Nonlinear mean-value formulas on fractal sets. (English) Zbl 1433.28022
Summary: In this paper we study the solutions to nonlinear mean-value formulas on fractal sets. We focus on the mean-value problem \(\frac{1}{2} \max_{q \in V_{m, p}} \{f(q) \} + \frac{1}{2} \min_{q \in V_{m, p}} \{f(q) \} - f(p) = 0\) in the Sierpiński gasket with prescribed values \(f(p_1), f(p_2)\) and \(f(p_3)\) at the three vertices of the first triangle. For this problem we show existence and uniqueness of a continuous solution and analyze some properties like the validity of a comparison principle, Lipschitz continuity of solutions (regularity) and continuous dependence of the solution with respect to the prescribed values at the three vertices of the first triangle.
MSC:
| 28A80 | Fractals |
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