TY - JOUR
T1 - DEFINING THE SPECTRAL POSITION OF A NEUMANN DOMAIN
AU - Band, Ram
AU - Cox, Graham
AU - Egger, Sebastian K.
N1 - Publisher Copyright:
© 2023 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). Open Access made possible by subscribing institutions via Subscribe to Open. All Rights Reserved.
PY - 2023/11/11
Y1 - 2023/11/11
N2 - A Laplacian eigenfunction on a two-dimensional Riemannian manifold provides a natural partition into Neumann domains, a.k.a. a Morse-Smale complex. This partition is generated by gradient flow lines of the eigenfunction, which bound the so-called Neumann domains. We prove that the Neumann Laplacian defined on a Neumann domain is self-adjoint and has a purely discrete spectrum. In addition, we prove that the restriction of an eigenfunction to any one of its Neumann domains is an eigenfunction of the Neumann Laplacian. By comparison, similar statements about the Dirichlet Laplacian on a nodal domain of an eigenfunction are basic and well-known. The difficulty here is that the boundary of a Neumann domain may have cusps and cracks, so standard results about Sobolev spaces are not available. Another very useful common fact is that the restricted eigenfunction on a nodal domain is the first eigenfunction of the Dirichlet Laplacian. This is no longer true for a Neumann domain. Our results enable the investigation of the resulting spectral position problem for Neumann domains, which is much more involved than its nodal analogue.
AB - A Laplacian eigenfunction on a two-dimensional Riemannian manifold provides a natural partition into Neumann domains, a.k.a. a Morse-Smale complex. This partition is generated by gradient flow lines of the eigenfunction, which bound the so-called Neumann domains. We prove that the Neumann Laplacian defined on a Neumann domain is self-adjoint and has a purely discrete spectrum. In addition, we prove that the restriction of an eigenfunction to any one of its Neumann domains is an eigenfunction of the Neumann Laplacian. By comparison, similar statements about the Dirichlet Laplacian on a nodal domain of an eigenfunction are basic and well-known. The difficulty here is that the boundary of a Neumann domain may have cusps and cracks, so standard results about Sobolev spaces are not available. Another very useful common fact is that the restricted eigenfunction on a nodal domain is the first eigenfunction of the Dirichlet Laplacian. This is no longer true for a Neumann domain. Our results enable the investigation of the resulting spectral position problem for Neumann domains, which is much more involved than its nodal analogue.
KW - Laplacian eigenfunctions
KW - Morse-Smale complexes
KW - Neumann domains
KW - Neumann lines
KW - nodal domains
UR - http://www.scopus.com/inward/record.url?scp=85178480574&partnerID=8YFLogxK
U2 - 10.2140/apde.2023.16.2147
DO - 10.2140/apde.2023.16.2147
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AN - SCOPUS:85178480574
SN - 2157-5045
VL - 16
SP - 2147
EP - 2171
JO - Analysis and PDE
JF - Analysis and PDE
IS - 9
ER -