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• W3185136273 abstract "Abstract We study and classify smooth bounded domains in an analytic Riemannian manifold which are critical for the heat content at all times $$t>0$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> . We do that by first computing the first variation of the heat content, and then showing that $$Omega $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>Ω</mml:mi> </mml:math> is critical if and only if it has the so-called constant flow property, so that we can use a previous classification result established in [33] and [34]. The outcome is that $$Omega $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>Ω</mml:mi> </mml:math> is critical for the heat content at time t , for all $$t>0$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> , if and only if $$Omega $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>Ω</mml:mi> </mml:math> admits an isoparametric foliation, that is, a foliation whose leaves are all parallel to the boundary and have constant mean curvature. Then, we consider the sequence of functionals given by the exit-time moments $$T_1(Omega ),T_2(Omega ),dots $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:msub> <mml:mi>T</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>Ω</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>T</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>Ω</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>,</mml:mo> <mml:mo>⋯</mml:mo> </mml:mrow> </mml:math> , which generalize the torsional rigidity $$T_1$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msub> <mml:mi>T</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> . We prove that $$Omega $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>Ω</mml:mi> </mml:math> is critical for all $$T_k$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:math> if and only if $$Omega $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>Ω</mml:mi> </mml:math> is critical for the heat content at every time t , and then we get a classification as well. The main purpose of the paper is to understand the variational properties of general isoparametric foliations and their role in PDE’s theory; in some respects they generalize the properties of the foliation of $$mathbf{R}^{n}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:math> by Euclidean spheres." @default.
• W3185136273 created "2021-08-02" @default.
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• W3185136273 date "2021-07-24" @default.
• W3185136273 modified "2023-09-26" @default.
• W3185136273 title "On the heat content functional and its critical domains" @default.
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