Resolvents for fractional-order operators with nonhomogeneous local boundary conditions
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Publication:6384266
DOI10.1016/J.JFA.2022.109815zbMath1509.35399arXiv2111.14763MaRDI QIDQ6384266
Publication date: 29 November 2021
Abstract: For $2a$-order strongly elliptic operators $P$ generalizing $(-Delta )^a$, $0<a<1$, the treatment of the homogeneous Dirichlet problem on a bounded open set $Omega subset R^n$ by pseudodifferential methods, has been extended in a recent joint work with Helmut Abels to nonsmooth settings, showing regularity theorems in $L_q$-Sobolev spaces $H_q^s$ for $1<q<infty $, when $Omega $ is $C^{ au +1}$ with a finite $ au >2a$. Presently, we study the $L_q$-Dirichlet realizations of $P$ and $P^*$, showing invertibility or Fredholmness, finding smoothness results for the kernels and cokernels, and establishing similar results for $P-lambda I$, $lambda in C$. The solution spaces equal $a$-transmission spaces $H_q^{a(s+2a)}(�arOmega)$. Similar results are shown for nonhomogeneous Dirichlet problems, prescribing the local Dirichlet trace $(u/d^{a-1})|_{partialOmega }$, $d(x)=dist(x,partialOmega)$. They are solvable in the larger spaces $H_q^{(a-1)(s+2a)}(�arOmega)$. Moreover, the nonhomogeneous problem with a spectral parameter $lambda in C$, $$ Pu-lambda u = f ext { in }Omega ,quad u=0 ext { in }R^nsetminus Omega ,quad (u/d^{a-1 })|_{partialOmega }=varphi ext{ on }partialOmega , $$ is for $q<(1-a)^{-1}$ shown to be uniquely resp. Fredholm solvable when $lambda $ is in the resolvent set resp. the spectrum of the $L_2$-Dirichlet realization. Finally, we show solvability results for evolution problems $Pu+d_tu= f(x,t)$ in $L_2$ and $L_q$-based spaces over $C^{1+ au}$-domains, including nonhomogeneous local boundary conditions.
Processes with independent increments; Lévy processes (60G51) Boundary value problems for second-order elliptic equations (35J25) Initial-boundary value problems for second-order parabolic equations (35K20) Boundary value problems for PDEs with pseudodifferential operators (35S15) Pseudodifferential operators (47G30) Fractional partial differential equations (35R11)
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