Remarks on the convexity of free boundaries (Scalar and system cases)

<p>Convexity is discussed for several free boundary value problems in exterior domains that are generally formulated as <disp-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Delta u equals f left-parenthesis u right-parenthesis in normal upper Omega minus upper D comma StartAbsoluteValue nabla u EndAbsoluteValue equals g on partial-differential normal upper Omega comma u greater-than-or-equal-to 0 in double-struck upper R Superscript n Baseline comma"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal">Δ<!-- Δ --></mml:mi> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>u</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mtext> </mml:mtext> <mml:mtext>in </mml:mtext> <mml:mi mathvariant="normal">Ω<!-- Ω --></mml:mi> <mml:mo class="MJX-variant">∖<!-- ∖ --></mml:mo> <mml:mi>D</mml:mi> <mml:mo>,</mml:mo> <mml:mspace width="1em" /> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi mathvariant="normal">∇<!-- ∇ --></mml:mi> <mml:mi>u</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mi>g</mml:mi> <mml:mtext> </mml:mtext> <mml:mtext> on </mml:mtext> <mml:mtext> </mml:mtext> <mml:mi mathvariant="normal">∂<!-- ∂ --></mml:mi> <mml:mi mathvariant="normal">Ω<!-- Ω --></mml:mi> <mml:mo>,</mml:mo> <mml:mspace width="1em" /> <mml:mi>u</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>0</mml:mn> <mml:mtext> </mml:mtext> <mml:mtext> in </mml:mtext> <mml:mtext> </mml:mtext> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\begin{equation*} \Delta u = f(u) \ \text {in } \Omega \setminus D, \quad |\nabla u | = g \ \text { on } \ \partial \Omega , \quad u\geq 0 \ \text { in } \ \mathbb {R}^n, \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="u"> <mml:semantics> <mml:mi>u</mml:mi> <mml:annotation encoding="application/x-tex">u</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is assumed to be continuous in <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:mi>n</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {R}^n</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="normal upper Omega equals left-brace u greater-than 0 right-brace"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal">Ω<!-- Ω --></mml:mi> <mml:mo>=</mml:mo> <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">\Omega = \{u > 0\}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (is unknown), <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u equals 1"> <mml:semantics> <mml:mrow> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">u=1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="partial-differential upper D"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal">∂<!-- ∂ --></mml:mi> <mml:mi>D</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\partial D</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="upper D"> <mml:semantics> <mml:mi>D</mml:mi> <mml:annotation encoding="application/x-tex">D</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a bounded domain in <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:mi>n</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {R}^n</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="n greater-than-or-equal-to 2"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">n\geq 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>). Here <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="g equals g left-parenthesis x right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>g</mml:mi> <mml:mo>=</mml:mo> <mml:mi>g</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">g= g(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a given smooth function that is either strictly positive (Bernoulli-type) or identically zero (obstacle type). Properties for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f"> <mml:semantics> <mml:mi>f</mml:mi> <mml:annotation encoding="application/x-tex">f</mml:annotation> </mml:semantics> </mml:math> </inline-formula> will be spelled out in exact terms in the text.</p> <p>The interest is in the particular case where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D"> <mml:semantics> <mml:mi>D</mml:mi> <mml:annotation encoding="application/x-tex">D</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is star-shaped or convex. The focus is on the case where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f left-parenthesis u right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>u</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">f(u)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> lacks monotonicity, so that the recently developed tool of quasiconvex rearrangement is not applicable directly. Nevertheless, such quasiconvexity is used in a slightly different manner, and in combination with scaling and asymptotic expansion of solutions at regular points. The latter heavily relies on the regularity theory of free boundaries.</p> <p>Also, convexity for several systems of equations in a general framework is discussed, and some ideas along with several open problems are presented.</p>

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