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.

本文研究了椭圆方程组的边界几何性质与边界正则性之间的关系。我们通过一种新的统一方法证明了在适当的几何条件下的点状边界Hölder正则性。“统一”表示我们的方法适用于拉普拉斯方程，发散和无散度形式的线性椭圆方程，完全非线性椭圆方程，p-拉普拉斯方程和分数拉普拉斯方程等。此外，这些几何条件非常通用。特别是对于局部方程，在相关边界点附近的区域的补数的度量可能为零。该方法的主要观察结果是，强大的最大原理意味着解的衰减，然后按比例缩放的论点导致了Hölder规律性。此外，我们还给出了一个几何条件，它保证了拉普拉斯方程Dirichlet问题的可解性。该条件的几何含义比维纳准则更明显。