\(\mathcal{C}^{1,\beta}\) regularity for Dirichlet problems associated to fully nonlinear degenerate elliptic equations. (English) Zbl 1315.35108
In the very interesting paper under review, the authors prove Hölder continuity up to the boundary for the gradient of solutions to some fully-nonlinear, degenerate elliptic equations, with degeneracy coming from the gradient. Precisely, let \(u\) be a viscosity solution of the Dirichlet problem
\[
\begin{cases} |\nabla u|^\alpha \big( F(D^2u)+h(u)\cdot \nabla u\big)=f & \text{in}\;\Omega,\\ u=\varphi & \text{on}\;\partial\Omega, \end{cases}
\]
where \(\Omega\subset\mathbb{R}^N\) is a bounded \(C^2\)-smooth domain, \(\alpha\geq0,\) \(F\) is uniformly elliptic, \(h,f\in C(\overline{\Omega})\) and \(\varphi\in C^{1,\beta_0}(\partial\Omega).\) The main result of the paper asserts existence of an exponent \(\beta,\) depending on the data of the problem, and a constant \(C=C(\beta)\) such that
\[
\|u\|_{C^{1,\beta}(\overline{\Omega})}\leq C \left( \|u\|_{L^{\infty}({\Omega})}+ \|\varphi\|_{C^{1,\beta_0}({\partial\Omega})}+ \|f\|_{L^{\infty}({\Omega})}^{\frac{1}{1+\alpha}} \right).
\]
Reviewer: Dian K. Palagachev (Bari)
MSC:
35J70 | Degenerate elliptic equations |
35J25 | Boundary value problems for second-order elliptic equations |
35P30 | Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs |
35D40 | Viscosity solutions to PDEs |