Quasiconvexity and Uniqueness of Stationary Points on a Space of Measure Preserving Maps
Abstract
<jats:p> Let <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\Omega \subset {\mathbb R}^n</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi mathvariant="normal">Ω</mml:mi> <mml:mo>⊂</mml:mo> <mml:msup> <mml:mi mathvariant="double-struck">R</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> be a bounded starshaped domain and consider the energy functional <jats:disp-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">{\mathbb F}[u; \Omega]:= \int_\Omega {\bf F}(\nabla u(x)) \, dx,</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"> <mml:mrow> <mml:mi mathvariant="double-struck">F</mml:mi> <mml:mo stretchy="false">[</mml:mo> <mml:mi>u</mml:mi> <mml:mo separator="true">;</mml:mo> <mml:mi mathvariant="normal">Ω</mml:mi> <mml:mo stretchy="false">]</mml:mo> <mml:mo>:</mml:mo> <mml:mo>=</mml:mo> <mml:msub> <mml:mo>∫</mml:mo> <mml:mi mathvariant="normal">Ω</mml:mi> </mml:msub> <mml:mi mathvariant="bold">F</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal">∇</mml:mi> <mml:mi>u</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> <mml:mtext> </mml:mtext> <mml:mi>d</mml:mi> <mml:mi>x</mml:mi> <mml:mo separator="true">,</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:disp-formula> over the space of measure preserving maps <jats:disp-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">{\mathcal A}_p(\Omega)=\bigg\{u \in \bar \xi x + W_0^{1,p}(\Omega, {\mathbb R}^n): \det \nabla u = 1 \text{ $a.e.$ in $\Omega$} \bigg\},</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"> <mml:mrow> <mml:msub> <mml:mi mathvariant="script">A</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal">Ω</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mo fence="false" stretchy="true" minsize="2.4em" maxsize="2.4em">{</mml:mo> <mml:mi>u</mml:mi> <mml:mo>∈</mml:mo> <mml:mover accent="true"> <mml:mi>ξ</mml:mi> <mml:mo>ˉ</mml:mo> </mml:mover> <mml:mi>x</mml:mi> <mml:mo>+</mml:mo> <mml:msubsup> <mml:mi>W</mml:mi> <mml:mn>0</mml:mn> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo separator="true">,</mml:mo> <mml:mi>p</mml:mi> </mml:mrow> </mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal">Ω</mml:mi> <mml:mo separator="true">,</mml:mo> <mml:msup> <mml:mi mathvariant="double-struck">R</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mo>:</mml:mo> <mml:mi>det</mml:mi> <mml:mo></mml:mo> <mml:mi mathvariant="normal">∇</mml:mi> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mrow> <mml:mtext> </mml:mtext> <mml:mstyle scriptlevel="0" displaystyle="false"> <mml:mi>a</mml:mi> <mml:mi mathvariant="normal">.</mml:mi> <mml:mi>e</mml:mi> <mml:mi mathvariant="normal">.</mml:mi> </mml:mstyle> <mml:mtext> in </mml:mtext> <mml:mstyle scriptlevel="0" displaystyle="false"> <mml:mi mathvariant="normal">Ω</mml:mi> </mml:mstyle> </mml:mrow> <mml:mo fence="false" stretchy="true" minsize="2.4em" maxsize="2.4em">}</mml:mo> <mml:mo separator="true">,</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:disp-formula> with <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">p \in [1, \infty[</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>∈</mml:mo> <mml:mo stretchy="false">[</mml:mo> <mml:mn>1</mml:mn> <mml:mo separator="true">,</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> <mml:mo stretchy="false">[</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> , <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\bar \xi \in {\mathbb M}_{n \times n}</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mover accent="true"> <mml:mi>ξ</mml:mi> <mml:mo>ˉ</mml:mo> </mml:mover> <mml:mo>∈</mml:mo> <mml:msub> <mml:mi mathvariant="double-struck">M</mml:mi> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>×</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\det \bar \xi =1</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>det</mml:mi> <mml:mo></mml:mo> <mml:mover accent="true"> <mml:mi>ξ</mml:mi> <mml:mo>ˉ</mml:mo> </mml:mover> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> . In this short note we address the question of uniqueness for solutions of the corresponding system of Euler-Lagrange equations. In particular we give a new proof of the celebrated result of R. J. Knops and C. A. Stuart [Arch. Rational Mech. Anal. 86, No. 3 (1984) 233–249] using a method based on comparison with homogeneous degree-one extensions as introduced by the second author in his recent paper "Quasiconvexity and uniqueness of stationary points in the multi-dimensional calculus of variations" [Proc. Amer. Math. Soc. 131, (2003) 3101–3107]