Abstract
AbstractIn this paper we study extremal behaviors of the mean to max ratio of the p-torsion function with respect to the geometry of the domain. For p larger than the dimension of the space N, we prove that the upper bound is uniformly below 1, contrary to the case $$p \in (1,N]$$
p
∈
(
1
,
N
]
. For $$p=+\infty $$
p
=
+
∞
, in two dimensions, we prove that the upper bound is asymptotically attained by a disc from which is removed a network of points consisting on the vertices of a tiling of the plane with regular hexagons of vanishing size.
Funder
Technische Universität München
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,Analysis
Reference27 articles.
1. Aurenhammer, F., Klein, R., Lee, D.T.: Voronoi diagrams and Delaunay triangulations, pp. 8–337. World Scientific Publishing Co. Pte. Ltd., Hackensack (2013)
2. Arnold, D.N., David, G., Filoche, M., Jerison, D., Mayboroda, S.: Localization of eigenfunctions via an effective potential. Comm. Partial Differ. Equ. 44(11), 1186–1216 (2019)
3. Bhattacharya, T., DiBenedetto, E., Manfredi, J.: Limits as $$p\rightarrow \infty $$ of $$\Delta _pu_p=f$$ and related extremal problems, Some topics in nonlinear PDEs (Turin, 1989). Rend. Sem. Mat. Univ. Politec. Torino 1989, Special Issue, 15–68 (1991)
4. Bucur, D., Fragalà, I., Velichkov, B., Verzini, G.: On the honeycomb conjecture for a class of minimal convex partitions. Trans. Amer. Math. Soc. 370(10), 7149–7179 (2018)
5. Bucur, D., Fragalà, I.: On the honeycomb conjecture for Robin Laplacian eigenvalues. Commun. Contemp. Math. 21(2), 1850007, 29 pp (2019)