DCDS
Boundary estimates for solutions of weighted semilinear elliptic equations
Claudia Anedda Giovanni Porru
Discrete & Continuous Dynamical Systems - A 2012, 32(11): 3801-3817 doi: 10.3934/dcds.2012.32.3801
Let $b(x)$ be a positive function in a bounded smooth domain $\Omega\subset R^N$, and let $f(t)$ be a positive non decreasing function on $(0,\infty)$ such that $\lim_{t\to\infty}f(t)=\infty$. We investigate boundary blow-up solutions of the equation $\Delta u=b(x)f(u)$. Under appropriate conditions on $b(x)$ as $x$ approaches $\partial\Omega$ and on $f(t)$ as $t$ goes to infinity, we find a second order approximation of the solution $u(x)$ as $x$ goes to $\partial\Omega$.
    We also investigate positive solutions of the equation $\Delta u+(\delta(x))^{2\ell}u^{-q}=0$ in $\Omega$ with $u=0$ on $\partial\Omega$, where $\ell\ge 0$, $q>3+2\ell$ and $\delta(x)$ denotes the distance from $x$ to $\partial\Omega$. We find a second order approximation of the solution $u(x)$ as $x$ goes to $\partial\Omega$.
keywords: blow-up solutions Singular elliptic equations second order estimates. weighted equations boundary estimates
PROC
Second order estimates for boundary blow-up solutions of elliptic equations
Claudia Anedda Giovanni Porru
Conference Publications 2007, 2007(Special): 54-63 doi: 10.3934/proc.2007.2007.54
We investigate blow-up solutions of the equation $\Deltau$ = $f(u)$ in a bounded smooth domain $\Omega \subset R^N$. Under appropriate growth conditions on $f(t)$ as $t$ goes to infinity we show how the mean curvature of the boundary $\partial\Omega$ appears in the second order term of the asymptotic expansion of the solution $u(x)$ as $x$ goes to $\partial\Omega$.
keywords: Blow-up solutions Elliptic equations Second order boundary estimates.
PROC
Symmetry breaking in problems involving semilinear equations
Lucio Cadeddu Giovanni Porru
Conference Publications 2011, 2011(Special): 219-228 doi: 10.3934/proc.2011.2011.219
This paper is concerned with two maximization problems where symmetry breaking arises. The rst one consists in the maximization of the energy integral relative to a homogeneous Dirichlet problem governed by the elliptic equation -$\deltau = XF^u^q$ in the annulus $B_(a,a+2)$ of the plane. Here 0  q < 1 and F is a varying subset of $B_(a,a+2)$, with a fixed measure. We prove that a subset which maximizes the corresponding energy integral is not symmetric whenever a is large enough. The second problem we consider is governed by the same equation in a disc $B_a+2$ when $F$ varies in the annulus Ba;a+2 keeping a xed measure. So, now we have a so called maximization problem with a constraint. As in the previous case, we prove that a subset which maximizes the corresponding energy integral is not symmetric whenever a is large enough.
keywords: Symmetry breaking Maximization Rearrangements Energy integral
PROC
Symmetry breaking and other features for Eigenvalue problems
Claudia Anedda Giovanni Porru
Conference Publications 2011, 2011(Special): 61-70 doi: 10.3934/proc.2011.2011.61
In the rst part of this paper we discuss a minimization problem where symmetry breaking arise. Consider the principal eigenvalue for the problem -$\Deltau = \lambdaxFu$ in the ball $B_(a+2) \subset\mathbb{R}^N$, where $N\>= 2$ and $F$ varies in the annulus $B_(a+2)\\B_a$, keeping a xed measure. We prove that, if a is large enough, the minimum of the corresponding principal eigenvalue is attained in a subset which is not symmetric. In the second part of the paper we show that the principal eigenvalue of an equation with an inde nite weight in a general bounded domain $\Omega$ can be approximated by a functional related to the energy integral of a suitable nonlinear equation; furthermore, we show that a solution of this nonlinear equation approximates, in the $H^1(\Omega)$ norm, the principal eigenfunction of our problem.
keywords: Minimization Symmetry breaking non- linear equations Rearrangements Eigenvalues
PROC
Optimization problems for the energy integral of p-Laplace equations
Antonio Greco Giovanni Porru
Conference Publications 2013, 2013(special): 301-310 doi: 10.3934/proc.2013.2013.301
We study maximization and minimization problems for the energy integral of a sub-linear $p$-Laplace equation in a domain $\Omega$, with weight $\chi_D$, where $D\subset\Omega$ is a variable subset with a fixed measure $\alpha$. We prove Lipschitz continuity for the energy integral of a maximizer and differentiability for the energy integral of the minimizer with respect to $\alpha$.
keywords: optimization regularity. Energy integral rearrangements

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