BIT obtains research result in absolute minimizers regularity of infinite variation

News Resource: School of Mathematics and Statistics

Editor: News Agency of BIT

Translator: Huang Yuwei, News Agency of BIT


Beijing Institute of Technology, September 21st, 2021: Recently, Miao Qianyun, associate research fellow from School of Mathematics and Statistics of Beijing Institute of Technology (BIT), and other collaborators published a research paper titled “Everywhere differentiability of absolute minimizers for locally strongly convex Hamiltonian H(p) ∈C1,1(Rn) with n ≥3n in Journal of Functional Analysis which is a top analytical journal.

What the paper studies is the everywhere differentiability of absolute minimizers for infinite variation corresponding to the general form of Hamilton function. Infinite variation originated from the research of mathematician Aronsson in the 1970s. The existence, uniqueness and especially regularity of  absolute minimizers are important problems, which have attracted the attention and in-depth research of famous mathematicians, such as Crandall, Evans, Jensen and Savin. When Hamilton function H(p)="|p|2 and space dimension n=2, the corresponding Euler-Lagrange variational equation is a famous infinite harmonic equation. Savin proves the absolute minimizers C1 is regularity . Evans and Savin prove the C1, α is regularity. So far, when the space dimension n≥3, it has still been a major unsolved problem whether the absolute minimizers of infinite variation C1 and C1, α is regularity or not. When H(p)=|p|2 and n≥3, Evans and Smart further prove the everywhere differentiability of absolute minimizers. Noting the explicit Hilbert structure of Hamilton function H(p)=|p|2 plays an important role in the proof. Associate research fellow Miao Qianyun and her collaborators overcome the difficulty that the general Hamilton function H(p) does not have an explicit structure. By introducing some new ideas, for space dimension n≥3 and H(p) ∈C1,1(Rn) which satisfies local strong convexity, the everywhere differentiability of absolute minimizers for infinite variation is proved. The reviewer comments: “Compared with the infinity Laplacian operator, general Aronsson is even harder to handle because general convex H lacks elegant structure of |p|2. Some non-trivial and new techniques/ideas are needed. I think this is a very nice progress in the theory of absolute minimizers and Aronsson equations.”

The research work is completed by associate research fellow Miao Qianyun in cooperation with professor Zhou Yuan of Beijing Normal University and Dr. Peng Fa of Beihang University. Associate research fellow Miao Qianyun is the corresponding author. The work is supported by Chinese National Natural Science Foundation and Beijing Institute of Technology Research Fund Program for Young Scholars.

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Research team and person profile attached:

Miao Qianyun is associate research fellow and a member of partial differential equation team from School of Mathematics and Statistics of BIT. She is mainly engaged in the mathematical theory research of infinite variation, p-variation and hydrodynamic equation. And, she has published several high-level academic papers in Arch. Rat. Mech. Anal., J. Functional Analysis, Calc. Var. PDE, Math. Mod. Meth. Appl. Sci. and other authoritative journals.