## Elliptic operators with variable coefficients generating fractional resolvent families.(English)Zbl 1155.45006

Let $${\mathbf B}(X)$$ be the Banach algebra of all bounded linear operators acting on a Banach space $$X$$, and $$A$$ a closed linear operator on $$X$$ with dense domain $$D(A)\subset X$$. Given $$\alpha>0$$, a family $$\{S_\alpha(t)\}_{t\geq 0}\subset{\mathbf B}(X)$$ is called an $$\alpha$$-times resolvent family for $$A$$ if: (a) $$S_\alpha(t)$$ is strongly continuous for $$t\geq 0$$ and $$S_\alpha(0)=I$$; (b) $$S_\alpha(t)D(A)\subset D(A)$$ and $$AS_\alpha(t)x=S_\alpha(t)Ax$$ for $$x\in D(A)$$ and $$t\geq 0$$; (c) for $$x\in D(A)$$ and $$t\geq 0$$,
$S_\alpha(t)x=x+\int_0^t\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)} S_\alpha(s)Ax\,ds.$
Let $${\mathcal A}^\alpha=\bigcup_{0<\theta\leq\pi/2}\bigcup_{\omega\geq 0}{\mathcal A}^\alpha(\theta,\omega)$$ where $$A\in{\mathcal A}^\alpha(\theta_0,\omega_0)$$ means that there is an $$\alpha$$-times resolvent family $$\{S_\alpha(t)\}_{t\geq 0}$$ for $$A$$ such that $$S_\alpha(t)$$ admits an analytic extension to a sector $$\Sigma_{\theta_0}:=\{z\in\mathbb{C}: | \arg z| <\theta_0\}$$ for some $$\theta_0\in(0,\pi/2]$$, and for each $$\theta<\theta_0$$ and $$\omega>\omega_0$$ there is an $$M=M(\theta,\omega)$$ such that $$\| S_\alpha(z)\| \leq Me^{\omega\text{ Re}\,z}$$ for $$z\in\Sigma_0$$.
Let $$\Omega$$ be a bounded domain in $$\mathbb{R}^n$$ with boundary of class $$C^{2m}$$ and $$X=L^2(\Omega)$$. Applying Gårding’s inequality and numerical ranges, the authors obtain sufficient conditions under which strongly elliptic differential operators $$A$$ of order $$2m$$ with $$D(A)\subset L^2(\Omega)$$ belong to $${\mathcal A}^\alpha$$, with possible specifications of ranges for $$\alpha$$. Such fractional resolvent families are related to Volterra equations and are generalizations of semigroups and cosine functions.

### MSC:

 45N05 Abstract integral equations, integral equations in abstract spaces 45D05 Volterra integral equations 47D06 One-parameter semigroups and linear evolution equations 47D09 Operator sine and cosine functions and higher-order Cauchy problems