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DCDS-S

The

-Laplacian operator

is not uniformly elliptic for any

and degenerates even more when

or

. In those two cases the Dirichlet and eigenvalue problems associated with the

-Laplacian lead to intriguing geometric questions, because their limits for

or

can be characterized by the geometry of

. In this little survey we recall some well-known results on eigenfunctions of the classical 2-Laplacian and elaborate on their extensions to general

. We report also on results concerning the normalized or game-theoretic

-Laplacian

$p$ |

$\Delta_pu={\rm div }\left(|\nabla u|^{p-2}\nabla u\right)$ |

$p\in(1,2)\cup(2,\infty)$ |

$p\to \infty$ |

$p\to 1$ |

$p$ |

$p\to\infty$ |

$p\to 1$ |

$\Omega$ |

$p\in[1,\infty]$ |

$p$ |

$\Delta_p^Nu:=\tfrac{1}{p}|\nabla u|^{2-p}\Delta_pu=\tfrac{1}{p}\Delta_1^Nu+\tfrac{p-1}{p}\Delta_\infty^Nu$ |

and its parabolic counterpart

. These equations are homogeneous of degree 1 and

is uniformly elliptic for any

. In this respect it is more benign than the

-Laplacian, but it is not of divergence type.

$u_t-\Delta_p^N u=0$ |

$\Delta_p^N$ |

$p\in (1,\infty)$ |

$p$ |

CPAA

We compare entire weak solutions $u$ and $v$ of quasilinear partial
differential inequalities on $R^n$ without any assumptions
on their behaviour at infinity and show among other things, that
they must coincide if they are ordered, i.e. if they satisfy $u\geq
v$ in $R^n$. For the particular case that $v\equiv 0$ we
recover some known Liouville type results. Model cases for the
equations involve the $p$-Laplacian operator for $p\in[1,2]$ and the
mean curvature operator.

CPAA

This special issue Emerging Trends in Nonlinear PDE of this journal was conceived during the Fall 2013 research semester “Evolutionary Problems” held at the Mittag-Leffler-Institute and its colophon conference Quasilinear PDEs and Game Theory held at Uppsala University in early December.
The editors of this special issue participated in these activities. Following several conversations with other participants, we solicited manuscripts from participants of the Mittag-Leffler special semester, the Uppsala conference, as well as from colleagues working in closely related fields. Seventeen papers (out of twenty-one) are authors by participants in the research program at the Mittag-Leffler or the conference at Uppsala.

keywords:

CPAA

We address the question if eigenfunctions of the
1-Laplacian, which are obtained through a variational argument, are also
viscosity solutions of the associated strongly degenerate formal Euler
equation. The answer is positive, but examples show also that there are many
more viscosity solutions than expected.

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