Abstract
In this note, we prove two monotonicity formulas for solutions of$\Delta _H f = c$and$\Delta _H f - \partial _t f = c$in Carnot groups. Such formulas involve the right-invariantcarré du champof a function and they are false for the left-invariant one. The main results, theorems 1.1 and 1.2, display a resemblance with two deep monotonicity formulas respectively due to Alt–Caffarelli–Friedman for the standard Laplacian, and to Caffarelli for the heat equation. In connection with this aspect we ask the question whether an ‘almost monotonicity’ formula be possible. In the last section, we discuss the failure of the nondecreasing monotonicity of an Almgren type functional.
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