Boundary behavior of solutions of elliptic equations in nondivergence form
The interesting paper under review deals with the boundary behaviour of the viscosity solutions to the Dirichlet problem \[ \begin{cases} -a^{ij}(x)\dfrac{\partial^2u(x)}{\partial x_i\partial x_j}=f(x) & \text{in}\;\Omega,\\ u(x)=g(x) & \text{on}\;\partial\Omega, \end{cases} \] where \(\Omega\subset \mathbb{R}^n\) is a bounded Lipschitz domain, the symmetric matrix \(\{a^{ij}(x)\}\) is uniformly elliptic, \(a^{ij},f\in C(\overline{\Omega})\) and \(g\in C(\partial\Omega)\). Introducing in a suitable way the notion of \(C^{1,\text{Dini}}\) regularity of \(\partial\Omega\) and \(g\) at a point \(x\in\partial\Omega,\) the authors prove that the solution of the above problem is Lipschitz continuous at \(x\in \partial\Omega\) if \(\partial\Omega\) and \(g\) are \(C^{1,\text{Dini}}\) in \(x.\) If, in addition, \(\partial\Omega\) is punctually \(C^1\) in \(x\) then the solution results differentiable at \(x\). The main tools used in the proofs rely on the Aleksandrov-Bakelman-Pucci maximum principle, the Harnack inequality and barrier technique.
Aleksandrov-Bakelman-Pucci maximum principle
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