9,214 research outputs found
Degenerate elliptic operators: capacity, flux and separation
Let be the semigroup generated on L_2(\Ri^d) by a
self-adjoint, second-order, divergence-form, elliptic operator with
Lipschitz continuous coefficients. Further let be an open subset of
\Ri^d with Lipschitz continuous boundary . We prove that
leaves invariant if, and only if, the capacity of the boundary
with respect to 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
Uniqueness of diffusion on domains with rough boundaries
Let be a domain in and
a
quadratic form on with domain where the
are real symmetric -functions with
for almost all . Further assume there are such that for where is the Euclidean
distance to the boundary of .
We assume that is Ahlfors -regular and if , the Hausdorff
dimension of , is larger or equal to we also assume a mild
uniformity property for in the neighbourhood of one . Then
we establish that is Markov unique, i.e. it has a unique Dirichlet form
extension, if and only if . The result applies to forms
on Lipschitz domains or on a wide class of domains with a self-similar
fractal. In particular it applies to the interior or exterior of the von Koch
snowflake curve in or the complement of a uniformly disconnected
set in .Comment: 25 pages, 2 figure
Degenerate elliptic operators in one dimension
Let be the symmetric second-order differential operator on L_2(\Ri)
with domain C_c^\infty(\Ri) and action where c\in
W^{1,2}_{\rm loc}(\Ri) is a real function which is strictly positive on
\Ri\backslash\{0\} but with . We give a complete characterization of
the self-adjoint extensions and the submarkovian extensions of . In
particular if where then has a unique self-adjoint extension if and only if and a unique submarkovian extension if and only if . In both cases the corresponding semigroup leaves
and invariant.
In addition we prove that for a general non-negative c\in W^{1,\infty}_{\rm
loc}(\Ri) the corresponding operator has a unique submarkovian extension.Comment: 28 page
Markov uniqueness of degenerate elliptic operators
Let be an open subset of \Ri^d and
a second-order partial
differential operator on with domain where
the coefficients are real symmetric and
is a strictly positive-definite matrix over .
In particular, is locally strongly elliptic.
We analyze the submarkovian extensions of , i.e. the self-adjoint
extensions which generate submarkovian semigroups. Our main result establishes
that 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
measured with respect to . The second main result establishes that
Markov uniqueness of is equivalent to the semigroup generated by the
Friedrichs extension of being conservative.Comment: 24 page
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