| Abstract: |
| In this talk, we study the boundary regularity for fully nonlinear elliptic equations. We show the boundary H\"{o}lder regularity on Reifenberg flat domains, the boundary Lipschitz regularity and the Hopf lemma.
For boundary H\"{o}lder regularity, we will prove that for any $\alpha\in (0,1) $, there exists a positive constant $\delta$ such that the solutions are $C^{\alpha}$ at $x_0\in \partial \Omega$ provided that $\Omega$ is $(\delta,R)$-Reifenberg flat at $x_0$. In particular, if $\partial \Omega\in C^1$, we have the boundary H\"{o}lder regularity with any exponent $\alpha\in (0,1)$. A similar result for the Poisson equation has been proved by Lemenant and Sire, where the Alt-Caffarelli-Friedman's monotonicity formula is used. Besides the generalization to fully nonlinear elliptic equations, our method is simple.
For the boundary Lipschitz regularity and the Hopf, we use a unified, simple method to prove that if the domain $\Omega$ satisfies the exterior $C^{1,\mathrm{Dini}}$ condition at $x_0\in \partial \Omega$, the solution is Lipschitz continuous at $x_0$; if $\Omega$ satisfies the interior $C^{1,\mathrm{Dini}}$ condition at $x_0$, the Hopf lemma holds at $x_0$. Moreover, we show that the $C^{1,\mathrm{Dini}}$ conditions are optimal. |
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