American Institute of Mathematical Sciences

Journals

CPAA
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:
CPAA
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:
CPAA
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.
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DCDS
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:
CPAA
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.
where
 $Ω$
is a bounded domain of
 $\mathbb{R}^{N}$
,
 $N≥ 2$
, with Lipschitz boundary,
 $1 < p < N$
,
 $\underline n$
is the outer unit normal to
 $\partial Ω$
, the datum
 $f$
belongs to
 $L^{(p^{*})'}(Ω)$
or to
 $L^{1}(Ω)$
and satisfies the compatibility condition
 $\int{{}}_Ω f \, dx = 0$
. Finally the coefficient
 $c(x)$
belongs to an appropriate Lebesgue space.
keywords: