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Schroedinger operators involving singular potentials and measure data
We study the existence of solutions of the Dirichlet problem for the
Schroedinger operator with measure data We characterize the finite measures
for which this problem has a solution for every nonnegative potential
in the Lebesgue space with . The full answer can
be expressed in terms of the capacity for , and the
(or Newtonian) capacity for . We then prove the existence of a solution
of the problem above when belongs to the real Hardy space and
is diffuse with respect to the capacity.Comment: Fixed a display problem in arxiv's abstract. Original tex file
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A Sublinear Variance Bound for Solutions of a Random Hamilton Jacobi Equation
We estimate the variance of the value function for a random optimal control
problem. The value function is the solution of a Hamilton-Jacobi
equation with random Hamiltonian
in dimension . It is known that homogenization occurs as , but little is known about the statistical fluctuations of .
Our main result shows that the variance of the solution is bounded
by . The proof relies on a modified Poincar\'e
inequality of Talagrand
The A-Stokes approximation for non-stationary problems
Let be an elliptic tensor. A function
is a solution to the non-stationary -Stokes problem iff
\begin{align}\label{abs} \int_Q v\cdot\partial_t\phi\,dx\,dt-\int_Q \mathcal
A(\varepsilon(v),\varepsilon(\phi))\,dx\,dt=0\quad\forall\phi\in
C^{\infty}_{0,div}(Q), \end{align} where ,
bounded. If the l.h.s. is not zero but small we talk about almost solutions. We
present an approximation result in the fashion of the -caloric
approximation for the non-stationary -Stokes problem. Precisely,
we show that every almost solution ,
, can be approximated by a solution in the
-sense for all . So, we extend the stationary -Stokes approximation by Breit-Diening-Fuchs to parabolic problems
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