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• W2009599229 abstract "In this paper we establish the existence of global weak solutions to the heat flow for surfaces of prescribed mean curvature, i.e. the existence for the Cauchy-Dirichlet problem to parabolic systems of the type <disp-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=StartLayout Enlarged left-brace 1st Row partial-differential Subscript t Baseline u minus normal upper Delta u equals minus 2 left-parenthesis upper H ring u right-parenthesis upper D 1 u times upper D 2 u in upper B times left-parenthesis 0 comma normal infinity right-parenthesis comma 2nd Row u equals u Subscript o Baseline on partial-differential Subscript par Baseline left-parenthesis upper B times left-parenthesis 0 comma normal infinity right-parenthesis right-parenthesis comma EndLayout> <mml:semantics> <mml:mrow> <mml:mo>{</mml:mo> <mml:mtable rowspacing=0.7em 0.4em columnspacing=1em> <mml:mtr> <mml:mtd> <mml:msub> <mml:mi mathvariant=normal>∂<!-- ∂ --></mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mi>u</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mi mathvariant=normal>Δ<!-- Δ --></mml:mi> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mo>−<!-- − --></mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy=false>(</mml:mo> <mml:mi>H</mml:mi> <mml:mo>∘<!-- ∘ --></mml:mo> <mml:mi>u</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:msub> <mml:mi>D</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mi>u</mml:mi> <mml:mo>×<!-- × --></mml:mo> <mml:msub> <mml:mi>D</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mi>u</mml:mi> <mml:mspace width=1em /> <mml:mstyle displaystyle=false scriptlevel=0> <mml:mtext>in </mml:mtext> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>B</mml:mi> <mml:mo>×<!-- × --></mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi mathvariant=normal>∞<!-- ∞ --></mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:mtext>,</mml:mtext> </mml:mstyle> </mml:mtd> </mml:mtr> <mml:mtr> <mml:mtd> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>o</mml:mi> </mml:msub> <mml:mspace width=1em /> <mml:mstyle displaystyle=false scriptlevel=0> <mml:mtext>on </mml:mtext> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi mathvariant=normal>∂<!-- ∂ --></mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mtext>par</mml:mtext> </mml:mrow> </mml:msub> <mml:mstyle scriptlevel=0> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo maxsize=1.2em minsize=1.2em>(</mml:mo> </mml:mrow> </mml:mstyle> <mml:mi>B</mml:mi> <mml:mo>×<!-- × --></mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi mathvariant=normal>∞<!-- ∞ --></mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mstyle scriptlevel=0> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo maxsize=1.2em minsize=1.2em>)</mml:mo> </mml:mrow> </mml:mstyle> </mml:mrow> </mml:mstyle> <mml:mo>,</mml:mo> </mml:mtd> </mml:mtr> </mml:mtable> <mml:mo fence=true stretchy=true symmetric=true /> </mml:mrow> <mml:annotation encoding=application/x-tex>begin{equation*} left { begin {array}{c} partial _t u-Delta u =-2 (Hcirc u)D_1utimes D_2uquad mbox {in $Btimes (0,infty )$,}[3pt] u=u_oquad mbox {on $partial _textrm {par} big (Btimes (0,infty )big )$}, end{array} right . end{equation*}</mml:annotation> </mml:semantics> </mml:math> </disp-formula> where <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper H colon double-struck upper R cubed right-arrow upper R> <mml:semantics> <mml:mrow> <mml:mi>H</mml:mi> <mml:mo>:<!-- : --></mml:mo> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>R</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> <mml:mo stretchy=false>→<!-- → --></mml:mo> <mml:mi>R</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>Hcolon mathbb {R}^3to R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a bounded continuous function satisfying an isoperimetric condition, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper B> <mml:semantics> <mml:mi>B</mml:mi> <mml:annotation encoding=application/x-tex>B</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the unit ball in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper R squared> <mml:semantics> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>R</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding=application/x-tex>mathbb {R}^2</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=u colon upper B times left-parenthesis 0 comma normal infinity right-parenthesis right-arrow double-struck upper R cubed> <mml:semantics> <mml:mrow> <mml:mi>u</mml:mi> <mml:mo>:<!-- : --></mml:mo> <mml:mi>B</mml:mi> <mml:mo>×<!-- × --></mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi mathvariant=normal>∞<!