Alt-Caffarelli-Friedman monotonicity formula and mean value properties in Carnot groups with applications. (English) Zbl 1542.35403
Summary: In this paper, we provide a different approach to the Alt-Caffarelli-Friedman monotonicity formula, reducing the problem to test the monotone increasing behavior of the mean value of a function involving the gradient’s norm. In particular, we show that our argument holds in the general framework of Carnot groups.
MSC:
| 35R03 | PDEs on Heisenberg groups, Lie groups, Carnot groups, etc. |
| 35B05 | Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs |
| 35R35 | Free boundary problems for PDEs |
Keywords:
Alt-Caffarelli-Friedman monotonicity formula; Carnot groups; two-phase free boundary problemsReferences:
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