Lower bounds of Dirichlet eigenvalues for a class of finitely degenerate Grushin type elliptic operators

2017 ◽  
Vol 37 (6) ◽  
pp. 1653-1664 ◽  
Author(s):  
Hua CHEN ◽  
Hongge CHEN ◽  
Yirui DUAN ◽  
Xin HU
Keyword(s):  
Lower Bounds ◽  
Elliptic Operators ◽  
Dirichlet Eigenvalues

Pointwise lower bounds on the heat kernels of higher order elliptic operators

1999 ◽  
Vol 125 (1) ◽  
pp. 105-111 ◽  
Author(s):  
E. B. DAVIES
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Space Dimension ◽  
Adjoint Operator ◽  
Integral Kernel ◽  
Divergence Form ◽  
Higher Order ◽  
Elliptic Operators ◽  
Heat Kernels ◽  
Second Order Elliptic Operators

Suppose that H=H*[ges ]0 on L2(X, dx) and that e−Ht has an integral kernel K(t, x, y) which is a continuous function of all three variables. It follows from the fact that e−Ht is a non-negative self-adjoint operator that K(t, x, x)[ges ]0 for all t>0 and x∈X. Our main abstract results, Theorems 2 and 3, provide a positive lower bound on K(t, x, x) under suitable general hypotheses. As an application we obtain a explicit positive lower bound on K(t, x, y) when x is close enough to y and H is a higher order uniformly elliptic operator in divergence form acting in L2(RN, dx); see Theorem 6.We emphasize that our results are not applicable to second order elliptic operators (except in one space dimension). For such operators much stronger lower bounds can be obtained by an application of the Harnack inequality. For higher order operators, however, we believe that our result is the first of its type which does not impose any continuity conditions on the highest order coefficients of the operators.

scholarly journals Fundamental tone estimates for elliptic operators in divergence form and geometric applications

2006 ◽  
Vol 78 (3) ◽  
pp. 391-404 ◽  
Author(s):  
Gregório P. Bessa ◽  
Luquésio P. Jorge ◽  
Barnabé P. Lima ◽  
José F. Montenegro
Keyword(s):  
Lower Bounds ◽  
Vector Fields ◽  
Divergence Form ◽  
Elliptic Operators ◽  
Fundamental Tone ◽  
Space Forms ◽  
Closed Hypersurfaces ◽  
Laplace Eigenvalues ◽  
Cheeger’S Constant ◽  
Lr Operator

We establish a method for giving lower bounds for the fundamental tone of elliptic operators in divergence form in terms of the divergence of vector fields. We then apply this method to the Lr operator associated to immersed hypersurfaces with locally bounded (r + 1)-th mean curvature Hr + 1 of the space forms Nn+ 1(c) of constant sectional curvature c. As a corollary we give lower bounds for the extrinsic radius of closed hypersurfaces of Nn+ 1(c) with Hr + 1 > 0 in terms of the r-th and (r + 1)-th mean curvatures. Finally we observe that bounds for the Laplace eigenvalues essentially bound the eigenvalues of a self-adjoint elliptic differential operator in divergence form. This allows us to show that Cheeger's constant gives a lower bounds for the first nonzero Lr-eigenvalue of a closed hypersurface of Nn+ 1(c).

Lower bounds for the first Dirichlet eigenvalue of the Laplacian for domains in hyperbolic space

2015 ◽  
Vol 160 (2) ◽  
pp. 191-208 ◽  
Author(s):  
SERGEI ARTAMOSHIN
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Negative Curvature ◽  
Connected Space ◽  
Constant Negative Curvature ◽  
Dirichlet Eigenvalue ◽  
Dirichlet Eigenvalues ◽  
First Dirichlet Eigenvalue ◽  
A New Technique ◽  
Simply Connected

AbstractWe consider domains in a simply connected space of constant negative curvature and develop a new technique that improves existing classical lower bound for Dirichlet eigenvalues obtained by H. P. McKean as well as the lower bounds recently obtained by A. Savo.

scholarly journals Estimates the upper bounds of Dirichlet eigenvalues for fractional Laplacian

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Hua Chen ◽  
Hong-Ge Chen
Keyword(s):  
Eigenvalue Problem ◽  
Lower Bounds ◽  
Fractional Laplacian ◽  
Upper Bounds ◽  
Dirichlet Eigenvalue ◽  
Dirichlet Eigenvalues ◽  
Dirichlet Eigenvalue Problem ◽  
Mean Function ◽  
Continuous Boundary ◽  
Dirichlet Heat Kernel

<p style='text-indent:20px;'>Let <inline-formula><tex-math id="M1">\begin{document}$ \Omega\subset\mathbb{R}^n \; (n\geq 2) $\end{document}</tex-math></inline-formula> be a bounded domain with continuous boundary <inline-formula><tex-math id="M2">\begin{document}$ \partial\Omega $\end{document}</tex-math></inline-formula>. In this paper, we study the Dirichlet eigenvalue problem of the fractional Laplacian which is restricted to <inline-formula><tex-math id="M3">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M4">\begin{document}$ 0&lt;s&lt;1 $\end{document}</tex-math></inline-formula>. Denoting by <inline-formula><tex-math id="M5">\begin{document}$ \lambda_{k} $\end{document}</tex-math></inline-formula> the <inline-formula><tex-math id="M6">\begin{document}$ k^{th} $\end{document}</tex-math></inline-formula> Dirichlet eigenvalue of <inline-formula><tex-math id="M7">\begin{document}$ (-\triangle)^{s}|_{\Omega} $\end{document}</tex-math></inline-formula>, we establish the explicit upper bounds of the ratio <inline-formula><tex-math id="M8">\begin{document}$ \frac{\lambda_{k+1}}{\lambda_{1}} $\end{document}</tex-math></inline-formula>, which have polynomially growth in <inline-formula><tex-math id="M9">\begin{document}$ k $\end{document}</tex-math></inline-formula> with optimal increase orders. Furthermore, we give the explicit lower bounds for the Riesz mean function <inline-formula><tex-math id="M10">\begin{document}$ R_{\sigma}(z) = \sum_{k}(z-\lambda_{k})_{+}^{\sigma} $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M11">\begin{document}$ \sigma\geq 1 $\end{document}</tex-math></inline-formula> and the trace of the Dirichlet heat kernel of fractional Laplacian.</p>

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