- Research interests: My research interests are mainly in the theory of partial differential equations and analysis. More specifically, I have been working on: fast diffusion equations, conformally invariant equations, integro-differential equations, p-Laplace equations, Monge-Ampère equations and reaction-diffusion equations.
- Publications:
- On the sharp constants in the regional fractional Sobolev inequalities (with Rupert L. Frank and Wei Wang), preprint.
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Regularity of solutions to the Dirichlet problem for fast diffusion equations (with Jingang Xiong), preprint.
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Optimal regularity and fine asymptotics for the porous medium equation in bounded domains (with Xavier Ros-Oton and Jingang Xiong), J. Reine Angew. Math. 809 (2024), 269--300
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Hölder regularity for the linearized porous medium equation in bounded domains (with Jingang Xiong), to appear in Anal. Theory Appl.
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Stability of the separable solutions for a nonlinear boundary diffusion problem (with Jingang Xiong and Xuzhou Yang), J. Math. Pures Appl. (9) 183 (2024), 1--43.
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Bubbling and extinction for some fast diffusion equations in bounded domains (with Jingang Xiong), Trans. Amer. Math. Soc. Ser. B 10 (2023), 1287--1332.
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Blow up limits of the fractional Laplacian and their applications to the fractional Nirenberg problem (with Xusheng Du, Jingang Xiong and Hui Yang), Proc. Amer. Math. Soc. 151 (2023), no. 11, 4693--4701.
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Existence of solutions to a conformally invariant integral equation involving Poisson-type kernels (with Xusheng Du and Hui Yang), J. Geom. Anal. 33 (2023), no. 9, Paper No. 286, 21 pp.
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Local estimates for conformal Q-curvature equations (with Hui Yang), J. Funct. Anal. 285 (2023), no. 2, Paper No. 109940.
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Optimal boundary regularity for fast diffusion equations in bounded domains (with Jingang Xiong), Amer. J. Math. 145 (2023), no. 1, 151--219.
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Singular extinction profiles of solutions to some fast diffusion equations (with Jingang Xiong), J. Funct. Anal. 283 (2022), no. 7, Paper No. 109595, 29 pp.
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On a Rayleigh-Faber-Krahn inequality for the regional fractional Laplacian (with Dennis Kriventsov and Jingang Xiong), Ann. Appl. Math. 37 (2021), no. 3, 363--393.
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Asymptotic symmetry and local behavior of solutions of higher order conformally invariant equations with isolated singularities (with Jingang Xiong), Ann. Inst. H. Poincaré Anal. Non Linéaire, 38 (2021), no. 4, 1167--1216.
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An existence theorem on the isoperimetric ratio over scalar-flat conformal classes (with Xuezhang Chen and Yuping Ruan), J. Differential Equations 269 (2020), no. 5, 4116--4136.
- Hölder gradient estimates for a class of singular or degenerate parabolic equations (with Cyril Imbert and Luis Silvestre), Adv. Nonlinear Anal. 8 (2019), no. 1, 845--867.
- On the isoperimetric quotient over scalar-flat conformal classes (with Jingang Xiong), Comm. Partial Differential Equations 43 (2018), no. 12, 1737--1760.
- Dini and Schauder estimates for nonlocal fully nonlinear parabolic equations with drifts (with Hongjie Dong and Hong Zhang), Anal. PDE 11 (2018), no. 6, 1487--1534.
- Minimizers for the fractional Sobolev inequality on domains (with Rupert L. Frank and Jingang Xiong), Calc. Var. Partial Differential Equations 57 (2018), no. 2, Art. 43, 31 pp.
- Potential singularity for a family of models of the axisymmetric incompressible flow (with Thomas Y. Hou and Pengfei Liu), J. Nonlinear Sci. 28 (2018), no. 6, 2217--2247.
- Schauder estimates for an integro-differential equation with applications to a nonlocal Burgers equation (with Cyril Imbert and Roman Shvydkoy), Ann. Fac. Sci. Toulouse Math. (6) 27 (2018), no. 4, 667--677.
