DCDS-S
On the geometry of the p-Laplacian operator
Bernd Kawohl Jiří Horák
The
$p$
-Laplacian operator
$\Delta_pu={\rm div }\left(|\nabla u|^{p-2}\nabla u\right)$
is not uniformly elliptic for any
$p\in(1,2)\cup(2,\infty)$
and degenerates even more when
$p\to \infty$
or
$p\to 1$
. In those two cases the Dirichlet and eigenvalue problems associated with the
$p$
-Laplacian lead to intriguing geometric questions, because their limits for
$p\to\infty$
or
$p\to 1$
can be characterized by the geometry of
$\Omega$
. In this little survey we recall some well-known results on eigenfunctions of the classical 2-Laplacian and elaborate on their extensions to general
$p\in[1,\infty]$
. We report also on results concerning the normalized or game-theoretic
$p$
-Laplacian
$\Delta_p^Nu:=\tfrac{1}{p}|\nabla u|^{2-p}\Delta_pu=\tfrac{1}{p}\Delta_1^Nu+\tfrac{p-1}{p}\Delta_\infty^Nu$
and its parabolic counterpart
$u_t-\Delta_p^N u=0$
. These equations are homogeneous of degree 1 and
$\Delta_p^N$
is uniformly elliptic for any
$p\in (1,\infty)$
. In this respect it is more benign than the
$p$
-Laplacian, but it is not of divergence type.
keywords: p-Laplacian viscosity solutions variational methods nodal lines eigenfunctions

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