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On the ground state eigenfunction of a convex domain in Euclidean space

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We study the first eigenfunction φ 1 of the Dirichlet Laplacian on a convex domain in Euclidean space. Elementary properties of Bessel functions yield that \(\left\| {\phi _1 } \right\|_\infty /\left\| {\phi _1 } \right\|_2 \to \infty\) if D is a sector in Euclidean plane with area 1 and the angle tends to 0. We aim to characterize those domains D such that \((vol(D))^{1/2} \left\| {\phi _1 } \right\|_\infty /\left\| {\phi _1 } \right\|_2\) is large in terms of the ratio of the first eigenvalue of D and the infimum of the first eigenvalues of all subdomains D≈ of D with given volume.

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Research supported by the Deutsche Forschungsgemeinschaft

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Kröger, P. On the ground state eigenfunction of a convex domain in Euclidean space. Potential Anal 5, 103–108 (1996). https://doi.org/10.1007/BF00276699

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Mathematics Subject Classifications (1991)

  • 35J05
  • 35B05
  • 35J25

Key words

  • Laplacian
  • eigenfunctions
  • first eigenvalue