rated torque rotary actuator is described. This drive is an integrated variable reluctance orbit motor-epicyclic transmission actuator. The performance goals were based on future control moment gyrotorquer applications and represent an advancement in the torque-to-weight ratio, backlash, inertia and response characteristics of electric rotary drives. The program accomplishments have been in two areas (1) the development of two high ratio (818:1) actuator configurations (breadboard and flightweight) and (2) the invention of a reliable proximity switch sensor system for self-commutation without use of optical or electrical brush techniques