Resolvents for fractional-order operators with nonhomogeneous local boundary conditions
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Resolvents for fractional-order operators with nonhomogeneous local boundary conditions. / Grubb, Gerd.
In: Journal of Functional Analysis, Vol. 284, No. 7, 109815, 2023.Research output: Contribution to journal › Journal article › Research › peer-review
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TY - JOUR
T1 - Resolvents for fractional-order operators with nonhomogeneous local boundary conditions
AU - Grubb, Gerd
N1 - Publisher Copyright: © 2022 The Author(s)
PY - 2023
Y1 - 2023
N2 - For 2a-order strongly elliptic operators P generalizing (−Δ)a, 0n has been widely studied. Pseudodifferential methods have been applied by the present author when Ω is smooth; this is extended in a recent joint work with Helmut Abels showing exact regularity theorems in the scale of Lq-Sobolev spaces Hqs for 1τ+1 with a finite τ>2a. We now develop this into existence-and-uniqueness theorems (or Fredholm theorems), by a study of the Lp-Dirichlet realizations of P and P⁎, showing that there are finite-dimensional kernels and cokernels lying in daCα(Ω‾) with suitable α>0, d(x)=dist(x,∂Ω). Similar results are established for P−λI, λ∈C. The solution spaces equal a-transmission spaces Hqa(t)(Ω‾). Moreover, the results are extended to nonhomogeneous Dirichlet problems prescribing the local Dirichlet trace (u/da−1)|∂Ω. They are solvable in the larger spaces Hq(a−1)(t)(Ω‾). Furthermore, the nonhomogeneous problem with a spectral parameter λ∈C, Pu−λu=f in Ω,u=0 in Rn∖Ω,(u/da−1)|∂Ω=φ on ∂Ω, is for q<(1−a)−1 shown to be uniquely resp. Fredholm solvable when λ is in the resolvent set resp. the spectrum of the L2-Dirichlet realization. The results open up for applications of functional analysis methods. Here we establish solvability results for evolution problems with a time-parameter t, both in the case of the homogeneous Dirichlet condition, and the case where a nonhomogeneous Dirichlet trace (u(x,t)/da−1(x))|x∈∂Ω is prescribed.
AB - For 2a-order strongly elliptic operators P generalizing (−Δ)a, 0n has been widely studied. Pseudodifferential methods have been applied by the present author when Ω is smooth; this is extended in a recent joint work with Helmut Abels showing exact regularity theorems in the scale of Lq-Sobolev spaces Hqs for 1τ+1 with a finite τ>2a. We now develop this into existence-and-uniqueness theorems (or Fredholm theorems), by a study of the Lp-Dirichlet realizations of P and P⁎, showing that there are finite-dimensional kernels and cokernels lying in daCα(Ω‾) with suitable α>0, d(x)=dist(x,∂Ω). Similar results are established for P−λI, λ∈C. The solution spaces equal a-transmission spaces Hqa(t)(Ω‾). Moreover, the results are extended to nonhomogeneous Dirichlet problems prescribing the local Dirichlet trace (u/da−1)|∂Ω. They are solvable in the larger spaces Hq(a−1)(t)(Ω‾). Furthermore, the nonhomogeneous problem with a spectral parameter λ∈C, Pu−λu=f in Ω,u=0 in Rn∖Ω,(u/da−1)|∂Ω=φ on ∂Ω, is for q<(1−a)−1 shown to be uniquely resp. Fredholm solvable when λ is in the resolvent set resp. the spectrum of the L2-Dirichlet realization. The results open up for applications of functional analysis methods. Here we establish solvability results for evolution problems with a time-parameter t, both in the case of the homogeneous Dirichlet condition, and the case where a nonhomogeneous Dirichlet trace (u(x,t)/da−1(x))|x∈∂Ω is prescribed.
KW - Evolution equation
KW - Fractional-order pseudodifferential operator
KW - Nonhomogeneous local Dirichlet condition
KW - Spectral parameter
UR - http://www.scopus.com/inward/record.url?scp=85146605299&partnerID=8YFLogxK
U2 - 10.1016/j.jfa.2022.109815
DO - 10.1016/j.jfa.2022.109815
M3 - Journal article
AN - SCOPUS:85146605299
VL - 284
JO - Journal of Functional Analysis
JF - Journal of Functional Analysis
SN - 0022-1236
IS - 7
M1 - 109815
ER -
ID: 371656461