New extremal domains for the first eigenvalue of the Laplacian in flat tori

@article{Sicbaldi2010NewED,
  title={New extremal domains for the first eigenvalue of the Laplacian in flat tori},
  author={Pieralberto Sicbaldi},
  journal={Calculus of Variations and Partial Differential Equations},
  year={2010},
  volume={37},
  pages={329-344}
}
  • Pieralberto Sicbaldi
  • Published 2010
  • Mathematics
  • Calculus of Variations and Partial Differential Equations
  • We prove the existence of nontrivial compact extremal domains for the first eigenvalue of the Laplacian in manifolds $${\mathbb{R}^{n}\times \mathbb{R}{/}T\, \mathbb{Z}}$$ with flat metric, for some T > 0. These domains are close to the cylinder-type domain $${B_1 \times \mathbb{R}{/}T\, \mathbb{Z}}$$, where B1 is the unit ball in $${\mathbb{R}^{n}}$$, they are invariant by rotation with respect to the vertical axe, and are not invariant by vertical translations. Such domains can be extended by… CONTINUE READING
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    References

    SHOWING 1-10 OF 17 REFERENCES
    Domain deformations and eigenvalues of the Dirichlet Laplacian in a Riemannian manifold
    • 37
    • PDF
    Elliptic Partial Differential Equations of Second Order
    • 8,347
    Extremal domains for the first eigenvalue of the Laplace-Beltrami operator
    • 31
    • PDF
    Bifurcation Theory: An Introduction with Applications to PDEs
    • 220
    • PDF
    Elliptic Partial Differential Equa-tions of Second Order
    • 9,596
    • PDF
    A symmetry problem in potential theory
    • 956
    Shock Waves and Reaction-Diffusion Equations
    • 3,654
    Perturbation theory for a linear operator
    • 64
    A Treatise on Bessel Functions: And Their Applications to Physics
    • 235
    Applied Mathematical Sciences
    • 4,103