An Almgren-type formula for planar $p$-harmonic functions

Abstract

For a nonconstant planar $p$-harmonic function $u$, with $1<p<\infty$, and according to previous results, most notably Aronsson's fundamental analysis of the hodograph representation in a neighborhood of an isolated critical point, we introduce the flux-normalized frequency $$N_{\ast }(x_0,r)=\frac{r{\int }_{B_r(x_0)}|Du{|}^{p}dx}{{\int }_{\partial B_r(x_0)}|Du{|}^{p-2}(u-u(x_0){)}^{2}dS}.$$ This ratio constitutes the natural counterpart of Almgren's frequency function in the nonlinear setting, as it recovers precisely the degree of homogeneity for every homogeneous $p$-harmonic profile. We further establish the corresponding gauge-corrected version of Almgren-type monotonicity and reinterpret, in terms of the frequency function, the classical planar unique continuation property. In addition, we derive a quantitative pinching estimate for the flux-normalized frequency and identify the specific obstruction that prevents a straightforward extension of the method to higher dimensions.

23 Pages. This manuscript is more of a small observation than a research article with new findings and has been under submission for months. The author initially decided against posting it on arXiv, but recently did so after some encouragement from Kai Xu