Asymptotic behavior of spectral functions of elliptic operators with Hölder continuous coefficients
In previous works the author improved the remainder estimate in the asymptotic formula for the counting function \(N(t)\) of a strongly elliptic operator \(A\) of order \(2m\) defined in a bounded domain \(\Omega\) of \(\mathbb{R}^n\), whose coefficients of top order are Hölder continuous of exponent \(\tau \in (0,\infty)\). The purpose of this paper is to obtain a similar remainder estimate for the spectral function \(e(t,x,x)\) when \(2m >n\). Improving Tsujimoto's estimate \[ e(t,x,x) =\mu_A(x) t^{n/2m}+O\bigl( \delta(x)^{-\theta} t^{(n-\theta)/2m}) \quad \text{as }t \to\infty \] with any \(\theta\in (0,\tau/(\tau+2))\), the author proves that the above formula holds with \(\theta= \tau/(\tau+1)\) if we replace \(\delta(x)^{-\theta}\) by \(\delta (x)^{- \theta} +\log_+ (\delta (x)t^{1/2m})\). Here \(\mu_A\) is the function defined by the principal term of \(A\) and \(\delta(x) =\min \{\text{dist} (x,\partial \Omega), 1\}\). Moreover the author also shows that a better remainder estimate holds if we consider the heat kernel \(U(t,x,x)\). Namely, we get \[ U(t,x,x) =c_{n,m} \mu_A(x) t^{-n/2m} +O\bigl(t^{(\tau-n)/2m} +\delta (x)^{-1} t^{(1- n)/2m} \bigr) \] at \(t\to +0\) when \(0< \tau\leq 1\).
scientific article; zbMATH DE number 1172207