اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدEigenvalues of collapsing domains and drift Laplacians
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By introducing a weight function to the Laplace operator, Bakry and Emery defined the drift Laplacian to study diffusion processes. Our first main result is that, given a Bakry-Emery manifold, there is a naturally associated family of graphs whose eigenvalues converge to the eigenvalues of the drift Laplacian as the graphs collapse to the manifold. Applications of this result include a new relationship between Dirichlet eigenvalues of domains in $R^n$ and Neumann eigenvalues of domains in $R^{n+1}$ and a new maximum principle. Using our main result and maximum principle, we are able to generalize emph{all the results in Riemannian geometry based on gradient estimates to Bakry-Emery manifolds}.
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For $alphain(0,pi)$, let $U_alpha$ denote the infinite planar sector of opening $2alpha$, [ U_alpha=big{ (x_1,x_2)inmathbb R^2: big|arg(x_1+ix_2) big|<alpha big}, ] and $T^gamma_alpha$ be the Laplacian in $L^2(U_alpha)$, $T^gamma_alpha u= -Delta u$,
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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 discuss several geometric conditions guaranteeing the finiteness or the infiniteness of the discrete spectrum for Robin Laplacians on conical domains.
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 $Omegasubsetmathbb{R}^n$ bel
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