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

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 alttext="script A"> encoding="application/x-tex">\mathcal {A} -capacity associated with elliptic PDE whose structure modeled on the alttext="p"> p encoding="application/x-tex">p -Laplace equation and solutions open set are called -harmonic. part article, prove capacity: [ ? ( ? E 1 + ?<!-- ? stretchy="false">) 2 ] n ?<!-- ? <mml:mspace width="thinmathspace" /> encoding="application/x-tex">\left [\operatorname {Cap}_\mathcal {A} ( \lambda E_1 + (1-\lambda ) E_2 )\right ]^{\frac {1}{(n-p)}} \geq \, \left (E_2 {1}{(n-p)}} \] when alttext="1 greater-than comma 0 &gt; 0 encoding="application/x-tex">1&gt;p&gt;n, &gt; 1, 2"> encoding="application/x-tex">E_1, E_2 compact sets positive -capacity. Moreover, if equality holds above some 1"> encoding="application/x-tex">E_1 encoding="application/x-tex">E_2, then under certain regularity structural assumptions {A}, show that these homothetic. second Minkowski measure E"> encoding="application/x-tex">E nonempty interior its -harmonic capacitary function complement . If alttext="mu ?<!-- ? encoding="application/x-tex">\mu _E denotes measure, consider setting that; given finite Borel alttext="mu"> encoding="application/x-tex">\mu alttext="double-struck S mathvariant="double-struck">S encoding="application/x-tex">\mathbb {S}^{n-1} , find necessary sufficient conditions which there exists as equals mu period"> = . _E = \mu . We existence exactly same volume well work Jerison \cite{J} electrostatic capacity. Using result from part, also has unique solution up to translation alttext="p not-equals ?<!-- ? encoding="application/x-tex">p\neq n- 1 dilation encoding="application/x-tex">p n-1