On the Plateau problem for surfaces with minimum moment of inertia

Abstract Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>α</m:mi> <m:mo>∈</m:mo> <m:mi>ℝ</m:mi> </m:mrow> </m:math> {\alpha\in\mathbb{R}} . In this paper, we study Plateau’s problem for the two-dimensional parametric energy <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mo largeop="true" symmetric="true">∫</m:mo> <m:mi>B</m:mi> </m:msub> <m:mrow> <m:msup> <m:mrow> <m:mo stretchy="false">|</m:mo> <m:mi>X</m:mi> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mi>α</m:mi> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">|</m:mo> <m:mrow> <m:msub> <m:mi>X</m:mi> <m:mi>u</m:mi> </m:msub> <m:mo>∧</m:mo> <m:msub> <m:mi>X</m:mi> <m:mi>v</m:mi> </m:msub> </m:mrow> <m:mo rspace="4.2pt" stretchy="false">|</m:mo> </m:mrow> <m:mo>⁢</m:mo> <m:mrow> <m:mo>𝑑</m:mo> <m:mpadded width="+1.7pt"> <m:mi>u</m:mi> </m:mpadded> </m:mrow> <m:mo>⁢</m:mo> <m:mrow> <m:mo>𝑑</m:mo> <m:mi>v</m:mi> </m:mrow> </m:mrow> </m:mrow> </m:math> {\int_{B}|X|^{\alpha}|X_{u}\wedge X_{v}|\,du\,dv} . Given a closed rectifiable Jordan curve <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="normal">Γ</m:mi> <m:mo>⊂</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mn>3</m:mn> </m:msup> </m:mrow> </m:math> {\Gamma\subset\mathbb{R}^{3}} contained in a ball <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mover accent="true"> <m:mi>B</m:mi> <m:mo>¯</m:mo> </m:mover> <m:mi>R</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mover accent="true"> <m:mi>m</m:mi> <m:mo>¯</m:mo> </m:mover> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {\overline{B}_{R}(\overline{m})} of center <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mover accent="true"> <m:mi>m</m:mi> <m:mo>¯</m:mo> </m:mover> <m:mo>∈</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mn>3</m:mn> </m:msup> </m:mrow> </m:math> {\overline{m}\in\mathbb{R}^{3}} and radius <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>R</m:mi> <m:mo>&gt;</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> {R&gt;0} , we prove the existence of a conformally parametrized analytic minimizer spanning Γ. The value of R and the point <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mover accent="true"> <m:mi>m</m:mi> <m:mo>¯</m:mo> </m:mover> </m:math> {\overline{m}} depend on α. We also prove necessary conditions for the existence of multiply connected stationary surfaces having prescribed disconnected boundaries.