The singular set in the Stefan problem
Alessio Figalli, Xavier Ros‐Oton, Joaquim Serra
Abstract
In this paper we analyze the singular set in the Stefan problem and prove the following results: The singular set has parabolic Hausdorff dimension at most <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n minus 1"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">n-1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . The solution admits a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Superscript normal infinity"> <mml:semantics> <mml:msup> <mml:mi>C</mml:mi> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">C^\infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -expansion at all singular points, up to a set of parabolic Hausdorff dimension at most <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n minus 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-2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . In <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R cubed"> <mml:semantics> <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:annotation encoding="application/x-tex">\mathbb {R}^3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , the free boundary is smooth for almost every time <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="t"> <mml:semantics> <mml:mi>t</mml:mi> <mml:annotation encoding="application/x-tex">t</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and the set of singular times <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper S subset-of double-struck upper R"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">S</mml:mi> </mml:mrow> <mml:mo> ⊂ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal S\subset \mathbb {R}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has Hausdorff dimension at most <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1 slash 2"> <mml:semantics> <mml:mrow> <mml:mn>1</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">1/2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . These results provide us with a refined understanding of the Stefan problem’s singularities and answer some long-standing open questions in the field.