The Hadamard variational formula and the Minkowski problem for p-capacity
TLDR
In this article, a Hadamard variational formula for p-capacity of convex bodies in R n is established when 1 p n and existence and regularity for 1 p 2.About:
This article is published in Advances in Mathematics.The article was published on 2015-11-05 and is currently open access. It has received 70 citations till now. The article focuses on the topics: Minkowski problem & Minkowski inequality.read more
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The Brunn-Minkowski inequality and A Minkowski problem for nonlinear capacity
TL;DR: This article studies two classical potential-theoretic problems in convex geometry and an inequality of Brunn-Minkowski type for a nonlinear capacity in Laplace equation and its solutions in an open set.
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The p -capacitary Orlicz–Hadamard variational formula and Orlicz–Minkowski problems
Han Hong,Deping Ye,Ning Zhang +2 more
TL;DR: In this paper, the p-capacitary Orlicz-Brunn-Minkowski theory was proposed and solved under some mild conditions on the involving functions and measures.
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The $p$-capacitary Orlicz-Hadamard variational formula and Orlicz-Minkowski problems
Han Hong,Deping Ye,Ning Zhang +2 more
TL;DR: In this paper, the authors developed the $p$-capacitary Orlicz-Brunn-Minkowski theory and proved the equivalence of these two inequalities under some mild conditions on the involving functions and measures.
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The Lp capacitary Minkowski problem for polytopes
Ge Xiong,Jiawei Xiong,Lu Xu +2 more
TL;DR: In this paper, sufficient conditions are given for the existence of solutions to the discrete L p Minkowski problem for p -capacity when 0 p 1 and 1 p 2, respectively.
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Geometric aspects of p-capacitary potentials
TL;DR: In this paper, the authors provide monotonicity formulas for solutions to the p-Laplace equation defined in the exterior of a convex domain, and a number of analytic and geometric consequences are derived.
References
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Book
Elliptic Partial Differential Equations of Second Order
David Gilbarg,Neil S. Trudinger +1 more
TL;DR: In this article, Leray-Schauder and Harnack this article considered the Dirichlet Problem for Poisson's Equation and showed that it is a special case of Divergence Form Operators.
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Measure theory and fine properties of functions
TL;DR: In this article, the authors define and define elementary properties of BV functions, including the following: Sobolev Inequalities Compactness Capacity Quasicontinuity Precise Representations of Soboleve Functions Differentiability on Lines BV Function Differentiability and Structure Theorem Approximation and Compactness Traces Extensions Coarea Formula for BV Functions isoperimetric inequalities The Reduced Boundary The Measure Theoretic Boundary Gauss-Green Theorem Pointwise Properties this article.
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Convex bodies : the Brunn-Minkowski theory
TL;DR: Inequalities for mixed volumes 7. Selected applications Appendix as discussed by the authors ] is a survey of mixed volumes with bounding boxes and quermass integrals, as well as a discussion of their applications.
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Nonlinear Potential Theory of Degenerate Elliptic Equations
TL;DR: In this paper, the existence of solutions for the obstacle problem is investigated and the John-Nirenberg lemma is shown to be true for nonlinear potential theory with respect to a super-harmonic function.