Workshop on Geometric PDEs

“Workshop on Geometric PDEs”  

 March27-March 29, 2024  

organized by  

Guohuan Qiu, Liming Sun  

All times are local to Beijing  

Titles and abstract:    

Bin Zhou (Peking University)    

Title: Regularity of variational problems with a convexity constraint  

Abstract: We establish the interior $C^{1,\alpha}$ regularity of minimizers of a class of functionals with a convexity constraint, which includes the principal-agent problems studied by Figalli-Kim-McCann. The  $C^{1,1}$ regularity was previously proved by  Caffarelli-Lions in an unpublished note when the cost is quadratic, and recently extended to the case where the cost is uniformly convex with respect to a general preference function by McCann-Rankin-Zhang(arXiv:2303.04937). Our main result does not require the uniform convexity assumption on the cost function. In particular, we show that the solutions  to the principal-agent problems with $q$-power cost are $C^{1,\frac{1}{q-1}}$ when $q > 2$ and $C^{1,1}$ when $1<q\leq 2$. Examples can show that this regularity is optimal when $q\geq 2$.  

Yuhua Sun (Nankai University)  

Title: A priori estimates and Liouville type results for quasilinear equations with gradient terms  

Abstract: We study local and global properties of positive solutions of quasilinear equations with gradient term, by using a direct Bernstein method combined with Keller–Osserman’s estimate, we obtain several a priori estimates as well as Liouville type theorems. This is joint work with Prof. Filippucci, and Yadong Zheng.  

Yuxing Deng ( Beijing Institute of Technology)    

Title: Rigidity of postively curved steady ricci solitons on manifolds and orbifolds.  

Abstract: Steady ricci solitons are important examples of singularities models. In higher dimensions, singlarity models can be steady Ricci solitons on orbifolds. In this talk, we will review some rigidity theorems on positively curved steady ricci solitons on manifolds. We will also classify positively curved noncollapsed steady ricci solitons on orbifolds that dimension recudce to quotients of spheres.  

Miaomiao Zhu (Shanghai Jiao Tong University)    

Title: Compensated integrability and applications to geometric variational problems  

Abstract: In this talk, we illustrate how the principle of compensated integrability can be applied to the analysis of critical elliptic systems and then present some recent progress on regularity and blow-up analysis for geometric variational problems.