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• W2564750176 abstract "We give the first proof, with independent smooth norms <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=phi Subscript i j Baseline comma> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>ϕ<!-- ϕ --></mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>i</mml:mi> <mml:mi>j</mml:mi> </mml:mrow> </mml:msub> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>phi _{ij},</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of the existence of surface energy minimizing partitions of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper R Superscript n> <mml:semantics> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>R</mml:mi> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>n</mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding=application/x-tex>mathbb {R}^{n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> into regions having prescribed volumes. Our existence proof significantly extends that of F. Almgren, who in 1976 gave the first such results for the special case in which each <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=phi Subscript i j> <mml:semantics> <mml:msub> <mml:mi>ϕ<!-- ϕ --></mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>i</mml:mi> <mml:mi>j</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding=application/x-tex>phi _{ij}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a scalar multiple of a fixed smooth <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=phi colon> <mml:semantics> <mml:mrow> <mml:mi>ϕ<!-- ϕ --></mml:mi> <mml:mo>:</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>phi :</mml:annotation> </mml:semantics> </mml:math> </inline-formula> <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=phi Subscript i j Baseline equals c Subscript i j Baseline phi> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>ϕ<!-- ϕ --></mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>i</mml:mi> <mml:mi>j</mml:mi> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>c</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>i</mml:mi> <mml:mi>j</mml:mi> </mml:mrow> </mml:msub> <mml:mi>ϕ<!-- ϕ --></mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>phi _{ij}=c_{ij}phi</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Most materials are polycrystalline and do not have surface energy density functions which are scalar multiples of one another, so it is important to extend the theory by removing this restriction, as we have done. We also discuss connections with polycrystalline evolution problems." @default.
• W2564750176 created "2017-01-06" @default.
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• W2564750176 date "2016-11-16" @default.
• W2564750176 modified "2023-09-27" @default.
• W2564750176 title "Existence of surface energy minimizing partitions of ℝⁿ satisfying volume constraints" @default.
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