The Brunn-Minkowski inequality and a Minkowski problem for nonlinear capacity

<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-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\operatorname {Cap}_{\mathcal {A}},</mml:annotation> </mml:semantics> </mml:math> </inline-formula> where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper A"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {A}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-capacity is associated with a nonlinear elliptic PDE whose structure is modeled on the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-Laplace equation and whose solutions in an open set are called <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper A"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {A}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-harmonic.</p> <p>In the first part of this article, we prove the Brunn-Minkowski inequality for this capacity: <disp-formula content-type="math/mathml"> \[ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket upper C a p Subscript script upper A Baseline left-parenthesis lamda upper E 1 plus left-parenthesis 1 minus lamda right-parenthesis upper E 2 right-parenthesis right-bracket Superscript StartFraction 1 Over left-parenthesis n minus p right-parenthesis EndFraction Baseline greater-than-or-equal-to lamda left-bracket upper C a p Subscript script upper A Baseline left-parenthesis upper E 1 right-parenthesis right-bracket Superscript StartFraction 1 Over left-parenthesis n minus p right-parenthesis EndFraction Baseline plus left-parenthesis 1 minus lamda right-parenthesis left-bracket upper C a p Subscript script upper A Baseline left-parenthesis upper E 2 right-parenthesis right-bracket Superscript StartFraction 1 Over left-parenthesis n minus p right-parenthesis EndFraction"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mrow> <mml:mo>[</mml:mo> <mml:msub> <mml:mi>Cap</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi> </mml:mrow> </mml:msub> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>λ<!-- λ --></mml:mi> <mml:msub> <mml:mi>E</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>+</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>−<!-- − --></mml:mo> <mml:mi>λ<!-- λ --></mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:msub> <mml:mi>E</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mo>]</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mi>p</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:mfrac> </mml:mrow> </mml:msup> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mi>λ<!-- λ --></mml:mi> <mml:mspace width="thinmathspace" /> <mml:msup> <mml:mrow> <mml:mo>[</mml:mo> <mml:msub> <mml:mi>Cap</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi> </mml:mrow> </mml:msub> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>E</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mo>]</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mi>p</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:mfrac> </mml:mrow> </mml:msup> <mml:mo>+</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>−<!-- − --></mml:mo> <mml:mi>λ<!-- λ --></mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:msup> <mml:mrow> <mml:mo>[</mml:mo> <mml:msub> <mml:mi>Cap</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi> </mml:mrow> </mml:msub> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>E</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mo>]</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mi>p</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:mfrac> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">\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)}}</mml:annotation> </mml:semantics> </mml:math> \] </disp-formula> when <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1 greater-than p greater-than n comma 0 greater-than lamda greater-than 1 comma"> <mml:semantics> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>&gt;</mml:mo> <mml:mi>p</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mi>n</mml:mi> <mml:mo>,</mml:mo> <mml:mn>0</mml:mn> <mml:mo>&gt;</mml:mo> <mml:mi>λ<!-- λ --></mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">1&gt;p&gt;n, 0 &gt; \lambda &gt; 1,</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E 1 comma upper E 2"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>E</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>E</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">E_1, E_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are convex compact sets with positive <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper A"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {A}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-capacity. Moreover, if equality holds in the above inequality for some <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E 1"> <mml:semantics> <mml:msub> <mml:mi>E</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">E_1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E 2 comma"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>E</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">E_2,</mml:annotation> </mml:semantics> </mml:math> </inline-formula> then under certain regularity and structural assumptions on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper A comma"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi> </mml:mrow> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {A},</mml:annotation> </mml:semantics> </mml:math> </inline-formula> we show that these two sets are homothetic.</p> <p>In the second part of this article we study a Minkowski problem for a certain measure associated with a compact convex set <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E"> <mml:semantics> <mml:mi>E</mml:mi> <mml:annotation encoding="application/x-tex">E</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with nonempty interior and its <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper A"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {A}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-harmonic capacitary function in the complement of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E"> <mml:semantics> <mml:mi>E</mml:mi> <mml:annotation encoding="application/x-tex">E</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. If <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu Subscript upper E"> <mml:semantics> <mml:msub> <mml:mi>μ<!-- μ --></mml:mi> <mml:mi>E</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\mu _E</mml:annotation> </mml:semantics> </mml:math> </inline-formula> denotes this measure, then the Minkowski problem we consider in this setting is that; for a given finite Borel measure <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu"> <mml:semantics> <mml:mi>μ<!-- μ --></mml:mi> <mml:annotation encoding="application/x-tex">\mu</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper S Superscript n minus 1"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">S</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {S}^{n-1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, find necessary and sufficient conditions for which there exists <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E"> <mml:semantics> <mml:mi>E</mml:mi> <mml:annotation encoding="application/x-tex">E</mml:annotation> </mml:semantics> </mml:math> </inline-formula> as above with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu Subscript upper E Baseline equals mu period"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>μ<!-- μ --></mml:mi> <mml:mi>E</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mi>μ<!-- μ --></mml:mi> <mml:mo>.</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mu _E = \mu .</mml:annotation> </mml:semantics> </mml:math> </inline-formula> We show that necessary and sufficient conditions for existence under this setting are exactly the same conditions as in the classical Minkowski problem for volume as well as in the work of Jerison in \cite{J} for electrostatic capacity. Using the Brunn-Minkowski inequality result from the first part, we also show that this problem has a unique solution up to translation when <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p not-equals n minus 1"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>≠<!-- ≠ --></mml:mo> <mml:mi>n</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">p\neq n- 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and translation and dilation when <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p equals n minus 1"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:mi>n</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">p = n-1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.</p>