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Register a new userSpectral isoperimetric inequalities for Robin Laplacians on 2-manifolds and unbounded cones
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We consider the problem of geometric optimization of the lowest eigenvalue for the Laplacian on a compact, simply-connected two-dimensional manifold with boundary subject to an attractive Robin boundary condition. We prove that in the sub-class of manifolds with the Gauss curvature bounded from above by a constant $K_circ ge 0$ and under the constraint of fixed perimeter, the geodesic disk of constant curvature $K_circ$ maximizes the lowest Robin eigenvalue. In the same geometric setting, it is proved that the spectral isoperimetric inequality holds for the lowest eigenvalue of the Dirichlet-to-Neumann operator. Finally, we adapt our methods to Robin Laplacians acting on unbounded three-dimensional cones to show that, under a constraint of fixed perimeter of the cross-section, the lowest Robin eigenvalue is maximized by the circular cone.
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We discuss several geometric conditions guaranteeing the finiteness or the infiniteness of the discrete spectrum for Robin Laplacians on conical domains.
Let $Omega$ be a curvilinear polygon and $Q^gamma_{Omega}$ be the Laplacian in $L^2(Omega)$, $Q^gamma_{Omega}psi=-Delta psi$, with the Robin boundary condition $partial_ u psi=gamma psi$, where $partial_ u$ is the outer normal derivative and $gamma>0$. We are interested in the behavior of the eigenvalues of $Q^gamma_Omega$ as $gamma$ becomes large. We prove that the asymptotics of the first eigenvalues of $Q^gamma_Omega$ is determined at the leading order by those of model operators associated with the vertices: the Robin Laplacians acting on the tangent sectors associated with $partial Omega$. In the particular case of a polygon with straight edges the first eigenpairs are exponentially close to those of the model operators. Finally, we prove a Weyl asymptotics for the eigenvalue counting function of $Q^gamma_Omega$ for a threshold depending on $gamma$, and show that the leading term is the same as for smooth domains.
This paper reviews many of the known inequalities for the eigenvalues of the Laplacian and bi-Laplacian on bounded domains in Euclidean space. In particular, we focus on isoperimetric inequalities for the low eigenvalues of the Dirichlet and Neumann Laplacians and of the vibrating clamped plate problem (i.e., the biharmonic operator with ``Dirichlet boundary conditions). We also discuss the known universal inequalities for the eigenvalues of the Dirichlet Laplacian and the vibrating clamped plate and buckling problems and go on to present some new ones. Some of the names associated with these inequalities are Rayleigh, Faber-Krahn, Szego-Weinberger, Payne-Polya-Weinberger, Sperner, Hile-Protter, and H. C. Yang. Occasionally, we will also comment on extensions of some of our inequalities to bounded domains in other spaces, specifically, S^n or H^n.
We study directed, weighted graphs $G=(V,E)$ and consider the (not necessarily symmetric) averaging operator $$ (mathcal{L}u)(i) = -sum_{j sim_{} i}{p_{ij} (u(j) - u(i))},$$ where $p_{ij}$ are normalized edge weights. Given a vertex $i in V$, we define the diffusion distance to a set $B subset V$ as the smallest number of steps $d_{B}(i) in mathbb{N}$ required for half of all random walks started in $i$ and moving randomly with respect to the weights $p_{ij}$ to visit $B$ within $d_{B}(i)$ steps. Our main result is that the eigenfunctions interact nicely with this notion of distance. In particular, if $u$ satisfies $mathcal{L}u = lambda u$ on $V$ and $$ B = left{ i in V: - varepsilon leq u(i) leq varepsilon right} eq emptyset,$$ then, for all $i in V$, $$ d_{B}(i) log{left( frac{1}{|1-lambda|} right) } geq log{left( frac{ |u(i)| }{|u|_{L^{infty}}} right)} - log{left(frac{1}{2} + varepsilonright)}.$$ $d_B(i)$ is a remarkably good approximation of $|u|$ in the sense of having very high correlation. The result implies that the classical one-dimensional spectral embedding preserves particular aspects of geometry in the presence of clustered data. We also give a continuous variant of the result which has a connection to the hot spots conjecture.
Using a correspondence between the spectrum of the damped wave equation and non-self-adjoint Schroedinger operators, we derive various bounds on complex eigenvalues of the former. In particular, we establish a sharp result that the one-dimensional damped wave operator is similar to the undamped one provided that the L^1 norm of the (possibly complex-valued) damping is less than 2. It follows that these small dampings are spectrally undetectable.
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