Infinity Laplacian equation with strong absorptions. (English) Zbl 1344.35034
Let \(\Delta_{\infty} u = (Du)^T D^2u Du\) denote the infinity Laplacian. In the work under review, the authors consider for a bounded domain \(\Omega \subset \mathbb{R}^n\) (\(n \geq 2\)) and continuous non-negative boundary data \(g \in C(\partial \Omega)\) the Dirichlet problem
\[
\begin{cases} \Delta_{\infty} u - \lambda (u^+)^{\gamma}= 0 & \qquad \text{in } \Omega, \\ u = g &\qquad \text{on } \partial \Omega, \end{cases}
\]
where \(\lambda > 0\) and \(\gamma \in [0,3)\) are constants. First, the authors establish existence and uniqueness of viscosity solutions for the problem. In the main part, it is then shown that the viscosity solution is pointwise of class \(C^{\frac{4}{3-\gamma}}\) along the boundary of the set \(\partial \{ u > 0 \}\).
Reviewer: Stephan Fackler (Ulm)
MSC:
| 35J60 | Nonlinear elliptic equations |
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 35D40 | Viscosity solutions to PDEs |
| 35K57 | Reaction-diffusion equations |
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