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

TL;DR

This paper investigates pattern formations in coupled elliptic PDE systems with transparent boundary conditions, revealing a new local pattern formation linked to PDE differences and domain geometry, with broad applications in inverse problems and wave scattering.

Contribution

It introduces a rigorous characterization of a novel local pattern formation in coupled PDEs with transparent boundaries, connecting PDE differences to domain geometry in product-type ends.

Findings

Established a quantitative relationship between PDE differences and geometric parameters.

Uncovered a new local pattern formation in coupled elliptic PDE systems.

Provided insights applicable to inverse boundary problems and wave scattering phenomena.

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.