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>></mml:mo> <mml:mi>p</mml:mi> <mml:mo>></mml:mo> <mml:mi>n</mml:mi> <mml:mo>,</mml:mo> <mml:mn>0</mml:mn> <mml:mo>></mml:mo> <mml:mi>λ<!-- λ --></mml:mi> <mml:mo>></mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">1>p>n, 0 > \lambda > 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" mathv

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