Fourier methods for fractional-order operators
Fuente:
arXiv
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2022
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866929746358566912 |
|---|---|
| author | Grubb, Gerd |
| author_facet | Grubb, Gerd |
| contents | This is a survey on the use of Fourier transformation methods in the treatment of boundary problems for the fractional Laplacian $(-Δ)^a$ (0<a<1), and pseudodifferential generalizations P, over a bounded open set $Ω$ in $R^n$. The presentation starts at an elementary level. Two points are explained in detail: 1) How the factor $d^a$, with $d(x)=dist(x,dΩ)$, comes into the picture, related to the fact that the precise solution spaces for the homogeneous Dirichlet problem are so-called a-transmission spaces. 2) The natural definition of a local nonhomogeneous Dirichlet condition $γ_0(u/d^{a-1})=φ$. We also give brief accounts of some further developments: Evolution problems (for $d_t u - r^+Pu = f(x,t)$) and resolvent problems (for $Pu-λu=f$), also with nonzero boundary conditions. Integration by parts, Green's formula. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_07175 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Fourier methods for fractional-order operators Grubb, Gerd Analysis of PDEs Spectral Theory 35S15, 47G30, 35J25, 35K05, 60G51 This is a survey on the use of Fourier transformation methods in the treatment of boundary problems for the fractional Laplacian $(-Δ)^a$ (0<a<1), and pseudodifferential generalizations P, over a bounded open set $Ω$ in $R^n$. The presentation starts at an elementary level. Two points are explained in detail: 1) How the factor $d^a$, with $d(x)=dist(x,dΩ)$, comes into the picture, related to the fact that the precise solution spaces for the homogeneous Dirichlet problem are so-called a-transmission spaces. 2) The natural definition of a local nonhomogeneous Dirichlet condition $γ_0(u/d^{a-1})=φ$. We also give brief accounts of some further developments: Evolution problems (for $d_t u - r^+Pu = f(x,t)$) and resolvent problems (for $Pu-λu=f$), also with nonzero boundary conditions. Integration by parts, Green's formula. |
| title | Fourier methods for fractional-order operators |
| topic | Analysis of PDEs Spectral Theory 35S15, 47G30, 35J25, 35K05, 60G51 |
| url | https://arxiv.org/abs/2208.07175 |