In this paper we prove uniform convergence of approximations to $p$-harmonic functions by using natural $p$-mean operators on bounded domains of the Heisenberg group $mathbb{H}$ which satisfy an intrinsic exterior corkscrew condition. These domains include Euclidean $C^{1,1}$ domains.
We consider the p-Laplacian in R^d perturbed by a weakly coupled potential. We calculate the asymptotic expansions of the lowest eigenvalue of such an operator in the weak coupling limit separately for p>d and p=d and discuss the connection with Sobolev interpolation inequalities.
We consider the Dirichlet problem for the nonlinear $p(x)$-Laplacian equation. For axially symmetric domains we prove that, under suitable assumptions, there exist Mountain-pass solutions which exhibit partial symmetry. Furthermore, we show that Semi
-stable or non-degenerate smooth solutions need to be radially symmetric in the ball.
We consider a Dirichlet type problem for a nonlinear, nonlocal equation driven by the degenerate fractional p-Laplacian, with a logistic type reaction depending on a positive parameter. In the subdiffusive and equidiffusive cases, we prove existence
and uniqueness of the positive solution when the parameter lies in convenient intervals. In the superdiffusive case, we establish a bifurcation result. A new strong comparison result, of independent interest, plays a crucial role in the proof of such bifurcation result.
We consider a pseudo-differential equation driven by the fractional $p$-Laplacian with $pge 2$ (degenerate case), with a bounded reaction $f$ and Dirichlet type conditions in a smooth domain $Omega$. By means of barriers, a nonlocal superposition pri
nciple, and the comparison principle, we prove that any weak solution $u$ of such equation exhibits a weighted Holder regularity up to the boundary, that is, $u/d^sin C^alpha(overlineOmega)$ for some $alphain(0,1)$, $d$ being the distance from the boundary.
We prove an equivalence between weighted Poincare inequalities and the existence of weak solutions to a Neumann problem related to a degenerate p- Laplacian. The Poincare inequalities are formulated in the context of degenerate Sobolev spaces defined
in terms of a quadratic form, and the associated matrix is the source of the degeneracy in the p-Laplacian.
Andras Domokos
,Juan J. Manfredi
,Diego Ricciotti
.
(2021)
.
"Convergence of Natural $p$-Means for the $p$-Laplacian in the Heisenberg Group"
.
Juan Manfredi
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