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Nonlocal problems for Laplace equation in Bochner spaces
Abstract
We study the Laplace equation posed in the unbounded rectangular domain Π= I × (0,∞) with I= (0,2π), and subject to nonlocal boundary conditions on ∂ Π in the trace sense. The analysis is carried out in the Bochner-Sobolev space W2p,1(Π;X), associated with the Bochner space Lp,1(Π;X), with $ p \in (1,\infty)$ and X is a suitable Banach space. To solve the problem, we employ a generalized spectral method. In particular, we introduce the notion of ⊗-basis generated by tensor products and extend the classical scheme known from the scalar case to the present setting. Moreover, we prove that the system of root functions of the corresponding nonlocal spectral problem forms a ⊗-basis in Lp(I;X).
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BibTeX
@misc{bilalov2026nonlocal,
title = {{Nonlocal problems for Laplace equation in Bochner spaces}},
author = {Bilal Bilalov and Sabina Sadigova and Pia Salerno and Lubomira Softova},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2605.25761},
eprint = {2605.25761},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.25761},
}
RIS
TY - UNPB
TI - Nonlocal problems for Laplace equation in Bochner spaces
AU - Bilal Bilalov
AU - Sabina Sadigova
AU - Pia Salerno
AU - Lubomira Softova
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2605.25761
UR - https://arxiv.org/abs/2605.25761
AB - We study the Laplace equation posed in the unbounded rectangular domain Π= I × (0,∞) with I= (0,2π), and subject to nonlocal boundary conditions on ∂ Π in the trace sense. The analysis is carried out in the Bochner-Sobolev space W^2_p,1(Π;X), associated with the Bochner space L^p,1(Π;X), with $ p \in (1,\infty)$ and X is a suitable Banach space. To solve the problem, we employ a generalized spectral method. In particular, we introduce the notion of ⊗-basis generated by tensor products and extend the classical scheme known from the scalar case to the present setting. Moreover, we prove that the system of root functions of the corresponding nonlocal spectral problem forms a ⊗-basis in L^p(I;X).
ER -