-- ∞ --></mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo stretchy=false>→<!-- → --></mml:mo> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>R</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>ucolon Btimes (0,infty )to mathbb {R}^3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. As one of the possible applications we show that the problem has a solution with values in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper B Subscript upper R Baseline subset-of double-struck upper R cubed> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>B</mml:mi> <mml:mi>R</mml:mi> </mml:msub> <mml:mo>⊂<!-- ⊂ --></mml:mo> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>R</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>B_Rsubset mathbb {R}^3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, whenever <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=u Subscript o Baseline left-parenthesis upper B right-parenthesis subset-of-or-equal-to upper B Subscript upper R> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>o</mml:mi> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mi>B</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>⊆<!-- ⊆ --></mml:mo> <mml:msub> <mml:mi>B</mml:mi> <mml:mi>R</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>u_o(B)subseteq B_R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and furthermore there holds <disp-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=integral Underscript StartSet xi element-of upper B Subscript upper R Baseline colon StartAbsoluteValue upper H left-parenthesis xi right-parenthesis EndAbsoluteValue greater-than-or-equal-to StartFraction 3 Over 2 upper R EndFraction EndSet Endscripts StartAbsoluteValue upper H EndAbsoluteValue cubed d xi greater-than StartFraction 9 pi Over 2 EndFraction comma StartAbsoluteValue upper H left-parenthesis a right-parenthesis EndAbsoluteValue less-than-or-equal-to StartFraction 1 Over upper R EndFraction for a element-of partial-differential upper B Subscript upper R Baseline period> <mml:semantics> <mml:mrow> <mml:msub> <mml:mo>∫<!-- ∫ --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo fence=false stretchy=false>{</mml:mo> <mml:mi>ξ<!-- ξ --></mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:msub> <mml:mi>B</mml:mi> <mml:mi>R</mml:mi> </mml:msub> <mml:mo>:</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>|</mml:mo> </mml:mrow> <mml:mi>H</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>ξ<!-- ξ --></mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>|</mml:mo> </mml:mrow> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mfrac> <mml:mn>3</mml:mn> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>R</mml:mi> </mml:mrow> </mml:mfrac> <mml:mo fence=false stretchy=false>}</mml:mo> </mml:mrow> </mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>|</mml:mo> </mml:mrow> <mml:mi>H</mml:mi> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>|</mml:mo> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> <mml:mspace width=thinmathspace /> <mml:mi>d</mml:mi> <mml:mi>ξ<!-- ξ --></mml:mi> <mml:mo>></mml:mo> <mml:mfrac> <mml:mrow> <mml:mn>9</mml:mn> <mml:mi>π<!-- π --></mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:mfrac> <mml:mo>,</mml:mo> <mml:mspace width=2em /> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>|</mml:mo> </mml:mrow> <mml:mi>H</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>a</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>|</mml:mo> </mml:mrow> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mstyle displaystyle=false scriptlevel=0> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mi>R</mml:mi> </mml:mfrac> </mml:mstyle> <mml:mspace width=1em /> <mml:mstyle displaystyle=false scriptlevel=0> <mml:mtext>for </mml:mtext> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>a</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mi mathvariant=normal>∂<!-- ∂ --></mml:mi> <mml:msub> <mml:mi>B</mml:mi> <mml:mi>R</mml:mi> </mml:msub> </mml:mrow> <mml:mtext>.</mml:mtext> </mml:mstyle> </mml:mrow> <mml:annotation encoding=application/x-tex>begin{equation*} int _{{ xi in B_R: |H(xi )|ge frac {3}{2R}}}|H|^3, dxi >frac {9pi }{2}, qquad |H(a)|le tfrac {1}{R}quad mbox {for $ain partial B_R$.} end{equation*}</mml:annotation> </mml:semantics> </mml:math> </disp-formula>" @default.
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• W2009599229 date "2013-04-09" @default.
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• W2009599229 title "Weak solutions to the heat flow for surfaces of prescribed mean curvature" @default.
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