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Register a new userThe Stability of the First Neumann Laplacian Eigenfunction Under Domain Deformations and Applications
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Authors
Nicholas F. Marshall
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The robustness of manifold learning methods is often predicated on the stability of the Neumann Laplacian eigenfunctions under deformations of the assumed underlying domain. Indeed, many manifold learning methods are based on approximating the Neumann Laplacian eigenfunctions on a manifold that is assumed to underlie data, which is viewed through a source of distortion. In this paper, we study the stability of the first Neumann Laplacian eigenfunction with respect to deformations of a domain by a diffeomorphism. In particular, we are interested in the stability of the first eigenfunction on tall thin domains where, intuitively, the first Neumann Laplacian eigenfunction should only depend on the length along the domain. We prove a rigorous version of this statement and apply it to a machine learning problem in geophysical interpretation.
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How close is the Dirichlet-to-Neumann (DtN) map to the square root of the corresponding boundary Laplacian? This question has been actively investigated in recent years. Somewhat surprisingly, a lot of techniques involved can be traced back to a newly rediscovered manuscript of Hormander from the 1950s. We present Hormanders approach and its applications, with an emphasis on eigenvalue estimates and spectral asymptotics. In particular, we obtain results for the DtN maps on non-smooth boundaries in the Riemannian setting, the DtN operators for the Helmholtz equation and the DtN operators on differential forms.
We consider harmonic Toeplitz operators $T_V = PV:{mathcal H}(Omega) to {mathcal H}(Omega)$ where $P: L^2(Omega) to {mathcal H}(Omega)$ is the orthogonal projection onto ${mathcal H}(Omega) = left{u in L^2(Omega),|,Delta u = 0 ; mbox{in};Omegaright}$, $Omega subset {mathbb R}^d$, $d geq 2$, is a bounded domain with $partial Omega in C^infty$, and $V: Omega to {mathbb C}$ is a suitable multiplier. First, we complement the known criteria which guarantee that $T_V$ is in the $p$th Schatten-von Neumann class $S_p$, by sufficient conditions which imply $T_V in S_{p, {rm w}}$, the weak counterpart of $S_p$. Next, we assume that $Omega$ is the unit ball in ${mathbb R}^d$, and $V = overline{V}$ is radially symmetric, and investigate the eigenvalue asymptotics of $T_V$ if $V$ has a power-like decay at $partial Omega$ or $V$ is compactly supported in $Omega$. Further, we consider general $Omega$ and $V geq 0$ which is regular in $Omega$, and admits a power-like decay of rate $gamma > 0$ at $partial Omega$, and we show that in this case $T_V$ is unitarily equivalent to a pseudo-differential operator of order $-gamma$, self-adjoint in $L^2(partial Omega)$. Using this unitary equivalence, we obtain the main asymptotic term of the eigenvalue counting function for the operator $T_V$. Finally, we introduce the Krein Laplacian $K geq 0$, self-adjoint in $L^2(Omega)$; it is known that ${rm Ker},K = {mathcal H}(Omega)$, and the zero eigenvalue of $K$ is isolated. We perturb $K$ by $V in C(overline{Omega};{mathbb R})$, and show that $sigma_{rm ess}(K+V) = V(partial Omega)$. Assuming that $V geq 0$ and $V{|partial Omega} = 0$, we study the asymptotic distribution of the eigenvalues of $K pm V$ near the origin, and find that the effective Hamiltonian which governs this distribution is the Toeplitz operator $T_V$.
We show that trapezoids with identical Neumann spectra are congruent up to rigid motions of the plane. The proof is based on heat trace invariants and some new wave trace invariants associated to certain diffractive billiard trajectories. The reason we can only treat the Neumann case is that the wave trace is more singular for the Neumann case compared to the Dirichlet case. This is a new observation which is interesting on its own.
We study the bilinear estimates in the Sobolev spaces with the Dirichlet and the Neumann boundary condition. The optimal regularity is revealed to get such estimates in the half space case, which is related to not only smoothness of functions and but also boundary behavior. The crucial point for the proof is how to handle boundary values of functions and their derivatives.
We give various estimates of the first eigenvalue of the $p$-Laplace operator on closed Riemannian manifold with integral curvature conditions.
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