Behaviour of $p$--Laplacian problems with Neumann boundary conditions when $p$ goes to 1
Anna Mercaldo Julio D. Rossi Sergio Segura de León Cristina Trombetti
Communications on Pure & Applied Analysis 2013, 12(1): 253-267 doi: 10.3934/cpaa.2013.12.253
We consider the solution $u_p$ to the Neumann problem for the $p$--Laplacian equation with the normal component of the flux across the boundary given by $g\in L^\infty(\partial\Omega)$. We study the behaviour of $u_p$ as $p$ goes to $1$ showing that they converge to a measurable function $u$ and the gradients $|\nabla u_p|^{p-2}\nabla u_p$ converge to a vector field $z$. We prove that $z$ is bounded and that the properties of $u$ depend on the size of $g$ measured in a suitable norm: if $g$ is small enough, then $u$ is a function of bounded variation (it vanishes on the whole domain, when $g$ is very small) while if $g$ is large enough, then $u$ takes the value $\infty$ on a set of positive measure. We also prove that in the first case, $u$ is a solution to a limit problem that involves the $1-$Laplacian. Finally, explicit examples are shown.
keywords: $1$--Laplacian. Neumann problem $p$--Laplacian
Uniqueness results for noncoercive nonlinear elliptic equations with two lower order terms
Olivier Guibé Anna Mercaldo
Communications on Pure & Applied Analysis 2008, 7(1): 163-192 doi: 10.3934/cpaa.2008.7.163
In the present paper we prove uniqueness results for weak solutions to a class of problems whose prototype is

$-d i v((1+|\nabla u|^2)^{(p-2)/2} \nabla u)-d i v(c(x) (1+|u|^2)^{(\tau+1)/2}) $

$+b(x) (1+|\nabla u|^2)^{(\sigma+1)/2}=f \ i n \ \mathcal D'(\Omega)\qquad\qquad\qquad\qquad\qquad\qquad\qquad$

$u\in W^{1,p}_0(\Omega)\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$

where $\Omega$ is a bounded open subset of $\mathbb R^N$ $(N\ge 2)$, $p$ is a real number $\frac{2N}{N+1}< p <+\infty$, the coefficients $c(x)$ and $b(x)$ belong to suitable Lebesgue spaces, $f$ is an element of the dual space $W^{-1,p'}(\Omega)$ and $\tau$ and $\sigma$ are positive constants which belong to suitable intervals specified in Theorem 2.1, Theorem 2.2 and Theorem 2.3.

keywords: nonlinear elliptic equations Uniqueness noncoercive problems weak solutions.
Gradient estimates and comparison principle for some nonlinear elliptic equations
Maria Francesca Betta Rosaria Di Nardo Anna Mercaldo Adamaria Perrotta
Communications on Pure & Applied Analysis 2015, 14(3): 897-922 doi: 10.3934/cpaa.2015.14.897
We consider a class of Dirichlet boundary problems for nonlinear elliptic equations with a first order term. We show how the summability of the gradient of a solution increases when the summability of the datum increases. We also prove comparison principle which gives in turn uniqueness results by strenghtening the assumptions on the operators.
keywords: first order terms. comparison principle Gradient estimates nonlinear elliptic operators uniqueness
Regularity and uniqueness results in grand Sobolev spaces for parabolic equations with measure data
Alberto Fiorenza Anna Mercaldo Jean Michel Rakotoson
Discrete & Continuous Dynamical Systems - A 2002, 8(4): 893-906 doi: 10.3934/dcds.2002.8.893
In this paper we prove some regularity and uniqueness results for a class of nonlinear parabolic problems whose prototype is

$\partial_t u - \Delta_N u=\mu$ in $\mathcal D'(Q) $

$u=0$ on $]0,T[\times\partial \Omega$

$u(0)=u_0$ in $ \Omega,$

where $Q$ is the cylinder $Q=(0,T)\times\Omega$, $T>0$, $\Omega\subset \mathbb R^n$, $N\ge 2$, is an open bounded set having $C^2$ boundary, $\mu\in L^1(0,T;M(\Omega))$ and $u_0$ belongs to $M(\Omega)$, the space of the Radon measures in $\Omega$, or to $L^1(\Omega)$. The results are obtained in the framework of the so-called grand Sobolev spaces, and represent an extension of earlier results on standard Sobolev spaces.

keywords: Grand Sobolev spaces uniqueness measure data parabolic equations. regularity
Uniqueness for Neumann problems for nonlinear elliptic equations
Maria Francesca Betta Olivier Guibé Anna Mercaldo
Communications on Pure & Applied Analysis 2019, 18(3): 1023-1048 doi: 10.3934/cpaa.2019050
In the present paper we prove uniqueness results for solutions to a class of Neumann boundary value problems whose prototype is
$\left\{ \begin{align} & -\text{div}({{(1+|\nabla u{{|}^{2}})}^{(p-2)/2}}\nabla u)-\text{div}(c(x)|u{{|}^{p-2}}u)=f\ \ \ \text{in}\ \Omega , \\ & \left( {{(1+|\nabla u{{|}^{2}})}^{(p-2)/2}}\nabla u+c(x)|u{{|}^{p-2}}u \right)\cdot \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{n}=0\ \ \ \text{on}\ \partial \Omega , \\ \end{align} \right.$
is a bounded domain of
$N≥ 2$
, with Lipschitz boundary,
$ 1 < p < N$
$\underline n$
is the outer unit normal to
$\partial Ω$
, the datum
belongs to
or to
and satisfies the compatibility condition
$\int{{}}_Ω f \, dx = 0$
. Finally the coefficient
belongs to an appropriate Lebesgue space.
keywords: Nonlinear elliptic equations Neumann problems renormalized solutions weak solutions uniqueness results

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