A research team from Shandong University, Tsinghua University, and Purdue University has made progress in the energy identity and bubble analysis of a class of fourth-order geometric elliptic systems in the critical dimension. The work, co-authored by Changyu Guo, Wenjuan Qi, Zhaomin Sun, and Changyou Wang, is published in Acta Mathematica Scientia.
The study focuses on the Lamm-Rivière system, a fourth-order geometric elliptic system including both extrinsic and intrinsic biharmonic maps into manifolds. In four dimension, due to the scaling invariance nature of system, a sequence of weak solutions may possibly lose compactness at singular points, which leads to energy concentration. To explore this "blow-up" process, the researchers investigated the inhomogeneous Lamm-Rivière system in four dimensions, particularly analyzing the asymptotic behavior of weak solutions when the inhomogeneous term merely belongs to LlogL.
The authors follow Laurain-Rivière's strategy and use the Lorentz duality method as a main tool. A major challenge is to establish the a priori L^{2,1} estimate for the second order angular derivatives in annular regions. A new observation is that the radial symmetric harmonic function has the minimal L^{2,1} norm among all harmonic functions with natural boundary conditions. Another new ingredient is that the authors successfully established the "bubble tree" decomposition, aking to harmonic maps, which completely characterized the asymptotic behavior of angular energy during the weak convergence and rules out any angular energy loss. Combining the above ingredients, the research team finally proved the angular energy identity for the fourth-order inhomogeneous Lamm-Rivière system.
This work extends the scope of research to inhomogeneous equations with borderline integrability, generalizing all previous works on energy identity for biharmonic maps. The results not only extend the regularity theory for elliptic systems and enrich the energy identity theory for approximate biharmonic maps, but also provide deeper understanding of concentrated-compactness phenomena in higher-order geometric variational problems.
See the article:
The Lamm-Rivière system II: energy identity
https://doi.org/10.1007/s10473-026-0410-0
