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• W2207414324 abstract "Å. Pleijel (1956) has proved that in the case of the Laplacian with Dirichlet condition, the equality in the Courant nodal theorem (Courant sharp situation) can only be true for a finite number of eigenvalues when the dimension is <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=greater-than-or-equal-to 2> <mml:semantics> <mml:mrow> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>geq 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Recently Polterovich extended the result to the Neumann problem in two dimensions in the case when the boundary is piecewise analytic. A question coming from the theory of spectral minimal partitions has motivated the analysis of the cases when one has equality in Courant’s theorem. We identify the Courant sharp eigenvalues for the Dirichlet and the Neumann Laplacians in balls in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper R Superscript d> <mml:semantics> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:annotation encoding=application/x-tex>mathbb R^d</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=d greater-than-or-equal-to 2> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>dgeq 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. It is the first result of this type holding in any dimension. The corresponding result for the Dirichlet Laplacian in the disc in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper R squared> <mml:semantics> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>R</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding=application/x-tex>mathbb R^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> was obtained by B. Helffer, T. Hoffmann-Ostenhof and S. Terracini." @default.
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• W2207414324 date "2016-04-20" @default.
• W2207414324 modified "2023-10-16" @default.
• W2207414324 title "On nodal domains in Euclidean balls" @default.
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• W2207414324 doi "https://doi.org/10.1090/proc/13098" @default.
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