ICCA2020:P000115

Recent results on fractional powers of vector operators and applications

*Fabrizio Colombo (Politecnico di Milano)

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
$$
T=\sum_{\ell=1}^3e_\ell a_\ell(x)\partial_{x_\ell}, \ \ \ x=(x_1,x_2,x_3)\in \overline{\Omega},
$$
where, $\Omega$ can be either a bounded or an unbounded domain in $\mathbb{R}^3$ whose boundary $\partial\Omega$ is considered suitably regular,
$\overline{\Omega}$ is the closure of $\Omega$
 and $e_\ell$, for $\ell=1,2,3$ are the imaginary units of the quaternions $\mathbb{H}$.
 The operators $T_\ell:=a_\ell(x)\partial_{x_\ell}$, for $\ell=1,2,3$, are called the components of $T$ and
 $a_1$, $a_2$, $a_3: \overline{\Omega} \subset\mathbb{R}^3\to \mathbb{R}$ are the coefficients of $T$.

In this talk we show some recent results on the generation of the fractional powers of $T$, 
denoted by $P_{\alpha}(T)$ for $\alpha\in(0,1)$,
 when the operators
$T_\ell$, for $\ell=1,2,3$ do not necessarily commute among themselves.
To define the fractional powers $P_{\alpha}(T)$ of $T$
we have to consider the weak formulation of a suitable boundary value problem
associated with the pseudo $S$-resolvent operator of $T$.
Some references for this talk are:


\begin{thebibliography}{10}


\bibitem{}
F. Colombo, J. Gantner,
{\em Quaternionic closed operators, fractional powers and fractional diffusion processes},
 Operator Theory: Advances and Applications, 274. Birkh\"auser/Springer, Cham, 2019. viii+322 pp.
 ISBN: 978--3--030--16408--9; 978--3--030--16409.

\bibitem{}
 F. Colombo, J. Gantner, D.P. Kimsey,
{\em Spectral theory on the $S$-spectrum for quaternionic operators},
Operator Theory: Advances and Applications, 270.
Birkh\"auser/Springer, Cham, 2018. ix+356 pp. 
ISBN: 978-3-030-03073-5; 978-3-030-03074-2 47-02.


\bibitem{}
F. Colombo, D. Deniz-Gonzales,  S. Pinton,
{\em Fractional powers of vector operators with first order boundary conditions},
  J. Geom. Phys., {\bf 151} (2020), 103618.

\bibitem{}
F. Colombo, M. Peloso, S. Pinton,
{\em The structure of the fractional powers of the noncommutative Fourier law},
 Math. Methods Appl. Sci., {\bf 42} (2019), no. 18, 6259--6276.


\end{thebibliography}

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 $$ T=\sum_{\ell=1}^3e_\ell a_\ell(x)\partial_{x_\ell}, \ \ \ x=(x_1,x_2,x_3)\in \overline{\Omega}, $$ where, $\Omega$ can be either a bounded or an unbounded domain in $\mathbb{R}^3$ whose boundary $\partial\Omega$ is considered suitably regular, $\overline{\Omega}$ is the closure of $\Omega$ and $e_\ell$, for $\ell=1,2,3$ are the imaginary units of the quaternions $\mathbb{H}$. The operators $T_\ell:=a_\ell(x)\partial_{x_\ell}$, for $\ell=1,2,3$, are called the components of $T$ and $a_1$, $a_2$, $a_3: \overline{\Omega} \subset\mathbb{R}^3\to \mathbb{R}$ are the coefficients of $T$. In this talk we show some recent results on the generation of the fractional powers of $T$, denoted by $P_{\alpha}(T)$ for $\alpha\in(0,1)$, when the operators $T_\ell$, for $\ell=1,2,3$ do not necessarily commute among themselves. To define the fractional powers $P_{\alpha}(T)$ of $T$ we have to consider the weak formulation of a suitable boundary value problem associated with the pseudo $S$-resolvent operator of $T$. Some references for this talk are: \begin{thebibliography}{10} \bibitem{} F. Colombo, J. Gantner, {\em Quaternionic closed operators, fractional powers and fractional diffusion processes}, Operator Theory: Advances and Applications, 274. Birkh\"auser/Springer, Cham, 2019. viii+322 pp. ISBN: 978--3--030--16408--9; 978--3--030--16409. \bibitem{} F. Colombo, J. Gantner, D.P. Kimsey, {\em Spectral theory on the $S$-spectrum for quaternionic operators}, Operator Theory: Advances and Applications, 270. Birkh\"auser/Springer, Cham, 2018. ix+356 pp. ISBN: 978-3-030-03073-5; 978-3-030-03074-2 47-02. \bibitem{} F. Colombo, D. Deniz-Gonzales, S. Pinton, {\em Fractional powers of vector operators with first order boundary conditions}, J. Geom. Phys., {\bf 151} (2020), 103618. \bibitem{} F. Colombo, M. Peloso, S. Pinton, {\em The structure of the fractional powers of the noncommutative Fourier law}, Math. Methods Appl. Sci., {\bf 42} (2019), no. 18, 6259--6276. \end{thebibliography}

The 12th International Conference on Clifford Algebras and Their Applications in Mathematical Physics

Recent results on fractional powers of vector operators and applications