Boundedness of imaginary powers of the Stokes operator in an infinite layer
The paper is devoted to the study of the Stokes operators \(A_q= -P_q\Delta\) on \(J_q(\Omega)\), with the domain \({\mathcal D}(A_q)= \{f\in W_q^2(\Omega)^n: {}^\gamma f- f|_{\partial\Omega}= 0\}\cap J_q(\Omega)\). In the above expresisons there were used the notations: \(\Omega= \mathbb{R}^{n-1}\times (-1,1)\), \(n\geq 2\), \(J_q(\Delta):= \{\overline {f\in C_0^\infty(\Omega)^n: \operatorname {div}f= 0\}}^{L_q(\Omega)}\), \(1< q< \infty\). \(\Omega\) is the infinite layer in \(\mathbb{R}^n\) and \(J_q(\Omega)\) the space of solenoidal vector fields in \(L_q(\Omega)^n\) with vanishing normal component on \(\partial\Omega\); \(p^q\) is, by definition, \(L_q(\Omega)^n\to J_q(\Omega)\), the Helmholtz projection. Based on results obtained by other authors, the author defines the fractional operator \(A_q^z\) for \(-1< \operatorname {Re}z< 0\) using the Dunford integral. For this fractional operator he proves a theorem, the main result of the paper, asserting that for \(0< a< 1/2\), for every \(\varepsilon>0\), there is a constant \(C_{\varepsilon,a}\) such that \(\|A_q^z\|\leq C_{\varepsilon,a} e^{\varepsilon |\operatorname {Im}z|}\) for all \(z\) with the property \(-a< \operatorname {Re}z< 0\); \(\|\cdot\|\) is the operator norm in \({\mathcal L}(J_q(\Omega))\). The proof of this theorem is made at the end of the paper after other preparations, where he proves many other theorems and lemmas; fundamental in these auxiliary tools is for him the Michlin's multiplier theorem. Proving these theorems and lemmas, he establishes many intermediate results with their own importance in the operator theory. The paper is useful, especially for the specialists in the domain, but, also, for the beginners, to become aquainted with the actual concerns in the field.
scientific article; zbMATH DE number 1921143