The following is an introduction to several selected papers.
1. Li, Dongsheng; Zhang, Kai; A note on the Harnack inequality for elliptic equations in divergence form. Proceedings of the American Mathematical Society 145 (2017), no. 1, 135–137.
In this note, we point out that the classical Harnack inequality for elliptic equations in divergence form (due to Moser) was hidden in the work of De Giorgi. That is, we can infer the Harnack inequality directly from the classical paper by De Giorgi, where the C^{α} regularity was obtained and hence Hilbert's 19th problem was solved. Here, we give a short proof of the Harnack inequality without using the John-Nirenberg inequality as the usual proof.
2. Li, Dongsheng; Zhang, Kai; Regularity for Fully Nonlinear Elliptic Equations with Oblique Boundary Conditions. Archive for Rational Mechanics and Analysis 228 (2018), no. 3, 923–967.
In this paper, we study the oblique derivative boundary value problems for fully nonlinear elliptic equations. We obtained a series of regularity results, including the ABP maximum principle, the Harnack inequality up to the boundary, boundary pointwise C^α, C^{1,α} and C^{2,α} regularity. In addition, we proved the uniqueness and solvability of the equations.
One feature of this paper is that we don’t need to flatten the boundary by a transformation as usual to prove the boundary regularity.
3. Lian, Yuanyuan; Zhang, Kai; Boundary pointwise C^{1,α} and C^{2,α} regularity for fully nonlinear elliptic equations. Journal of Differential Equations 269 (2020), no. 2, 1172–1191.
In this paper, we prove the sharp pointwise boundary C^{1,α} and C^{2,α} regularity for fully nonlinear uniformly elliptic equations. This is the counterpart of the classical interior regularity obtained by Caffarelli in 1989. The boundary of the domain does not need to be a graph of a function locally. Hence, our results are new, even for the Laplace equation. Another new observation is that if |Du(0)|=0 where 0∈∂Ω, then ∂Ω∈C^{1,α}(0) implies u∈C^{2,α}(0). That is, we can lift the regulairty one order than the usual if |Du(0)|=0.
4. Lian, Yuanyuan; Zhang, Kai; Li, Dongsheng; Hong, Guanghao; Boundary Hölder regularity for elliptic equations. Journal de Mathématiques Pures et Appliquées 143 (2020), 311–333.
This paper concerns the boundary Hölder (C^α) regularity for various elliptic equations. It is well-known that the exterior cone condition implies the boundary Hölder regularity. Usually, it is proved by constructing a barrier or by applying the Harnack inequality up to the boundary. In this paper, we give an elementary proof of this result based on the following observation: the strong maximum principle with scaling invariance implies the boundary Hölder regularity.
We can prove the boundary Hölder regularity in a unified manner under rather weak geometrical conditions (e.g. the Reifenberg flat domains, Corkscrew domains and NTA domains) for various elliptic equations (e.g. the Laplace equation, linear elliptic equations in divergence and non-divergence form, fully nonlinear elliptic equations, the p−Laplace equations and the fractional Laplace equations).
5. Wu, Duan; Lian, Yuanyuan; Zhang, Kai; Pointwise Boundary Differentiability for Fully Nonlinear Elliptic Equations, Israel Journal of Mathematics 258 (2023), no. 1, 375–401.
This paper proves the pointwise boundary differentiability for fully nonlinear uniformly elliptic equations. All previous results concerning the boundary differentiability are special cases of the result in this paper. Moreover, we provide a relatively simple proof.
6. Li, Dongsheng; Li, Xuemei; Zhang, Kai; W^{2,p} Estimates for Elliptic Equations on C^{1,α} Domains, Mathematische Annalen 387 (2023), no. 1-2, 57–78.
In this paper, we obtain the global W^{2,p} estimates for elliptic equations (including fully nonlinear uniformly elliptic equations) on C^{1,α} domains (0<α<1). Usually, we require ∂Ω∈C^{1,1} to obtain the W^{2,p} estimates since we need to flatten the boundary by some transformation. Here, based on a delicately designed Whitney decomposition, interior W^{2,p} estimate and boundary pointwise C^{1,α} regularity, we can obtain the global W^{2,p} estimate without flattening the boundary.
7. Lian, Yuanyuan; Zhang, Kai; Boundary pointwise regularity and applications to the regularity of free boundaries, Calculus of Variations and Partial Differential Equations 62 (2023), no. 8, Paper No. 230, 32 pp.
In this paper, we further develop the observation made in our previous paper 3(JDE, 2020). We show that if u(0)=|Du(0)|· · ·=|D^k u(0)|=0 where 0∈∂Ω, then ∂Ω∈C^{m,α}(0) (m≥1,0<α<1) implies u∈C^{k+m,α}(0). That is, we can lift the regulairty k order than the usual if u(0)=|Du(0)|· · ·=|D^k u(0)|=0.
As an application, we give a direct and short proof of the higher regularity of free boundaries in obstacle problems. That is, if the free boundary ∂Ω∈C^{1,α}(0<α<1) and u is a viscosity solution, then ∂Ω∈C^{∞} and u∈C^{∞}. Kinderlehrer and Nirenberg first prove this classical result by using complicated partial hodograph and Legendre transformations. However, even for the Laplace equation, it becomes a fully nonlinear elliptic equation after transformation. Our proof is more direct, and the equation has not been changed.
We also prove similar regularity results for oblique derivative boundary value problems. As an application, we give a direct and short proof of the higher regularity of free boundaries in one-phase problems.
8. Lian, Yuanyuan; Zhang, Kai; A note on the BMO and Calderón-Zygmund estimate, Collectanea Mathematica 75 (2024), no. 1, 1–8.
Consider the following Poisson equation:
Δu=f in B_1
In this note, we give a short proof of the classical W^{2,BMO} regularity. That is, if f∈BMO, then u∈W^{2,BMO}. We do not use any harmonic analysis tool as the classical approach. Instead, we prove it in the same way as the Schauder estimate. The key observation is that the W^{2,BMO} regularity is a kind of pointwise regularity, which is similar to the C^{k,α} regularity.
Based on above result and the interpolation inequality, we can also give a short proof of the classical W^{2,p}(1<p<∞) regularity.
9. Lian, Yuanyuan; Wang, Lihe; Zhang, Kai; Pointwise Regularity for Fully Nonlinear Elliptic Equations in General Forms, arXiv:2012.00324.
In this paper, we systematically develop the interior and boundary pointwise C^{k}, C^{k,α} and C^{k,logLipschitz} (k≥1, 0<α<1) regularity for fully nonlinear uniformly elliptic equations in a general form. The regularity results are (almost) the optimal and the proofs are relatively simple.
10. Lian, Yuanyuan; Zhang, Kai; Pointwise regularity for locally uniformly elliptic equations and applications, arXiv:2405.07199
In this paper, we develop the interior pointwise C^{k,α}(k≥1, 0<α<1) regularity for locally uniformly elliptic equations (called small perturbation regularity by Savin, etc.). Based on this theory, we prove a series of regularity in a unified way under the smallness condition for various non-uniformly elliptic equations, including the prescribed mean curvature equation, the Monge-Ampère equation, the k-Hessian equation, the Hessian quotient equation and the Lagrangian mean curvature equation.
In addition, the counterexamples show that the smallness condition is necessary. We may infer that regarding the prescribed mean curvature equations etc. as locally uniformly elliptic equations is essential, and roughly speaking, the regularity originates from the uniform ellipticity.