This thesis studies the extension problem for higherorder fractional powers of the heat operator $H=\partial_t  \Delta$ in $\mathbb{R}^{n+1}$. Specifically, given $s>0$ and indicating with $[s]$ its integral part, we study the following degenerate partial differential equation in the thick space $\mathbb{R}^{n+1}\times \mathbb{R}_y^+$, \begin{equation} \label{a:1} \mathscr{H}^{[s]+1}= \left( \partial_{yy} +\frac{a}{y}\partial_y H \right)^{[s]+1}U=0. \end{equation} The connection between the Bessel parameter $a$ in \eqref{a:1} and the fractional parameter $s>0$ is given by the equation \begin{equation*} a= 12(s[s]). \end{equation*} When $s\in(0,1)$ this equation reduces to the wellknown relation $a=12s$, and in such case \eqref{a:1} becomes the famous CaffarelliSilvestre extension problem. Generalising their result, in this thesis we show that the nonlocal operator $H^s$ can be realised as the DirichlettoNeumann map associated with the solution $U$ of the extension equation \eqref{a:1}. In this thesis we systematically exploit the evolutive semigroup $\{P_{\tau}^H \}_{\tau>0}$, associated with the Cauchy problem \begin{equation*} \begin{cases} \partial_{\tau}u+Hu=0\\ u((x,t),0)=f(x,t). \end{cases} \end{equation*} This approach provides a powerful tool in analysis, and it has the twofold advantage of allowing an independent treatment of several complex calculations involving the Fourier transform, while at same time extending to frameworks where the Fourier transform is not available.
This thesis studies the extension problem for higherorder fractional powers of the heat operator $H=\partial_t  \Delta$ in $\mathbb{R}^{n+1}$. Specifically, given $s>0$ and indicating with $[s]$ its integral part, we study the following degenerate partial differential equation in the thick space $\mathbb{R}^{n+1}\times \mathbb{R}_y^+$, \begin{equation} \label{a:1} \mathscr{H}^{[s]+1}= \left( \partial_{yy} +\frac{a}{y}\partial_y H \right)^{[s]+1}U=0. \end{equation} The connection between the Bessel parameter $a$ in \eqref{a:1} and the fractional parameter $s>0$ is given by the equation \begin{equation*} a= 12(s[s]). \end{equation*} When $s\in(0,1)$ this equation reduces to the wellknown relation $a=12s$, and in such case \eqref{a:1} becomes the famous CaffarelliSilvestre extension problem. Generalising their result, in this thesis we show that the nonlocal operator $H^s$ can be realised as the DirichlettoNeumann map associated with the solution $U$ of the extension equation \eqref{a:1}. In this thesis we systematically exploit the evolutive semigroup $\{P_{\tau}^H \}_{\tau>0}$, associated with the Cauchy problem \begin{equation*} \begin{cases} \partial_{\tau}u+Hu=0\\ u((x,t),0)=f(x,t). \end{cases} \end{equation*} This approach provides a powerful tool in analysis, and it has the twofold advantage of allowing an independent treatment of several complex calculations involving the Fourier transform, while at same time extending to frameworks where the Fourier transform is not available.
The extension problem for fractional powers of higher order of some evolutive operators
GALLATO, PIETRO
2021/2022
Abstract
This thesis studies the extension problem for higherorder fractional powers of the heat operator $H=\partial_t  \Delta$ in $\mathbb{R}^{n+1}$. Specifically, given $s>0$ and indicating with $[s]$ its integral part, we study the following degenerate partial differential equation in the thick space $\mathbb{R}^{n+1}\times \mathbb{R}_y^+$, \begin{equation} \label{a:1} \mathscr{H}^{[s]+1}= \left( \partial_{yy} +\frac{a}{y}\partial_y H \right)^{[s]+1}U=0. \end{equation} The connection between the Bessel parameter $a$ in \eqref{a:1} and the fractional parameter $s>0$ is given by the equation \begin{equation*} a= 12(s[s]). \end{equation*} When $s\in(0,1)$ this equation reduces to the wellknown relation $a=12s$, and in such case \eqref{a:1} becomes the famous CaffarelliSilvestre extension problem. Generalising their result, in this thesis we show that the nonlocal operator $H^s$ can be realised as the DirichlettoNeumann map associated with the solution $U$ of the extension equation \eqref{a:1}. In this thesis we systematically exploit the evolutive semigroup $\{P_{\tau}^H \}_{\tau>0}$, associated with the Cauchy problem \begin{equation*} \begin{cases} \partial_{\tau}u+Hu=0\\ u((x,t),0)=f(x,t). \end{cases} \end{equation*} This approach provides a powerful tool in analysis, and it has the twofold advantage of allowing an independent treatment of several complex calculations involving the Fourier transform, while at same time extending to frameworks where the Fourier transform is not available.File  Dimensione  Formato  

Gallato_Pietro.pdf
accesso aperto
Dimensione
662.66 kB
Formato
Adobe PDF

662.66 kB  Adobe PDF  Visualizza/Apri 
The text of this website © Università degli studi di Padova. Full Text are published under a nonexclusive license. Metadata are under a CC0 License
https://hdl.handle.net/20.500.12608/35546