Recently, the renowned international mathematics journal "Mathematische Annalen"(Math. Ann.) published a collaborative research paper titled “The free boundary for the singular obstacle problem with logarithmic forcing term” by Dr. Zhou Yi, a postdoctoral researcher from the Differential Equations and Applications Research Team at the School of Mathematical Sciences, Shenzhen University, and Professor Du Lili from Sichuan University. This paper investigates the regularity of the free boundary for a class of obstacle problems with logarithmic singularities. Unlike classical obstacle problems, the logarithmic term here tends to negative infinity at the free boundary, exhibiting singularity, and the functional lacks scaling invariance—the growth rate of the solution near the free boundary is r²|log r|, rather than simply r². These inherent difficulties make the analysis of the free boundary extremely challenging. The paper innovatively introduces a Weiss-type energy with a variable parameter and an epiperimetric inequality with an integrable correction term, thereby achieving energy decay and ensuring the uniqueness of the blow-up limit. Based on the energy decay estimate and the uniqueness of the blow-up limit, it is ultimately proven that, in the vicinity of a regular point, the free boundary is a C^{1,β} smooth surface.

Journal Introduction: “Mathematische Annalen” (Mathematical Annals), founded in 1869 by German mathematicians Alfred Clebsch and Carl Neumann, is a top-tier international comprehensive mathematics journal published by Springer in Germany. Renowned for its rigorous manuscript selection process and outstanding academic reputation, it is one of the oldest and most influential journals in the mathematical community. Past editors-in-chief and editorial board members include scientific giants such as Felix Klein, David Hilbert, and physicist Albert Einstein.

Biography: Zhou Yi is a postdoctoral researcher at the School of Mathematical Sciences, Shenzhen University. He received his Ph.D. in 2026 from the School of Mathematics at Sichuan University, specializing in the theory and applications of nonlinear partial differential equations. Since June 2026, he has been conducting postdoctoral research at the Postdoctoral Research Station in Mathematics at Shenzhen University, focusing primarily on partial differential equations and free boundary problems, with a particular emphasis on the mathematical theory of barrier-type free boundary problems.

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https://doi.org/10.1007/s00208-026-03494-6