[Paper Review] On the Mean Value Property for the p-Laplace equation in the plane

Peter Lindqvist, Juan J. Manfredi|arXiv (Cornell University)|Aug 31, 2014

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TL;DR

This paper establishes a pointwise mean value property for solutions of the p-Laplace equation in the plane for 1 < p < p₀ ≈ 9.525, using the hodograph representation and quasiregular mapping theory. It proves that the asymptotic mean value formula holds directly for solutions themselves, eliminating the need for viscosity solutions or test functions, by analyzing the behavior of critical points via complex gradient structure and power series expansions in the complex plane.

ABSTRACT

We study the p-Laplace equation in the plane and prove that the mean value property holds directly for the solutions themselves. This removes the need to interpret the formula in the viscosity sense via test functions. The method is based on the hodograph representation.

Motivation & Objective

• To establish a pointwise mean value property for solutions of the p-Laplace equation in the plane, valid without viscosity interpretation.
• To remove the reliance on test functions in the asymptotic mean value formula by proving it holds directly for solutions.
• To analyze the structure of critical points of p-harmonic functions in two dimensions using complex analysis.
• To determine the maximal range of p for which the mean value formula holds pointwise via asymptotic expansion and power series analysis.
• To extend the validity of the mean value property beyond the classical viscosity framework in the planar case.

Proposed method

• Utilizes the hodograph representation derived from the Stoilow factorization of the complex gradient of a p-harmonic function.
• Applies the theory of quasiregular mappings to show that critical points of p-harmonic functions in the plane are isolated unless the function is constant.
• Expands the inverse mapping of the complex gradient in a power series around a critical point, separating the dominant term from higher-order remainders.
• Estimates the remainder terms using Cauchy's inequality and asymptotic behavior of the exponents λₖ, which depend on p.
• Derives a sufficient condition for the error term to be o(ε²) by ensuring the exponent of the leading error term exceeds 2.
• Verifies the mean value formula by showing that the average, max+min/2, and the function value all agree up to o(ε²) under the derived p-condition.

Experimental results

Research questions

• RQ1 Can the asymptotic mean value property for the p-Laplace equation be interpreted directly for solutions, without viscosity solutions or test functions, in the plane?
• RQ2 What is the maximal range of p for which the mean value formula holds pointwise in two dimensions?
• RQ3 How does the structure of critical points of p-harmonic functions in the plane affect the validity of the mean value property?
• RQ4 Can the hodograph method and quasiregular mapping theory be used to derive pointwise asymptotic expansions for p-harmonic functions near critical points?
• RQ5 What role do the exponents λₖ in the power series expansion of the inverse complex gradient play in determining the regularity and mean value behavior?

Key findings

• The mean value formula for the p-Laplace equation holds pointwise for solutions in the plane when 1 < p < p₀ ≈ 9.525, without requiring viscosity interpretation.
• The critical exponent p₀ arises as the root of a sixth-degree algebraic equation derived from the condition that the error term in the asymptotic expansion is o(ε²).
• For n=1 (simple critical points), the condition λ₃ / (λ₂)² > 1 ensures the error term is O(r²⁺ᵅ) for some α > 0, which implies o(ε²) in the mean value formula.
• The method relies on the fact that the complex gradient is a quasiregular mapping, implying isolated critical points, which allows local power series expansions.
• The result extends to higher-order critical points (n ≥ 2), where the same method yields a larger valid range of p, though p₀ remains the limiting threshold.
• The mean value formula is verified by showing that the average of u, the average of max and min u, and u(0,0) all agree up to o(ε²), confirming the identity.