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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
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