BIFURCATING EXTREMAL DOMAINS FOR THE FIRST EIGENVALUE OF THE LAPLACIAN
By F. Schlenk, P. Sicbaldi. 2011
70 citations recorded by the source corpus.
Abstract We prove the existence of a smooth family of non-compact domains Ω s ⊂ R n + 1 , n ⩾ 1 , bifurcating from the straight cylinder B n × R for which the first eigenfunction of the Laplacian with 0 Dirichlet boundary condition also has constant Neumann data at the boundary: For each s ∈ ( − e , e ) , the overdetermined system { Δ u + λ u = 0 in Ω s , u = 0 on ∂ Ω s , 〈 ∇ u , ν 〉 = const on ∂ Ω s has a bounded positive solution. The domains Ω s are rotationally symmetric and periodic with respect to the R -axis of the cylinder; they are of the form Ω s = { ( x , t ) ∈ R n × R | ‖ x ‖ 1 + s cos ( 2 π T s t ) + O ( s 2 ) } where T s = T 0 + O ( s ) and T 0 is a positive real number depending on n . For n ⩾ 2 these domains provide a smooth family of counter-examples to a conjecture of Berestycki, Caffarelli and Nirenberg. We also give rather precise upper and lower bounds for the bifurcation period T 0 . This work improves a recent result of the second author.
Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/6deea8d347bfd061cd01e31afe0a1211bed48b36
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In Applied Mathematics
In Computational Theory and Mathematics
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Citations in the corpus
- Geometry and Topology of some overdetermined elliptic problems (2012, background): “Sicbaldi in [28] are examples of such kind of extremal domains.”
- Serrin’s overdetermined problem and constant mean curvature surfaces (2013, methodology): “In the two-dimensional case the same construction can be done in perioidic perturbations of a strip, see Schlenk and Sicbaldi [24].”
- Curves and surfaces with constant nonlocal mean curvature: Meeting Alexandrov and Delaunay (2015, methodology): “[24] P. Sicbaldi, New extremal domains for the first eigenvalue of the Laplacian in flat tori, Calc.”
- Solutions to overdetermined elliptic problems in nontrivial exterior domains (2016, background): “Indeed, the counterexample to the BCN Conjecture built in [25] belongs to a smooth one-parameter family that can be seen as a counterpart of the family of Delaunay surfaces, see [24].”