arXiv subjects
Publications and source records attributed to Yuanyuan Lian.
In this paper, we prove a boundary pointwise regularity for fully nonlinear elliptic equations on cones. In addition, based on this regularity, we give simple proofs of the Liouville theorems on cones.
In this note, we give a simple proof of the pointwise BMO estimate for Poisson's equation. Then the Calderón-Zygmund estimate follows by the interpolation and duality.
In this paper, we develop a series of boundary pointwise regularity for Dirichlet problems and oblique derivative problems. As applications, we give direct and simple proofs of the higher regularity of the free boundaries in obstacle-type problems and one phase problems.
In this paper, we prove the pointwise boundary differentiability for viscosity solutions of fully nonlinear elliptic equations. This generalizes the previous related results for linear equations. The geometrical conditions in this paper are pointwise and more general than before. Moreover, our proofs are relatively simple.
This paper investigates the relation between the boundary geometric properties and the boundary regularity of the solutions of elliptic equations. We prove by a new unified method the pointwise boundary Hölder regularity under proper geometric conditions. "Unified" means that our method is applicable for 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 etc. In addition, these geometric conditions are quite general. In particular, for local equations, the measure of the complement of the domain near the boundary point concerned could be zero. The key observation in the method is that the strong maximum principle implies a decay for the solution, then a scaling argument leads to the Hölder regularity. Moreover, we also give a geometric condition, which guarantees the solvability of the Dirichlet problem for the Laplace equation. The geometric meaning of this condition is more apparent than that of the Wiener criterion.
In this paper, we prove the boundary Lipschitz regularity and the Hopf Lemma by a unified method on Reifenberg domains for fully nonlinear elliptic equations. Precisely, if the domain $Ω$ satisfies the exterior Reifenberg $C^{1,\mathrm{Dini}}$ condition at $x_0\in \partial Ω$ (see Definition 1.3), the solution is Lipschitz continuous at $x_0$; if $Ω$ satisfies the interior Reifenberg $C^{1,\mathrm{Dini}}$ condition at $x_0$ (see Definition 1.4), the Hopf lemma holds at $x_0$. Our paper extends the results under the usual $C^{1,\mathrm{Dini}}$ condition.
In this note, we give a simple proof that the values of the trigonometric functions at any nonzero rational number are transcendental numbers.
In this paper, we obtain the boundary pointwise $C^{1,α}$ and $C^{2,α}$ regularity for viscosity solutions of fully nonlinear elliptic equations. I.e., If $\partial Ω$ is $C^{1,α}$ (or $C^{2,α}$) at $x_0\in \partial Ω$, the solution is $C^{1,α}$ (or $C^{2,α}$) at $x_0$. Our results are new even for the Laplace equation. Moreover, our proofs are simple.
In this note, we prove the boundary Hölder regularity for the infinity Laplace equation under a proper geometric condition. This geometric condition is quite general, and the exterior cone condition, the Reifenberg flat domains, and the corkscrew domains (including the non-tangentially accessible domains) are special cases. The key idea, following [3], is that the strong maximum principle and the scaling invariance imply the boundary Hölder regularity.