Bounds for eigenvalues of the Dirichlet problem for the logarithmic Laplacian
Huyuan Chen, Лаурент Верон
Abstract
Abstract We provide bounds for the sequence of eigenvalues <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:mrow> <m:msub> <m:mi>λ</m:mi> <m:mi>i</m:mi> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi mathvariant="normal">Ω</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy="false">}</m:mo> </m:mrow> <m:mi>i</m:mi> </m:msub> </m:math> {\{\lambda_{i}(\Omega)\}_{i}} of the Dirichlet problem <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mrow> <m:mrow> <m:msub> <m:mi>L</m:mi> <m:mi mathvariant="normal">Δ</m:mi> </m:msub> <m:mo></m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:mi>λ</m:mi> <m:mo></m:mo> <m:mi>u</m:mi> <m:mo></m:mo> <m:mtext> in </m:mtext> <m:mo></m:mo> <m:mi mathvariant="normal">Ω</m:mi> </m:mrow> </m:mrow> <m:mo rspace="12.5pt">,</m:mo> <m:mrow> <m:mi>u</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mrow> <m:mn>0</m:mn> <m:mo></m:mo> <m:mtext> in </m:mtext> <m:mo></m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mi>N</m:mi> </m:msup> </m:mrow> <m:mo>∖</m:mo> <m:mi mathvariant="normal">Ω</m:mi> </m:mrow> </m:mrow> </m:mrow> <m:mo>,</m:mo> </m:mrow> </m:math> L_{\Delta}u=\lambda u\text{ in }\Omega,\quad u=0\text{ in }\mathbb{R}^{N}% \setminus\Omega, where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>L</m:mi> <m:mi mathvariant="normal">Δ</m:mi> </m:msub> </m:math> {L_{\Delta}} is the logarithmic Laplacian operator with Fourier transform symbol <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>2</m:mn> <m:mo></m:mo> <m:mrow> <m:mi>ln</m:mi> <m:mo></m:mo> <m:mrow> <m:mo fence="true" stretchy="false">|</m:mo> <m:mi>ζ</m:mi> <m:mo fence="true" stretchy="false">|</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> {2\ln\lvert\zeta\rvert} . The logarithmic Laplacian operator is not positively defined if the volume of the domain is large enough. In this article, we obtain the upper and lower bounds for the sum of the first k eigenvalues by extending the Li–Yau method and Kröger’s method, respectively. Moreover, we show the limit of the quotient of the sum of the first k eigenvalues by <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>k</m:mi> <m:mo></m:mo> <m:mrow> <m:mi>ln</m:mi> <m:mo></m:mo> <m:mi>k</m:mi> </m:mrow> </m:mrow> </m:math> {k\ln k} is independent of the volume of the domain. Finally, we discuss the lower and upper bounds of the k -th principle eigenvalue, and the asymptotic behavior of the limit of eigenvalues.