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August  2017, 10(4): 799-813. doi: 10.3934/dcdss.2017040

## On the geometry of the p-Laplacian operator

 1 Mathematisches Institut, Universität zu Köln, 50923 Köln, Germany 2 Fakultät Maschinenbau, TH Ingolstadt, Postfach 21 04 54,85019 Ingolstadt, Germany

* Corresponding author: Bernd Kawohl

Received  April 2016 Revised  August 2016 Published  April 2017

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.
Citation: Bernd Kawohl, Jiří Horák. On the geometry of the p-Laplacian operator. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 799-813. doi: 10.3934/dcdss.2017040
##### References:

show all references

##### References:
The positive viscosity solution of (4.4)
Conceivable nodal lines of the second eigenfunction for $p=\infty$ in the disc
Illustration of (5.4) and (5.5)
Numerical simulation of $u_{15}$ and side view in diagonal direction
Numerical simulation of $u_p$: normalized values along half of the diagonal for $p=2, 3, 4, 6, 8, 10, 15$ (left), and for $p=15$ compared to the line $y=x$ (right)
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