A note on the critical set of harmonic functions near the boundary

<p>Let <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> be a harmonic function in a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Superscript 1"> <mml:semantics> <mml:msup> <mml:mi>C</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">C^1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> domain <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D subset-of double-struck upper R Superscript d"> <mml:semantics> <mml:mrow> <mml:mi>D</mml:mi> <mml:mo>⊂</mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">D\subset {\mathbb {R}}^d</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, which vanishes on an open subset of the boundary. In this note we study its critical set <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper C left-parenthesis u right-parenthesis colon equals StartSet x element-of upper D overbar colon nabla u left-parenthesis x right-parenthesis equals 0 EndSet"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">C</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>u</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>:=</mml:mo> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mi>x</mml:mi> <mml:mo>∈</mml:mo> <mml:mover> <mml:mi>D</mml:mi> <mml:mo accent="false">¯</mml:mo> </mml:mover> <mml:mo>:</mml:mo> <mml:mi mathvariant="normal">∇</mml:mi> <mml:mi>u</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:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {C}(u): = \{x \in \overline {D}: \nabla u(x) = 0 \}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. When <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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Superscript 1 comma alpha"> <mml:semantics> <mml:msup> <mml:mi>C</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi>α</mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">C^{1,\alpha }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> domain for some <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha element-of left-parenthesis 0 comma 1 right-bracket"> <mml:semantics> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>∈</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\alpha \in (0,1]</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, we give an upper bound on the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis d minus 2 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>d</mml:mi> <mml:mo>−</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(d-2)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-dimensional Hausdorff measure of the critical set by the frequency function. We also discuss possible ways to extend such estimate to all <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Superscript 1"> <mml:semantics> <mml:msup> <mml:mi>C</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">C^1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-Dini domains, the optimal class of domains for which analogous estimates have been shown to hold for the singular set <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper S left-parenthesis u right-parenthesis colon equals StartSet x element-of upper D overbar colon u left-parenthesis x right-parenthesis equals 0 equals StartAbsoluteValue nabla u left-parenthesis x right-parenthesis EndAbsoluteValue EndSet"> <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 stretchy="false">(</mml:mo> <mml:mi>u</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>:=</mml:mo> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mi>x</mml:mi> <mml:mo>∈</mml:mo> <mml:mover> <mml:mi>D</mml:mi> <mml:mo accent="false">¯</mml:mo> </mml:mover> <mml:mo>:</mml:mo> <mml:mi>u</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:mo>=</mml:mo> <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:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {S}(u): = \{x \in \overline {D}: u(x) = 0 = |\nabla u(x)| \}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (see [Arch. Ration. Mech. Anal. 245 (2022), pp. 1–88] and [Adv. Nonlinear Stud. 23 (2023)]).</p>