My research is primarily concerned with Nonlinear Partial Differential Equations. In particular, my interests include but are not limited to the following topics:
Nonlocal geometric variational problems such as block copolymer morphology,
Pattern formation problems modeling morphogenesis in cell development,
Nonlinear elliptic partial differential equations especially of Allen-Cahn type,
Mean field type equations on Torus and Sphere.
My PhD thesis title: Disc Assemblies and Spike Assemblies in Inhibitory Systems.
Publications:
with Changfeng Gui and Weihong Xie: Improved Beckner's inequality for axially symmetric functions on Sn, accepted by J. Funct. Anal.
with Aleks Jevnikar and Weihong Xie: Infinitely many solutions for Schrodinger-Newton equation, accepted by Commun. Contemp. Math.
with Changfeng Gui and Weihong Xie: Improved Beckner's inequality for axially symmetric functions on S4, submitted.
with Changfeng Gui and Qinfeng Li: Extensions and Traces of BV functions on Rough domains and Generalized Cheeger sets, preprint.
with Daniele Bartolucci, Aleks Jevnikar and Wen Yang: Generic properties of free boundary problems in plasma physics, accepted by Nonlinearity.
with Changfeng Gui and Qinfeng Li: On smooth interior approximation of sets of finite perimeter, accepted by Proc. Amer. Math. Soc.
with Mostafa Fazly and Wen Yang: On stable and finite Morse index solutions of the nonlocal Hénon-Gelfand-Liouville equation, Calc. Var. Partial Differential Equations 60 (2021), no. 1, Paper No. 11, 26 pp.
with Guangze Gu, Changfeng Gui and Qinfeng Li: Uniqueness and symmetry results of a mean field equation on arbitrary flat tori, accepted by Int. Math. Res. Not. IMRN.
with Daniele Bartolucci, Changfeng Gui, Aleks Jevnikar and Wen Yang: Mean field equation on tori: existence and uniqueness of evenly symmetric blow-up solutions, Discrete Contin. Dyn. Syst. 40(6):3093-3116, 2020.
with Changfeng Gui: Non-axially symmetric solutions of a mean field equation on S2, Adv. Calc. Var. 14(3):419-439, 2021.
with Ze Cheng and Changfeng Gui: Blow-up solutions for a mean field equation on a flat torus, Indiana Univ. Math. J. 69(2):453-485, 2020.
with Ze Cheng and Changfeng Gui: The existence of solution to supercritical Hardy-Littlewood-Sobolev system with fractional laplacian, Discrete Contin. Dyn. Syst. , 39(3):1345-1358, 2019.
with Chao-Nien Chen, Yung-Sze Choi and Xiaofeng Ren: Higher dimensional bubble profiles in a singular limit of the FitzHugh-Nagumo system, SIAM J. Math. Anal. 50(5):5072-5095, 2018.
Interior and Boundary Spikes for the Two-dimensional Gierer-Meinhardt System, J. Differential Equations , 262(11):5363 - 5408, 2017.
Stationary Disc Assemblies on Inhibitory Vesicles, preprint.