The authors prove resolvent estimates for a fractional elasticity problem written as \(\partial _{t}^{2}u-\partial _{x}\sigma =f\) with \(\sigma =(C+\partial _{t}^{\alpha }D)\partial _{x}u\), where \(C\) and \(D\) are nonnegative functions of the spatial variable \(x\in \mathbb{R}\), \(\alpha \in (0,1)\), and \(\partial _{t}^{\alpha }\) is the fractional derivative with respect to time. The problem intends to model the behavior of a viscoelastic material. The authors first recall the well-posedness of this problem as presented by R. Picard et al. [Math. Methods Appl. Sci. 38, No. 15, 3141–3154 (2015; Zbl 1351.35254)]. They here need the Fourier-Laplace operator which also allows defining the fractional time derivative operator \(\partial _{t}^{\alpha }\).
In the present paper, the authors consider functions \(C\) and \(D\) which are \( \varepsilon \)-periodic and they intend to prove resolvent estimates for the assocated fractional time derivative problem in terms of \(\varepsilon \). They introduce the spaces \(L_{\#}^{\infty }(\mathbb{R})=\{f\in L^{\infty }( \mathbb{R}):f(1)=f(0)\}\) and \(\mathcal{M}_{\gamma }=\{M\in (L_{\#}^{\infty }( \mathbb{R}))^{2\times 2}:ReM(x)\geq \gamma 1_{2\times 2}\) a.e. \(x\in \mathbb{R}\}\) for \(\gamma >0\), the averaging operator \(av:\mathcal{M} _{\gamma }\rightarrow C^{2\times 2}\) defined through \(av(M)=\int_{0}^{1}M\), the operator \(\partial _{\#}:L^{2}(0,1)\supseteq H_{\#}^{1}(0,1)\ni f\rightarrow f^{\prime }\in L^{2}(0,1)\), where \(H_{\#}^{1}(0,1)=\{f\in H^{1}(0,1):f(0)=f(1)\}\) and the operator \(A_{\tau }=\begin{pmatrix} 0 & \partial _{\#}+i\tau \\ \partial _{\#}+i\tau & 0 \end{pmatrix}\). The authors also introduce the Gelfand transform \(\mathcal{G} _{\varepsilon }:L^{2}(\mathbb{R})\rightarrow L^{2}(\varepsilon ^{-1}Q\times Q^{\prime })\) with \(Q=[0,1)\) and \(Q^{\prime }=[-\pi ,\pi )\) as the continuous extension to \(L^{2}(\mathbb{R})\) of the mapping defined on \( C_{c}^{\infty }(\mathbb{R})\) through
\[ (\mathcal{G}_{\varepsilon }f)(\theta ,y)=\frac{\varepsilon }{\sqrt{2\pi }}\underset{n\in \mathbb{Z}}{\sum } f(\varepsilon (y+n))e^{-i\varepsilon \theta (y+n)} \]
for \(\theta \in \varepsilon ^{-1}Q^{\prime }\) and \(y\in Q\). The main result proves the existence of \(\varepsilon ^{\prime }>0\) and \(K>0\) such that
\[ \left\| (M-\varepsilon ^{-1}A_{\varepsilon \theta })^{-1}-(av(M)-\varepsilon ^{-1}A_{\varepsilon \theta })^{-1}\right\| _{\mathbb{L} (L^{2}(0,1)^{2})}\leq K\varepsilon \] for every \(\gamma >0\), \(M\in \mathcal{M} _{\gamma }\), \(\varepsilon \leq \varepsilon ^{\prime }\) and \(\theta \in \varepsilon ^{-1}Q^{\prime }\). The authors prove an estimate on \(K\) when \(M\) is given with a special structure. For the proof, the authors first establish properties of the operator \(\partial _{\#}\), then of the operator \(av\). The paper ends with the presentation of more precise results in the case of a fractional elasticity problem in the case of a viscoelasticity material.