- The Nirenberg problem and its generalizations: A unified approach (with YanYan Li and Jingang Xiong), Math. Ann. 369 (2017), no. 1-2, 109--151.
- Hölder gradient estimates for parabolic homogeneous p-Laplacian equations (with Luis Silvestre), J. Math. Pures Appl. (9) 108 (2017), no. 1, 63--87.
- On local behavior of singular positive solutions to nonlocal elliptic equations (with Olivaine S. de Queiroz, Yannick Sire and Jingang Xiong), Calc. Var. Partial Differential Equations 56 (2017), no. 1, Paper No. 9, 25 pp.
- Solutions of some Monge-Ampère equations with isolated and line singularities (with Jingang Xiong), Adv. Math. 289 (2016), 114--141.
- Propagation in a non local reaction diffusion equation with spatial and genetic trait structure (with Henri Berestycki and Luis Silvestre), Nonlinearity 29 (2016), no. 4, 1434--1466.
- Schauder estimates for nonlocal fully nonlinear equations (with Jingang Xiong), Ann. Inst. H. Poincaré Anal. Non Linéaire 33 (2016), no. 5, 1375--1407.
- Schauder estimates for solutions of linear parabolic integro-differential equations (with Jingang Xiong), Discrete Contin. Dyn. Syst. -A. 35 (2015), no 12, 5977--5998.
- A sharp Sobolev trace inequality involving the mean curvature on Riemannian manifolds (with Jingang Xiong), Trans. Amer. Math. Soc. 367 (2015), no. 9, 6751--6770.
- On a fractional Nirenberg problem, part II: existence of solutions (with YanYan Li and Jingang Xiong), Int. Math. Res. Not. IMRN 2015, no. 6, 1555--1589.
- Existence and asymptotics for solutions of a non-local Q-curvature equation in dimension three (with Ali Maalaoui, Luca Martinazzi and Jingang Xiong), Calc. Var. Partial Differential Equations 52 (2015), no. 3-4, 469--488.
- Local analysis of solutions of fractional semi-linear elliptic equations with isolated singularities (with Luis Caffarelli, Yannick Sire and Jingang Xiong), Arch. Ration. Mech. Anal. 213 (2014), no. 1, 245--268.
- On a fractional Nirenberg problem, part I: blow up analysis and compactness of solutions (with YanYan Li and Jingang Xiong), J. Eur. Math. Soc. 16 (2014), no. 6, 1111--1171.
- A Liouville theorem for solutions of degenerate Monge-Ampère equations (with Jingang Xiong), Comm. Partial Differential Equations 39 (2014), no. 2, 306--320.
- A fractional Yamabe flow and some applications (with Jingang Xiong), J. Reine Angew. Math. 696 (2014), 187--223.
- Sharp constants in weighted trace inequalities on Riemannian manifolds (with Jingang Xiong),
Calc. Var. Partial Differential Equations 48 (2013), no. 3-4, 555--585.
- Symmetry and nonexistence of positive solutions of elliptic equations and systems with Hardy terms, Ann. Inst. H. Poincaré Anal. Non Linéaire 28 (2011), no. 6, 965--981.
- Pathological solutions to elliptic problems in divergence form with continuous coefficients (with Vladimir Maz'ya and Jean Van Schaftingen), C. R. Math. Acad. Sci. Paris 347 (2009), no. 13-14, 773--778.
- Grants:
- 2023 Hong Kong Mathematical Society Young Scholar Award.
- 2022 Hong Kong RGC grant GRF 16303822.
- 2021 NSFC Excellent Young Scientist (Hong Kong & Macau).
- 2020 Hong Kong RGC grant GRF 16306320.
- 2019 Hong Kong RGC grant GRF 16302519.
- 2018 Hong Kong RGC grant GRF 16306918.
- 2017 Hong Kong RGC grant GRF 16302217.
- 2016 Hong Kong RGC grant ECS 26300716.
- 2014 NSF grant DMS-1362525.
- 2013 AMS-Simons travel grant.
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