Asymptotic behavior of spectral functions of elliptic operators with Hölder continuous coefficients
Yôichi Miyazaki · Journal of the Mathematical Society of Japan · 1997
In the previous papers [9], [10], [11] we 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 $R^{n}$ , whose coefficients of top order are H\"older continuous of exponent $\tau$ .The notation will be given in the next section.Let $2m>n$ .We obtained (0.1)with $\theta=\tau/(\tau+1)$ when $02$ ), and that (0.3) holds with $\theta=1$ when $\tau=\infty$ (Tsujimoto [16] and Br\"uning [4]) under some additional assumptions.Improving the above result, we will obtain the asymptotic formula (0.4) $e(t, x, x)=\mu A(x)t^{n/2m}+O(\{\delta(x)^{-\theta}+\log_{+}(\delta(x)t^{1/2m})\}t^{(n-\theta)/2m})$ as $tarrow\infty$ $x eq y)$ are bounded for $|\alpha|=k$ .We set $|u|_{0.\Omega}= \sup_{x\in\Omega}|u(x)|$ , $|u|_{\tau.\Omega}= \sum_{|\alpha|\leq k}|D^{a}u|_{0.\Omega}+\sum_{|a|=k}xy\in\Omega\sup_{\dot{x} eqy}\frac{|D^{\alpha}u(x)-D^{a}u(y)|}{|x-y|^{\theta}}$ .We consider a symmetric integro-differential sesquilinear form of order $m$ :and a closed subspace $V$ of $H^{m}(\Omega)$ , and assume the following conditions.(HO) $2m>n$ ; and the boundary $\partial\Omega$ of $\Omega$ is minimally smooth (see [13]).(H1) $H_{0}^{m}(\Omega)\subset V\subset H^{m}(\Omega)$ .(H2) There are constants $C_{0}\geqq 0$ and $\delta>0$ such thatfor any $u\in V$ .(H3) The coefficients $a_{\alpha\beta}(x)(|\alpha|\leqq m, |\beta|\leqq m)$ are bounded on 9, and for some $\tau>0$ the coefficients of top order satisfy $a_{\alpha\beta}\in B^{\tau}(\Omega)$ $(|\alpha|=|\beta|=m)$ .Let $A$ be the self-adjoint operator associated with the variational triple $\{B, V, L^{2}(\Omega)\}$ , that is, $u\in V$ belongs to $D(A)$ , the domain of $A$ , and $Au=f$ if and only if $B[u, v]=(f, v)$ is valid for any $v\in V$ .Here $(, )$ denotes the inner product in $L^{2}(\Omega)$ .We use the following notation.$c_{n.m}= \int_{0}^{\infty}t^{n/2m}e^{-}{}^{t}dt$ , $a(x, \xi)=\sum_{|\alpha|=\mathfrak{l}\beta|=m}a_{a\beta}(x)\xi^{\alpha+\beta}$ ,for $t>0$ .THEOREM A. Let $00$ .