Fractional powers of higher-order vector operators on bounded and unbounded domains
نویسندگان
چکیده
Abstract Using the $H^{\infty }$ -functional calculus for quaternionic operators, we show how to generate fractional powers of some densely defined differential operators order $m\geq 1$ , acting on right linear Hilbert space $L^{2}(\Omega,\mathbb {C}\otimes \mathbb {H})$ . The that consider are type \begin{align*} T=i^{m-1}\left(a_1(x) e_1\partial_{x_1}^{m}+a_2(x) e_2\partial_{x_2}^{m}+a_3(x) e_3\partial_{x_3}^{m}\right), \quad x=(x_1,\, x_2,\, x_3)\in \overline{\Omega}, \end{align*} where $\overline {\Omega is closure either a bounded domain $\Omega$ with $C^{1}$ boundary, or an unbounded in $\mathbb {R}^{3}$ sufficiently regular which satisfy so-called property $(R)$ (see Definition 1.3), $e_1,\, e_2,\, e_3\in {H}$ pairwise anticommuting imaginary units, $a_1,\,a_2,\, a_3: \overline } \subset {R}^{3}\to {R}$ coefficients $T$ In particular, it will be given sufficient conditions denoted by $P_{\alpha }(T)$ $\alpha \in (0,1)$ when components i.e. $T_l:=a_l\partial _{x_l}^{m}$ do not commute among themselves. This kind result understood more general setting diffusion problems. method used construct power operator generalization developed Balakrishnan.
منابع مشابه
Bounded Domains and Unbounded Domains
First, notions of inside components and outside components are introduced for any subset of n-dimensional Euclid space. Next, notions of the bounded domain and the unbounded domain are defined using the above components. If the dimension is larger than 1, and if a subset is bounded, a unbounded domain of the subset coincides with an outside component (which is unique) of the subset. For a spher...
متن کاملHigher Order Cauchy Problems in Bounded Domains
Abstract. We study solutions of a class of higher order partial differential equations in bounded domains. These partial differential equations appeared first time in the papers of Baeumer, Meerschaert and Nane [10] and Meerschaert, Nane and Vellaisamy [31], and Nane [36]. We express the solutions by subordinating a killed Markov process by a hitting time of a stable subordinator of index 0 < β...
متن کاملFractional Cauchy problems on bounded domains
Fractional Cauchy problems replace the usual first-order time derivative by a fractional derivative. This paper develops classical solutions and stochastic analogues for fractional Cauchy problems in a bounded domain D ⊂ Rd with Dirichlet boundary conditions. Stochastic solutions are constructed via an inverse stable subordinator whose scaling index corresponds to the order of the fractional ti...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Proceedings of the Edinburgh Mathematical Society
سال: 2022
ISSN: ['1464-3839', '0013-0915']
DOI: https://doi.org/10.1017/s0013091522000396