Sharp extension problem characterizations for higher fractional power operators in Banach spaces
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2023
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866929319892221952 |
|---|---|
| author | Biswas, A. Stinga, P. R. |
| author_facet | Biswas, A. Stinga, P. R. |
| contents | We prove sharp characterizations of higher order fractional powers $(-L)^s$, where $s>0$ is noninteger, ofgenerators $L$ of uniformly bounded $C_0$-semigroups on Banach spaces via extension problems, which in particular include results of Caffarelli-Silvestre, Stinga-Torrea and Galé-Miana-Stinga when $0<s<1$. More precisely, we prove existence and uniqueness of solutions $U(y)$, $y\geq0$, to initial value problems for both higher order and second order extension problems and characterizations of $(-L)^su$, $s>0$, in terms of boundary derivatives of $U$ at $y=0$, under the sharp hypothesis that $u$ is in the domain of $(-L)^s$. Our results resolve the question of setting up the correct initial conditions that guarantee well-posedness of both extension problems. Furthermore, we discover new explicit subordination formulas for the solution $U$ in terms of the semigroup $\{e^{tL}\}_{t\geq0}$ generated by $L$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_12512 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Sharp extension problem characterizations for higher fractional power operators in Banach spaces Biswas, A. Stinga, P. R. Analysis of PDEs Classical Analysis and ODEs Functional Analysis We prove sharp characterizations of higher order fractional powers $(-L)^s$, where $s>0$ is noninteger, ofgenerators $L$ of uniformly bounded $C_0$-semigroups on Banach spaces via extension problems, which in particular include results of Caffarelli-Silvestre, Stinga-Torrea and Galé-Miana-Stinga when $0<s<1$. More precisely, we prove existence and uniqueness of solutions $U(y)$, $y\geq0$, to initial value problems for both higher order and second order extension problems and characterizations of $(-L)^su$, $s>0$, in terms of boundary derivatives of $U$ at $y=0$, under the sharp hypothesis that $u$ is in the domain of $(-L)^s$. Our results resolve the question of setting up the correct initial conditions that guarantee well-posedness of both extension problems. Furthermore, we discover new explicit subordination formulas for the solution $U$ in terms of the semigroup $\{e^{tL}\}_{t\geq0}$ generated by $L$. |
| title | Sharp extension problem characterizations for higher fractional power operators in Banach spaces |
| topic | Analysis of PDEs Classical Analysis and ODEs Functional Analysis |
| url | https://arxiv.org/abs/2309.12512 |