TY - JOUR
T1 - Local asymmetry and the inner radius of nodal domains
AU - Mangoubi, Dan
PY - 2008/9
Y1 - 2008/9
N2 - Let M be a closed Riemannian manifold of dimension n. Let φλ be an eigenfunction of the Laplace-Beltrami operator corresponding to an eigenvalue λ. We show that the volume of {φλ > 0} ∪ B is ≥ C|B|/λn, where B is any ball centered at a point of the nodal set. We apply this result to prove that each nodal domain contains a ball of radius ≥ C/λn. The results in this paper extend previous results of Nazarov, Polterovich, Sodin and of the author.
AB - Let M be a closed Riemannian manifold of dimension n. Let φλ be an eigenfunction of the Laplace-Beltrami operator corresponding to an eigenvalue λ. We show that the volume of {φλ > 0} ∪ B is ≥ C|B|/λn, where B is any ball centered at a point of the nodal set. We apply this result to prove that each nodal domain contains a ball of radius ≥ C/λn. The results in this paper extend previous results of Nazarov, Polterovich, Sodin and of the author.
KW - Asymmetry
KW - Growth of eigenfunctions
KW - Inner radius
KW - Nodal domains
UR - http://www.scopus.com/inward/record.url?scp=49949117378&partnerID=8YFLogxK
U2 - 10.1080/03605300802038577
DO - 10.1080/03605300802038577
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AN - SCOPUS:49949117378
SN - 0360-5302
VL - 33
SP - 1611
EP - 1621
JO - Communications in Partial Differential Equations
JF - Communications in Partial Differential Equations
IS - 9
ER -