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.


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