On the Dirichlet problem for solutions of a restricted nonlinear mean value property. (Q265183)

The authors are concerned with the study of the operator NEWLINE\[NEWLINET_\alpha u(x)=\frac{\alpha}{2}(\sup_{B_x}u+\inf_{B_x} u)+\frac{1-\alpha}{|B_x|}\int_{B_x}u,NEWLINE\]NEWLINE where \(0\leq \alpha<1\), \(u\in C(\overline\Omega)\) with \(\Omega\subset\mathbb{R}^d\) a bounded domain, \(B_x=B(x,r(x))\subset\Omega\) a ball centred at \(x\). The main result of the paper states that if \(\Omega\) is strictly convex and \(f\in C(\partial\Omega)\), then, there exists a solution \(u\in C(\overline\Omega)\) of \(T_\alpha u=u\) in \(\Omega\) and \(u=f\) on \(\partial\Omega\).NEWLINENEWLINE The proofs rely on appropriate iteration techniques and equicontinuity in metric spaces.