Pattern formations of coupled PDEs with transparent boundary conditions in product-type ends and applications
Abstract
This paper studies pattern formations in coupled elliptic PDE systems governed by transparent boundary conditions. Such systems unify diverse areas, including inverse boundary problems (via a single passive/active boundary measurement), spectral geometry of transmission eigenfunctions, and geometric characterization of invisibility phenomena and inverse shape problems in wave scattering. We uncover and rigorously characterize a novel local pattern formation, establishing a sharp quantitative relationship between the difference in the PDEs' lower-order terms and the geometric/regularity parameters within a generic domain's product-type ends-structures characterized by high extrinsic curvature. This foundational result yields new findings with novel physical insights and practical implications across these fields.
Keywords
Cite
@article{arxiv.2507.14779,
title = {Pattern formations of coupled PDEs with transparent boundary conditions in product-type ends and applications},
author = {Huaian Diao and Hongyu Liu and Qingle Meng and Li Wang},
journal= {arXiv preprint arXiv:2507.14779},
year = {2025}
}