154 research outputs found
Schauder theorems for Ornstein-Uhlenbeck equations in infinite dimension
We prove Schauder type estimates for stationary and evolution equations
driven by the classical Ornstein-Uhlenbeck operator in a separable Banach
space, endowed with a centered Gaussian measure
Traces of Sobolev functions on regular surfaces in infinite dimensions
In a Banach space endowed with a nondegenerate Gaussian measure, we
consider Sobolev spaces of real functions defined in a sublevel set of a Sobolev nondegenerate function . We define
the traces at of the elements of for , as
elements of where is the surface measure of Feyel
and de La Pradelle. The range of the trace operator is contained in
for and even in under
further assumptions. If is a suitable halfspace, the range is characterized
as a sort of fractional Sobolev space at the boundary.
An important consequence of the general theory is an integration by parts
formula for Sobolev functions, which involves their traces at
Maximal regularity for Dirichlet problems in Hilbert spaces
We consider the Dirichlet problem in
\mathcal{O}, U=0 on . Here
where is a nondegenerate centered Gaussian measure in a Hilbert space
, is an Ornstein-Uhlenbeck operator, and is an
open set in with good boundary. We address the problem whether the weak
solution belongs to the Sobolev space . It is
well known that the question has positive answer if ; if
we give a sufficient condition in terms of geometric
properties of the boundary . The results are quite
different with respect to the finite dimensional case, for instance if
\mathcal{O} is the ball centered at the origin with radius we prove that
only for small
Strong convergence of solutions to nonautonomous Kolmogorov equations
We study a class of nonautonomous, linear, parabolic equations with unbounded
coefficients on which admit an evolution system of measures. It
is shown that the solutions of these equations converge to constant functions
as . We further establish the uniqueness of the tight evolution
system of measures and treat the case of converging coefficients
Surface measures in infinite dimension
We construct surface measures associated to Gaussian measures in separable
Banach spaces, and we prove several properties including an integration by
parts formula
Semilinear nonautonomous parabolic equations with unbounded coefficients in the linear part
We study the Cauchy problem for the semilinear nonautonomous parabolic
equation in ,
, in the spaces and in . Here is a Borel measure defined via a
tight evolution system of measures for the evolution operator
associated to the family of time depending second order uniformly elliptic
operators . Sufficient conditions for existence in the large
and stability of the null solution are also given in both and
contexts. The novelty with respect to the literature is that the coefficients
of the operators are allowed to be unbounded
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