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CPAA

The 6th european conference on elliptic and parabolic problems took place in Gaeta from May 25 to May 29, 2009. It brought together more than 170 participants. This volume collects some of the papers presented there.

This meeting could not have been possible without the support of the Dipartimento di Matematica e Applicazioni, Università di Napoli Federico II, the Università di Cassino, the Accademia Pontaniana (Napoli), the Istituto Italiano per gli Studi Filosoci (Napoli), the GNAMPA, the Université de Haute Alsace (Mulhouse), the Universität Zürich, the MeMoMat Sapienza Università di Roma, the IAC CNR, the Comune di Gaeta and the partial support of the ERC grant 207573-2 Vectorial Problems. We thank all these institutions for their help.

We would like also to thank DCDS and especially Professor Shouchuan Hu for having accepted to publish these articles.

This meeting could not have been possible without the support of the Dipartimento di Matematica e Applicazioni, Università di Napoli Federico II, the Università di Cassino, the Accademia Pontaniana (Napoli), the Istituto Italiano per gli Studi Filosoci (Napoli), the GNAMPA, the Université de Haute Alsace (Mulhouse), the Universität Zürich, the MeMoMat Sapienza Università di Roma, the IAC CNR, the Comune di Gaeta and the partial support of the ERC grant 207573-2 Vectorial Problems. We thank all these institutions for their help.

We would like also to thank DCDS and especially Professor Shouchuan Hu for having accepted to publish these articles.

keywords:

CPAA

The paper is concerned with the existence of solutions to an integrodifferential
problem arising in the neutron transport theory. By an anisotropic
singular perturbations method we show that solutions of such a problem exist
and are close to those of some nonlocal elliptic problem. The existence of the
solutions of the nonlocal elliptic problem with bounded data is ensured by the
Schauder fixed point theorem. Then an asymptotic method is applied in the
general case. The limits of the solutions of the nonlocal elliptic problems are
solutions of our integro-differential problem.

DCDS

We study the asymptotic behavior of solutions to variational
inequalities with pointwise constraint on the value and gradient
of the functions as the domain becomes unbounded. First, as a
model problem, we consider the case when the constraint is
only on the value of the functions. Then we consider the more general case of
constraint also on the gradient. At the end we consider the
case when there is no force term
which corresponds to Saint-Venant
principle for linear problems.

CPAA

In this paper, we consider anitropic singular perturbations of some
elliptic boundary value problems. We study the asymptotic behavior as $\varepsilon \rightarrow 0$
for the solution. Strong convergence in some Sobolev spaces is proved and the
rate of convergence in cylindrical domains is given.

DCDS

Let $\Omega_\ell = \ell\omega_1 \times \omega_2$ where $\omega_1 \subset \mathbb{R}^p$ and $\omega_2 \subset \mathbb{R}^{n-p}$ are assumed to be open and
bounded. We consider the following minimization problem:
$$E_{\Omega_\ell}(u_\ell) = \min_{u\in W_0^{1,q}(\Omega_\ell)}E_{\Omega_\ell}(u)$$
where $E_{\Omega_\ell}(u) = \int_{\Omega_\ell}F(\nabla u)-fu$, $F$ is a convex function and $f\in L^{q'}(\omega_2)$. We are
interested in studying the asymptotic behavior of the solution $u_\ell$ as $\ell$ tends to infinity.

CPAA

The paper addresses the Dirichlet problem for the doubly nonlinear
parabolic equation with nonstandard growth conditions:
\begin{eqnarray}
u_{t}=div(a(x,t,u)|u|^{\alpha(x,t)}|\nabla
u|^{p(x,t)-2} \nabla u) +f(x,t)
\end{eqnarray}
with given variable exponents $\alpha(x,t)$ and $p(x,t)$. We
establish conditions on the data which guarantee the comparison
principle and uniqueness of bounded weak solutions in suitable
function spaces of Orlicz-Sobolev type.

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