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Register a new userSpectral asymptotics for Robin Laplacians on polygonal domains
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Magda Khalile
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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.
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We study the discrete spectrum of the Robin Laplacian $Q^{Omega}_alpha$ in $L^2(Omega)$, [ umapsto -Delta u, quad dfrac{partial u}{partial n}=alpha u text{ on }partialOmega, ] where $Omegasubset mathbb{R}^{3}$ is a conical domain with a regular cross-section $Thetasubset mathbb{S}^2$, $n$ is the outer unit normal, and $alpha>0$ is a fixed constant. It is known from previous papers that the bottom of the essential spectrum of $Q^{Omega}_alpha$ is $-alpha^2$ and that the finiteness of the discrete spectrum depends on the geometry of the cross-section. We show that the accumulation of the discrete spectrum of $Q^Omega_alpha$ is determined by the discrete spectrum of an effective Hamiltonian defined on the boundary and far from the origin. By studying this model operator, we prove that the number of eigenvalues of $Q^{Omega}_alpha$ in $(-infty,-alpha^2-lambda)$, with $lambda>0$, behaves for $lambdato0$ as [ dfrac{alpha^2}{8pi lambda} int_{partialTheta} kappa_+(s)^2d s +oleft(frac{1}{lambda}right), ] where $kappa_+$ is the positive part of the geodesic curvature of the cross-section boundary.
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We study spectral properties for $H_{K,Omega}$, the Krein--von Neumann extension of the perturbed Laplacian $-Delta+V$ defined on $C^infty_0(Omega)$, where $V$ is measurable, bounded and nonnegative, in a bounded open set $Omegasubsetmathbb{R}^n$ belonging to a class of nonsmooth domains which contains all convex domains, along with all domains of class $C^{1,r}$, $r>1/2$. In particular, in the aforementioned context we establish the Weyl asymptotic formula [ #{jinmathbb{N} | lambda_{K,Omega,j}leqlambda} = (2pi)^{-n} v_n |Omega| lambda^{n/2}+Obig(lambda^{(n-(1/2))/2}big) {as} lambdatoinfty, ] where $v_n=pi^{n/2}/ Gamma((n/2)+1)$ denotes the volume of the unit ball in $mathbb{R}^n$, and $lambda_{K,Omega,j}$, $jinmathbb{N}$, are the non-zero eigenvalues of $H_{K,Omega}$, listed in increasing order according to their multiplicities. We prove this formula by showing that the perturbed Krein Laplacian (i.e., the Krein--von Neumann extension of $-Delta+V$ defined on $C^infty_0(Omega)$) is spectrally equivalent to the buckling of a clamped plate problem, and using an abstract result of Kozlov from the mid 1980s. Our work builds on that of Grubb in the early 1980s, who has considered similar issues for elliptic operators in smooth domains, and shows that the question posed by Alonso and Simon in 1980 pertaining to the validity of the above Weyl asymptotic formula continues to have an affirmative answer in this nonsmooth setting.
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