Fractional powers of vector operators with first order boundary conditions
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Cited by 18 publications
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The aim of this paper is to give an overview of the spectral theories associated with the notions of holomorphicity in dimension greater than one. A first natural extension is the theory of several complex variables whose Cauchy formula is used to define the holomorphic functional calculus for n-tuples of operators $$(A_1,\ldots ,A_n)$$ ( A 1 , … , A n ) . A second way is to consider hyperholomorphic functions of quaternionic or paravector variables. In this case, by the Fueter-Sce-Qian mapping theorem, we have two different notions of hyperholomorphic functions that are called slice hyperholomorphic functions and monogenic functions. Slice hyperholomorphic functions generate the spectral theory based on the S-spectrum while monogenic functions induce the spectral theory based on the monogenic spectrum. There is also an interesting relation between the two hyperholomorphic spectral theories via the F-functional calculus. The two hyperholomorphic spectral theories have different and complementary applications. We finally discuss how to define the fractional Fourier’s law for nonhomogeneous materials using the spectral theory on the S-spectrum.
The aim of this paper is to give an overview of the spectral theories associated with the notions of holomorphicity in dimension greater than one. A first natural extension is the theory of several complex variables whose Cauchy formula is used to define the holomorphic functional calculus for n-tuples of operators $$(A_1,\ldots ,A_n)$$ ( A 1 , … , A n ) . A second way is to consider hyperholomorphic functions of quaternionic or paravector variables. In this case, by the Fueter-Sce-Qian mapping theorem, we have two different notions of hyperholomorphic functions that are called slice hyperholomorphic functions and monogenic functions. Slice hyperholomorphic functions generate the spectral theory based on the S-spectrum while monogenic functions induce the spectral theory based on the monogenic spectrum. There is also an interesting relation between the two hyperholomorphic spectral theories via the F-functional calculus. The two hyperholomorphic spectral theories have different and complementary applications. We finally discuss how to define the fractional Fourier’s law for nonhomogeneous materials using the spectral theory on the S-spectrum.
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.
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