Asymptotic approximation for the solution to a semi-linear elliptic problem in a thin aneurysm-type domain

• Main Authors: Mel’nyk Taras A., Klevtsovskiy Arsen V.
• Format: Article
• Language: English
• Published: De Gruyter 2017-11-01
• Series: Open Mathematics
• Subjects: multiscale analysis; thin aneurysm-type domains; asymptotic approximation; semi-linear elliptic problem; 35b25; 35j65; 35b40; 74k30
• Online Access: http://www.degruyter.com/view/j/math.2017.15.issue-1/math-2017-0114/math-2017-0114.xml?format=INT

A semi-linear boundary-value problem with nonlinear Robin boundary conditions is considered in a thin 3D aneurysm-type domain that consists of thin curvilinear cylinders that are joined through an aneurysm of diameter 𝓞(ε). Using the multi-scale analysis, the asymptotic approximation for the solution is constructed and justified as the parameter ε → 0. Namely, we derive the limit problem (ε = 0) in the corresponding graph, define other terms of the asymptotic approximation and prove energetic and uniform pointwise estimates. These estimates allow us to observe the impact of the aneurysm on some properties of the solution.