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Bounds in spaces of Morrey under Cordes type conditions
No full textIn the study of boundary value problems for linear elliptic
equations in nondivergence form with discontinuous coefficients we consider the class of discontinuity of Cordes type.
In particular we state some local and non local a priori bounds for solutions of Dirichlet problem in unbounded domains.
The coefficients of lower terms in the differential operator belong to Morrey spaces and the principal coefficients are 'near' to functions satisfying a condition of Cordes type.
Our results are based on embedding theorems which allow us to require a summability lower
than n for the coefficients of the operator L.
We introduce a modulus of continuity of the functions in Morrey spaces to obtain the dependence of the constants in the estimates. We state also a result about
the multiplication operator from W^1(\Omega) in L^2(\Omega)
A class of weighted Hardy type inequalities in R^N
No full textIn the paper we prove the weighted Hardy type inequality
\begin{equation}
\int_{{\mathbb R}^N}V\varphi^2 \mu(x)dx\le \int_{\R^N}|\nabla \varphi|^2\mu(x)dx
+K\int_{\R^N}\varphi^2\mu(x)dx,
\end{equation}
for functions in a weighted Sobolev space ,
for a wider class of potentials than inverse square potentials
and for weight functions of a quite general type.
The case is included.
To get the result we introduce a generalized vector field method.
The estimates apply to evolution problems with Kolmogorov operators
\begin{equation*}
Lu=\Delta u+\frac{\nabla \mu}{\mu}\cdot\nabla u
\end{equation*}
perturbed by singular potentials
Existence and uniqueness results in Morrey type spaces
No full textWe consider boundary value problems for linear elliptic
equations in nondivergence form with discontinuous coefficients when the class of discontinuity is of Chicco type.
The main result we state is an existence ad uniqueness theorem for solutions of Dirichlet problem in unbounded domains.
To this aim we introduce a regularization of the principal coefficients of the differential operator.
Basic tools to prove the result are some a priori bounds stated in a previous paper
An embedding result
No full textIn unbounded subset in R^n we study the multiplication operator as an operator defined in Sobolev spaces and which takes values in L^p. The functions g belong to wider spaces of L^p connected with the Morrey type spaces.
The main result is an embedding theorem from which we an deduce a Fefferman type inequality
A higher integrability theorem from reverse Holder weighted inequalities with different supports
No full textOn some results in weighted spaces under Chicco type conditions
No full textIn the present paper we consider weighted spaces where the weight is related to the distance function from a fixed subset S of \partial\Omega.
In unbounded domains we study Dirichlet problem for linear elliptic equations in nondivergence form with discontinuous coefficients when the class of discontinuity is of Chicco type.
In particular we state some local and non local a priori bounds
and study the dependence of the constants in the estimates.
The coefficients of lower terms in the differential operator belong to weighted spaces and the principal coefficients are 'near' to functions satisfying a condition of Chicco type.
The conditions we impose on the coefficients allow us to apply embedding results to get local estimates
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