The basis of our discussion is that if \(\lambda _0^D(M) > 0\), then there exists a natural Dirichlet-to-Neumann map on M, as illustrated in the following.
For a Riemannian manifold M with boundary, we denote by \(\nu \) the outward pointing unit normal to the boundary, and by \(\lambda _0^D(M)\) the bottom of the Dirichlet spectrum of M. In this paper, we focus on a certain Dirichlet-to-Neumann map on Riemannian manifolds with boundary and bounded geometry, in the sense of. For instance, on the half-line, we have that nonzero, constant functions on the boundary do not admit square-integrable harmonic extensions, while on a half-open, bounded interval, a constant function on the boundary admits different bounded and square-integrable harmonic extensions. Indeed, there may exist compactly supported smooth functions on the boundary of such a manifold which do not admit unique harmonic extension even under constraints, such as square-integrability or boundedness, or the normal derivative of the harmonic extension does not satisfy integrability conditions to give rise to an operator in a Hilbert space.
This is reasonable, since even the definition of Dirichlet-to-Neumann maps is quite more complicated in this case. However, the Steklov spectrum of non-compact Riemannian manifolds has not attracted that much attention yet.
for instance the survey, or and the references therein). During the last years, the Steklov spectrum of compact Riemannian manifolds has been extensively studied, and analogues of various results on the Dirichlet and the Neumann spectrum have been established (cf.
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