The Brunn-Minkowski Inequality and A Minkowski Problem for Nonlinear Capacity
The Brunn-Minkowski Inequality and A Minkowski Problem for Nonlinear Capacity
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DOI：
10.1090/memo/1348
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发表时间：
2017-09
影响因子：
1.9
通讯作者：
M. Akman;Jasun Gong;Jay Hineman;Johnny M. Lewis;A. Vogel
中科院分区：
文献类型：
--
作者：
M. Akman;Jasun Gong;Jay Hineman;Johnny M. Lewis;A. Vogel
本文研究了凸几何中的两个经典势论问题。第一个问题是关于一个非线性容量（Cap a, \operatorname Cap＿{}{\mathcal a）的Brunn-Minkowski型不等式，其中一个{}}\mathcal a{ -容量与一个非线性椭圆偏微分方程相关联，其结构以p p -拉普拉斯方程建模，其在开集中的解称为}\mathcal a{ -调和。在本文的第一部分中，我们证明了该容量的Brunn-Minkowski不等式：}\[ [ Cap A ( λ E 1 + ( 1 − λ ) E 2 ) ] 1 ( n − p ) ≥ λ [ Cap A ( E 1 ) ] 1 ( n − p ) + ( 1 − λ ) [ Cap A ( E 2 ) ] 1 ( n − p ) \left [\operatorname {Cap}_\mathcal {A} ( \lambda E_1 + (1-\lambda ) E_2 )\right ]^{\frac {1}{(n-p)}} \geq \lambda \, \left [\operatorname {Cap}_\mathcal {A} ( E_1 )\right ]^{\frac {1}{(n-p)}} + (1-\lambda ) \left [\operatorname {Cap}_\mathcal {A} (E_2 )\right ]^{\frac {1}{(n-p)}} \]当1>p>n, 0 > λ > 1, 1>p>n, 0 > \lambda > 1，以及e1， e2 E＿1， E＿2是正A的凸紧集\mathcal A{ -容量。此外，对于某些e1e＿1和e2e＿2，如果上述不等式成立，则在<mml:mi class=" mjx - text - caligrapic " mathv的一定正则性和结构假设下}
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, Cap A , \operatorname {Cap}_{\mathcal {A}}, where A \mathcal {A} -capacity is associated with a nonlinear elliptic PDE whose structure is modeled on the p p -Laplace equation and whose solutions in an open set are called A \mathcal {A} -harmonic. In the first part of this article, we prove the Brunn-Minkowski inequality for this capacity: \[ [ Cap A ( λ E 1 + ( 1 − λ ) E 2 ) ] 1 ( n − p ) ≥ λ [ Cap A ( E 1 ) ] 1 ( n − p ) + ( 1 − λ ) [ Cap A ( E 2 ) ] 1 ( n − p ) \left [\operatorname {Cap}_\mathcal {A} ( \lambda E_1 + (1-\lambda ) E_2 )\right ]^{\frac {1}{(n-p)}} \geq \lambda \, \left [\operatorname {Cap}_\mathcal {A} ( E_1 )\right ]^{\frac {1}{(n-p)}} + (1-\lambda ) \left [\operatorname {Cap}_\mathcal {A} (E_2 )\right ]^{\frac {1}{(n-p)}} \] when 1 > p > n , 0 > λ > 1 , 1>p>n, 0 > \lambda > 1, and E 1 , E 2 E_1, E_2 are convex compact sets with positive A \mathcal {A} -capacity. Moreover, if equality holds in the above inequality for some E 1 E_1 and E 2 , E_2, then under certain regularity and structural assumptions on <mml:mi class="MJX-tex-caligraphic" mathv