2020
DOI: 10.48550/arxiv.2012.13933
|View full text |Cite
Preprint
|
Sign up to set email alerts
|

Anisotropic $p$-capacity and anisotropic Minkowski inequality

Abstract: In this paper, we prove a sharp anisotropic L p Minkowski inequality involving the total L p anisotropic mean curvature and the anisotropic p-capacity, for any bounded domains with smooth boundary in R n . As consequences, we obtain an anisotropic Willmore inequality, a sharp anisotropic Minkowski inequality for outward F -minimising sets and a sharp volumetric anisotropic Minkowski inequality. For the proof, we utilize a nonlinear potential theoretic approach which has been recently developed in [2].

Help me understand this report

Search citation statements

Order By: Relevance

Paper Sections

Select...
3
1
1

Citation Types

0
6
0

Year Published

2021
2021
2021
2021

Publication Types

Select...
2

Relationship

0
2

Authors

Journals

citations
Cited by 2 publications
(6 citation statements)
references
References 38 publications
0
6
0
Order By: Relevance
“…and we have got some completely new results, e.g., the equality cases in Theorems 9 and 12. Besides, to the best of our knowledge, so far there have been few estimates on the anisotropic p-capacity in addition to the classical ones in [17,21] and the recent ones in [31]. We hope our results would be a nice stimulation in the field of estimates on the anisotropic p-capacity.…”
Section: Introductionmentioning
confidence: 61%
See 3 more Smart Citations
“…and we have got some completely new results, e.g., the equality cases in Theorems 9 and 12. Besides, to the best of our knowledge, so far there have been few estimates on the anisotropic p-capacity in addition to the classical ones in [17,21] and the recent ones in [31]. We hope our results would be a nice stimulation in the field of estimates on the anisotropic p-capacity.…”
Section: Introductionmentioning
confidence: 61%
“…See e.g. [31,Theorem 1.2] as a corollary of the main result there. Here we present a direct proof for the readers' convenience.…”
Section: Proof Of Theoremmentioning
confidence: 91%
See 2 more Smart Citations
“…Starting from this observation, Colding [15] and Colding-Minicozzi [16,17] discovered in recent years analogous monotonic quantities on manifolds with nonnegative Ricci curvature, where the level set flow of the distance function is replaced by the level set flow of a harmonic function. This new class of monotonicity formulas, together with their extension to the case of p-harmonic functions, 1 < p < n, has revealed to be extremely flexible and powerful, leading to the proof of new geometric inequalities [1,8,37] as well as to a new proof of classical results [4,2]. More in general, level set methods for harmonic functions have been recently employed to investigate the geometry of asymptotically flat initial data in general relativity [3,6,13,22].…”
Section: A Monotonic Quantity For the Green's Functionsmentioning
confidence: 99%