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arXiv · 2507.14779

Pattern formations of coupled PDEs with transparent boundary conditions in product-type ends and applications

Huaian Diao·Hongyu Liu·Qingle Meng·Li Wang
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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.

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Huaian Diao, Hongyu Liu, Qingle Meng, Li Wang. 2025-07-20. Pattern formations of coupled PDEs with transparent boundary conditions in product-type ends and applications. https://arxiv.org/abs/2507.14779

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