Existence of bounded solutions to some nonlinear degenerate elliptic systems
Francesco Leonetti Pier Vincenzo Petricca
Discrete & Continuous Dynamical Systems - B 2009, 11(1): 191-203 doi: 10.3934/dcdsb.2009.11.191
We prove existence of bounded weak solutions $u: \Omega \subset \R^{n} \to \R^{N}$ for the Dirichlet problem

-div $( a(x, u(x), Du(x) ) ) = f(x),$ $ x \in \Omega$;
$u(x) = 0, $ $ x \in \partial\Omega$

where $\Omega$ is a bounded open set, $a$ is a suitable degenerate elliptic operator and $f$ has enough integrability.

keywords: nonlinear system bounded degenerate solution Existence elliptic coercivit
On the asymptotic behavior of symmetric solutions of the Allen-Cahn equation in unbounded domains in $\mathbb{R}^2$
Giorgio Fusco Francesco Leonetti Cristina Pignotti
Discrete & Continuous Dynamical Systems - A 2017, 37(2): 725-742 doi: 10.3934/dcds.2017030

We consider a Dirichlet problem for the Allen-Cahn equation in a smooth, bounded or unbounded, domain $Ω\subset\mathbb{R}^n.$ Under suitable assumptions, we prove an existence result and a uniform exponential estimate for symmetric solutions. In dimension $n=2$ an additional asymptotic result is obtained. These results are based on a pointwise estimate obtained for local minimizers of the Allen-Cahn energy.

keywords: Allen-Cahn equation symmetric solutions minimization pointwise estimates asymptotic behavior

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