Variational problems with two phases and their free boundaries

1984/01/01 by Hans Wilhelm Alt, Luis A. Caffarelli, Luis Caffarelli +1 · 44 citations
Mathematics · Computer Science · #Nonlinear Partial Differential Equations #Contact Mechanics and Variational Inequalities #Advanced Mathematical Modeling in Engineering

paper · pdf · doi:10.1090/s0002-9947-1984-0732100-6

Abstract

The problem of minimizing <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral left-bracket nabla upsilon vertical-bar Superscript 2 Baseline plus q squared left-parenthesis x right-parenthesis lamda squared left-parenthesis upsilon right-parenthesis right-bracket d x"> <mml:semantics> <mml:mrow> <mml:mo> ∫ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">[</mml:mo> <mml:mi mathvariant="normal"> ∇ </mml:mi> <mml:mi> υ </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:mrow> <mml:mo>+</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>q</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi> λ </mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi> υ </mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">]</mml:mo> <mml:mi>d</mml:mi> <mml:mi>x</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">∫ [∇ υ |² + q²(x)λ ²(υ )]dx</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in an appropriate class of functions <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upsilon"> <mml:semantics> <mml:mi> υ </mml:mi> <mml:annotation encoding="application/x-tex">υ</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is considered. Here <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q left-parenthesis x right-parenthesis not-equals 0"> <mml:semantics> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo> ≠ </mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">q(x) ≠ 0</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="lamda squared left-parenthesis upsilon right-parenthesis equals lamda 1 squared"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi> λ </mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi> υ </mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:msubsup> <mml:mi> λ </mml:mi> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:msubsup> </mml:mrow> <mml:annotation encoding="application/x-tex">λ ²(υ ) = λ ₁²</mml:annotation> </mml:semantics> </mml:math> </inline-formula> if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upsilon greater-than 0 comma equals lamda 2 squared"> <mml:semantics> <mml:mrow> <mml:mi> υ </mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mo>=</mml:mo> <mml:msubsup> <mml:mi> λ </mml:mi> <mml:mn>2</mml:mn> <mml:mn>2</mml:mn> </mml:msubsup> </mml:mrow> <mml:annotation encoding="application/x-tex">υ &gt; 0, = λ ₂²</mml:annotation> </mml:semantics> </mml:math> </inline-formula> if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upsilon greater-than 0"> <mml:semantics> <mml:mrow> <mml:mi> υ </mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">υ &gt; 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Any minimizer <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u"> <mml:semantics> <mml:mi>u</mml:mi> <mml:annotation encoding="application/x-tex">u</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is harmonic in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace u not-equals 0 right-brace"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false"></mml:mo> <mml:mi>u</mml:mi> <mml:mo> ≠ </mml:mo> <mml:mn>0</mml:mn> <mml:mo fence="false" stretchy="false"></mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\ u ≠ 0\</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="StartAbs

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