B. Brighi Michel Chipot A. Corbo Esposito G. Mingione C. Sbordone I. Shafrir V. Valente G. Vergara Caffarelli
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.
On a class of integro-differential problems
Michel Chipot Senoussi Guesmia
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.
keywords: anisotropic singular perturbations Integro-partial differential equations asymptotic behaviour elliptic problems. Schauder fixed point theorem neutron transport
On the asymptotic behavior of variational inequalities set in cylinders
Michel Chipot Karen Yeressian
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.
keywords: asymptotic behavior. Variational inequality
On the asymptotic behavior of elliptic, anisotropic singular perturbations problems
Michel Chipot Senoussi Guesmia
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.
keywords: asymptotic behavior Anisotropic singular perturbations elliptic problems.
On some variational problems set on domains tending to infinity
Michel Chipot Aleksandar Mojsic Prosenjit Roy
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.
keywords: cylindrical domains. asymptotic analysis Variational methods
Uniqueness and comparison theorems for solutions of doubly nonlinear parabolic equations with nonstandard growth conditions
Stanislav Antontsev Michel Chipot Sergey Shmarev
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.
keywords: double nonlinearity Nonlinear parabolic equations nonstandard growth conditions.

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