Limits for Monge-Kantorovich mass transport problems
Jesus Garcia Azorero Juan J. Manfredi I. Peral Julio D. Rossi
In this paper we study the limit of Monge-Kantorovich mass transfer problems when the involved measures are supported in a small strip near the boundary of a bounded smooth domain, $\Omega$. Given two absolutely continuos measures (with respect to the surface measure) supported on the boundary $\partial \Omega$, by performing a suitable extension of the measures to a strip of width $\varepsilon$ near the boundary of the domain $\Omega$ we consider the mass transfer problem for the extensions. Then we study the limit as $\varepsilon$ goes to zero of the Kantorovich potentials for the extensions and obtain that it coincides with a solution of the original mass transfer problem. Moreover we look for the possible approximations of these problems by solutions to equations involving the $p-$Laplacian for large values of $p$.
keywords: quasilinear elliptic equations Mass transport Neumann boundary conditions.
On quasilinear elliptic equations related to some Caffarelli-Kohn-Nirenberg inequalities
B. Abdellaoui I. Peral
The present work is devoted to analyze the Dirichlet problem for quasilinear elliptic equation related to some Caffarelli-Kohn-Nirenberg inequalities. Precisely the problem under study is,

-div $( |x|^{-p\gamma}|\nabla u|^{p-2}\nabla u)=f(x, u)\in L^1(\Omega),\quad x\in \Omega$

$u(x)=0$ on $\partial \Omega,$

where $-\infty<\gamma<\frac{N-p}{p}$, $\Omega$ is a bounded domain in $\mathbb R^N$ such that $0\in\Omega$ and $f(x,u)$ is a Caratheodory function under suitable conditions that will be stated in each section.

keywords: Weyl type lemmas. Cafarelli-Kohn-Nirenberg inequalities existence and uniqueness blow-up Degenerate and singular elliptic equations
Optimal reaction exponent for some qualitative properties of solutions to the $p$-heat equation
Susana Merchán Luigi Montoro I. Peral
This article deals with the following quasilinear parabolic problem \begin{eqnarray} u_t-\Delta_p u=h(x)u^{q}, u\geq 0 & \text{in} \,\, \Omega\times (0,\infty),\\ u(x,t)=0 & \text{on}\,\, \partial \Omega\times (0,\infty), \\ u(x,0)=f(x), \,\, f\geq 0 & \text{in} \,\, \Omega, \end{eqnarray} where $-\Delta_p u=-div(|\nabla u|^{p-2}\nabla u)$, $p>1$, $q>0$, $h(x)>0$ and $f(x)\geq 0$ are non negative functions satisfying suitable hypotheses. We assume the domain $\Omega$ is either a bounded regular domain or the whole $\mathbb{R}^N$. The main contribution of this work is to prove that the optimal exponent in the reaction term in order to prove existence of a global positive solution is $q_0=\min\{1,(p-1)\}$. More precisely, we obtain the following conclusions
If $1 < p < 2$ and $0 < q < p-1$, there is no finite extinction time.

If $p > 2$ and $0 < q< 1$, there is no finite speed of propagation.
In both cases the result is optimal.
keywords: reaction optimal exponent finite time extinction. existence finite speed of propagation p-heat equation
Existence and nonexistence results for a class of parabolic equations with mixed boundary conditions
B. Abdellaoui E. Colorado I. Peral
We study the following parabolic problem

$u_t-$ div $(|x|^{-p\gamma}|\nabla u|^{p-2}\nabla u) = \lambda f(x,u), u\ge 0$ in $\Omega\times (0,T)$,

$ B(u) = 0$ on $\partial\Omega\times (0,T),$

$ u(x,0) = \varphi (x)\quad$ if $x\in\Omega$,

where $\Omega\subset\mathbb R^N$ is a smooth bounded domain with $0\in\Omega$,

$B(u)\equiv u\chi_{\Sigma_1\times(0,T)}+|x|^{-p\gamma} |\nabla u|^{p-2}\frac{\partial u}{\partial \nu}\chi_{\Sigma_2 \times (0,T)}$

and $-\infty<\gamma<\frac{N-p}{p}$. The boundary conditions over $\partial\Omega\times (0,T)$ verify hypotheses that will be precised in each case.
Mainly, we will consider the second member $f(x,u)=\frac{u^{\alpha}}{|x|^{p(\gamma+1)}}$ with $ \alpha\ge p-1$, as a model case. The main points under analysis are some existence, nonexistence and complete blow-up results related to some Hardy-Sobolev inequalities and a weak version of Harnack inequality, that holds for $p\ge 2$ and $\gamma+1>0$.

keywords: Harnack inequality Hardy-Sobolev inequalities Quasilinear parabolic equations blow-up mixed Dirichlet-Neumann boundary conditions.

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