The Brunn-Minkowski Inequality and A Minkowski Problem for Nonlinear Capacity

@article{Akman2022TheBI,
  title={The Brunn-Minkowski Inequality and A Minkowski Problem for Nonlinear Capacity},
  author={Murat Akman and Jasun Gong and Jay Hineman and Johnny M. Lewis and Andrew Vogel},
  journal={Memoirs of the American Mathematical Society},
  year={2022}
}
<p>In this article we study two classical potential-theoretic problems in convex geometry. The first problem is an inequality of Brunn-Minkowski type for a nonlinear capacity, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C a p Subscript script upper A Baseline comma"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>Cap</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX… 
On the Discrete Orlicz Electrostatic q -Capacitary Minkowski Problem
<jats:p>We establish the existence of solutions to the Orlicz electrostatic <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M2"> <mi>q</mi> </math>
The Lp Minkowski problem for q-capacity
  • Zhengmao Chen
  • Mathematics
    Proceedings of the Royal Society of Edinburgh: Section A Mathematics
  • 2021
In the present paper, we first introduce the concepts of the Lp q-capacity measure and Lp mixed q-capacity and then prove some geometric properties of Lp q-capacity measure and a Lp Minkowski
The Brunn-Minkowski inequality and a Minkowski problem for $\mathcal{A}$-harmonic Green's function
In this article we study two classical problems in convex geometry associated to $\mathcal{A}$-harmonic PDEs, quasi-linear elliptic PDEs whose structure is modeled on the $p$-Laplace equation. Let
The Brunn--Minkowski inequality and a Minkowski problem for 𝒜-harmonic Green's function
Abstract In this article we study two classical problems in convex geometry associated to 𝒜{\mathcal{A}}-harmonic PDEs, quasi-linear elliptic PDEs whose structure is modelled on the p-Laplace
The p-capacitary Orlicz–Hadamard variational formula and Orlicz–Minkowski problems
In this paper, combining the p-capacity for $$p\in (1, n)$$p∈(1,n) with the Orlicz addition of convex domains, we develop the p-capacitary Orlicz–Brunn–Minkowski theory. In particular, the Orlicz
A Lp Brunn-Minkowski Theory for Logarithmic Capacity
In the present paper, we first introduce the concepts of the Lpn-dimensional Logarithmic-Capacity measure and Lp mixed n-dimensional Logarithmic-Capacity and then prove some geometric properties of
Minkowski problem for the electrostatic p-capacity for p > n
The setting for this article is the Euclidean n-space R. A convex body in R is a compact convex set with nonempty interior. A polytope in R is the convex hull of a finite set of points in R, provided
Note on an eigenvalue problem with applications to a Minkowski type regularity problem in $${\mathbb {R}}^{n}$$
We consider existence and uniqueness of homogeneous solutions $ u > 0 $ to certain PDE of $p$-Laplace type, $ p $ fixed, $ n - 1 \cos \alpha \, | x| \} \quad \mbox{for fixed}\, \, \alpha \in (0, \pi
Note on an eigenvalue problem for an ODE originating from a homogeneous $p$-harmonic function
We discuss what is known about homogeneous solutions $ u $ to the p-Laplace equation, $ p $ fixed, $1 or $ u > 0 $ is p-harmonic in the cone, \[ K(\alpha) = \{ x = (x_1, \dots, x_n ) : x_1 > \cos
NOTE ON AN EIGENVALUE PROBLEM WITH APPLICATIONS TO A MINKOWSKI TYPE REGULARITY PROBLEM IN R
We consider existence and uniqueness of homogeneous solutions u > 0 to certain PDE of p-Laplace type, p fixed, n − 1 < p < ∞, n ≥ 2, when u is a solution in K(α) ⊂ R where K(α) := {x = (x1, . . . ,
...
1
2
3
...

References

SHOWING 1-10 OF 65 REFERENCES
On the Case of Equality in the Brunn-Minkowski Inequality for Capacity
Suppose that Ω and Ω1 are convex, open subsets of Rn. Denote their convex combination by The Brunn-Minkowski inequality says that (vol Ω)t≥ (1 -t) vol Ω0 1/N +t Vol Ω for 0≤t ≤ l. Moreover, if there
Capacitary inequalities of the Brunn-Minkowski type
The main purpose of this paper is to show how so-called :a :-concave capacities may be built up from Newton capacity c, in lR"(n > 3) in much the same way as :~ :concave measures from Lebesgue
The Brunn-Minkowski inequality
In 1978, Osserman [124] wrote an extensive survey on the isoperimetric inequality. The Brunn-Minkowski inequality can be proved in a page, yet quickly yields the classical isoperimetric inequality
Nonlinear Potential Theory of Degenerate Elliptic Equations
Introduction. 1: Weighted Sobolev spaces. 2: Capacity. 3: Supersolutions and the obstacle problem. 4: Refined Sobolev spaces. 5: Variational integrals. 6: A-harmonic functions. 7: A superharmonic
The Brunn-Minkowski inequality for p-capacity of convex bodies
Abstract.In this paper we prove the Brunn-Minkowski inequality for the p-capacity of convex bodies (i.e convex compact sets with non-empty interior) in Rn, for every p(1,n). Moreover we prove that
The Brunn–Minkowski Inequality for the n-dimensional Logarithmic Capacity of Convex Bodies
Abstract We define the n-dimensional logarithmic capacity for convex bodies in Rn, with n≥2; then, for this quantity, we prove a Brunn–Minkowski type inequality, and we characterize the corresponding
REGULARITY AND FREE BOUNDARY REGULARITY FOR THE p LAPLACIAN IN LIPSCHITZ AND C1 DOMAINS
In this paper we study regularity and free boundary regularity, below the continuous threshold, for the p Laplace equation in Lipschitz and C 1 domains. To formulate our results we let Ω C R n be a
...
1
2
3
4
5
...