The Noncommutative Fractional Fourier Law in Bounded and Unbounded Domains

Abstract

Using the spectral theory on the S-spectrum it is possible to define the fractional powers of a large class of vector operators. This possibility leads to new fractional diffusion and evolution problems that are of particular interest for nonhomogeneous materials where the Fourier law is not simply the negative gradient operator but it is a nonconstant coefficients differential operator of the form $$\begin{aligned} T=\sum _{\ell =1}^3e_\ell a_\ell (x)\partial _{x_\ell }, \ \ \ x=(x_1,x_2,x_3)\in \overline{\Omega }, \end{aligned}$$ T = ∑ ℓ = 1 3 e ℓ a ℓ ( x ) ∂ x ℓ , x = ( x 1 , x 2 , x 3 ) ∈ Ω ¯ , where, $$\Omega $$ Ω can be either a bounded or an unbounded domain in $$\mathbb {R}^3$$ R 3 whose boundary $$\partial \Omega $$ ∂ Ω is considered suitably regular, $$\overline{\Omega }$$ Ω ¯ is the closure of $$\Omega $$ Ω and $$e_\ell $$ e ℓ , for $$\ell =1,2,3$$ ℓ = 1 , 2 , 3 are the imaginary units of the quaternions $$\mathbb {H}$$ H . The operators $$T_\ell :=a_\ell (x)\partial _{x_\ell }$$ T ℓ : = a ℓ ( x ) ∂ x ℓ , for $$\ell =1,2,3$$ ℓ = 1 , 2 , 3 , are called the components of T and $$a_1$$ a 1 , $$a_2$$ a 2 , $$a_3: \overline{\Omega } \subset \mathbb {R}^3\rightarrow \mathbb {R}$$ a 3 : Ω ¯ ⊂ R 3 → R are the coefficients of T. In this paper we study the generation of the fractional powers of T, denoted by $$P_{\alpha }(T)$$ P α ( T ) for $$\alpha \in (0,1)$$ α ∈ ( 0 , 1 ) , when the operators $$T_\ell $$ T ℓ , for $$\ell =1,2,3$$ ℓ = 1 , 2 , 3 do not commute among themselves. To define the fractional powers $$P_{\alpha }(T)$$ P α ( T ) of T we have to consider the weak formulation of a suitable boundary value problem associated with the pseudo S-resolvent operator of T. In this paper we consider two different boundary conditions. If $$\Omega $$ Ω is unbounded we consider Dirichlet boundary conditions. If $$\Omega $$ Ω is bounded we consider the natural Robin-type boundary conditions associated with the generation of the fractional powers of T represented by the operator $$\sum _{\ell =1}^3a_\ell ^2(x)n_\ell (x) \partial _{x_\ell }+a(x)I$$ ∑ ℓ = 1 3 a ℓ 2 ( x ) n ℓ ( x ) ∂ x ℓ + a ( x ) I , for $$x\in \partial \Omega $$ x ∈ ∂ Ω , where I is the identity operator, $$a:\partial \Omega \rightarrow \mathbb {R}$$ a : ∂ Ω → R is a given function and $$n=(n_1,n_2,n_3)$$ n = ( n 1 , n 2 , n 3 ) is the outward unit normal vector to $$\partial \Omega $$ ∂ Ω . The Robin-type boundary conditions associated with the generation of the fractional powers of T are, in general, different from the Robin boundary conditions associated to the heat diffusion problem which leads to operators of the type $$ \sum _{\ell =1}^3a_\ell (x)n_\ell (x) \partial _{x_\ell }+b(x)I$$ ∑ ℓ = 1 3 a ℓ ( x ) n ℓ ( x ) ∂ x ℓ + b ( x ) I , $$x\in \partial \Omega . $$ x ∈ ∂ Ω . For this reason we also discuss the conditions on the coefficients $$a_1$$ a 1 , $$a_2$$ a 2 , $$a_3: \overline{\Omega } \subset \mathbb {R}^3\rightarrow \mathbb {R}$$ a 3 : Ω ¯ ⊂ R 3 → R of T and on the coefficient $$b:\partial \Omega \rightarrow \mathbb {R}$$ b : ∂ Ω → R so that the fractional powers of T are compatible with the physical Robin boundary conditions for the heat equations.

Fractional powers of higher-order vector operators on bounded and unbounded domains

“…x l u , has to control the L 2 -norm of u and the L 2 -norms of all the partial derivatives of u up to order m − 1. In § 3, when Ω is a bounded domain of R 3 , through an iterated use of Poincaré's inequality, we will show that the conditions: (see [24]).…”

“…Using the S-functional calculus, in the series of papers [15,[21][22][23][24], we defined the fractional powers of a class of vector operators with non-constant coefficients. In this paper, we consider the quaternionic differential operators of the form…”

Fractional powers of higher order vector operators on bounded and unbounded domains

“…For this reason, the other positive term in b s (u, u), that is 3 l=1 a l ∂ m x l u , has to control the L 2 -norm of u and the L 2 -norms of all the partial derivatives of u up to order m − 1. In Section 3, when Ω is a bounded domain of R 3 , through an iterated use of Poincaré's inequality we will show that the conditions: When Ω is an unbounded domain of R 3 and m = 1, the role of the Poincaré inequalities is replaced by the Gagliardo-Nirenberg estimates and a condition of integrability on the first derivatives of the coefficients is sufficient to get the coercivity of b s (•, •) (see [21]).…”

“…Using the S-functional calculus, in the series of papers [14], [18], [19], [20], [21], we defined the fractional powers of a class of vector operators with non-constant coefficients. In this paper we consider the quaternionic differential operators of the form…”

Axially harmonic functions and the harmonic functional calculus on the S-spectrum

“…Nowadays there are several research directions in the area of the spectral theory on the Sspectrum, and without claiming completeness we mention: the characteristic operator function, see [1], slice hyperholomorphic Schur analysis, see [5], and several applications to fractional powers of vector operators that describe fractional Fourier's laws for nonhomogeneous materials, see for example [6,17,18]. These results on the fractional powers are based on the H ∞ -functional calculus (see the seminal papers [4], [14]).…”