[Paper Review] Stable and singular solutions of the equation $Δu = 1/u$

Alexander M. Meadows|arXiv (Cornell University)|Apr 22, 2004

Nonlinear Partial Differential Equations3 citations

TL;DR

This paper investigates stable and singular solutions to the semilinear elliptic equation Δu = 1/u in ℝⁿ, establishing the nonexistence of singular stable solutions in dimensions n ≤ 6 and deriving an upper bound on the Hausdorff dimension of the singular set as n − 4 − 2√2. The analysis combines variational methods, stability inequalities, and geometric measure theory to characterize the regularity and structure of solutions in low dimensions.

ABSTRACT

We study properties of the semilinear elliptic equation $Δu = 1/u$ on domains in $R^n$, with an eye toward nonnegative singular solutions as limits of positive smooth solutions. We prove the nonexistence of such solutions in low dimensions when we also require them to be stable for the corresponding variational problem. The problem of finding singular solutions is related to the general study of singularities of minimal hypersurfaces in Euclidean space.

Motivation & Objective

• To understand the existence and regularity of nonnegative singular solutions to the equation Δu = 1/u as limits of positive smooth solutions.
• To characterize the stability of solutions via the second variation of the functional 𝒥(u) = ∫(½|Du|² + log u)dx.
• To determine the sharp dimension threshold below which singular stable solutions cannot exist.
• To estimate the Hausdorff dimension of the singular set where u = 0 in the limit of positive stable solutions.

Proposed method

• Uses the variational formulation of Δu = 1/u as the Euler–Lagrange equation for the functional 𝒥(u) = ∫(½|Du|² + log u)dx.
• Defines stable solutions via the second variation inequality: ∫(ζ²/u²)dx ≤ ∫|Dζ|²dx for all test functions ζ.
• Applies Lp estimates and Calderón–Zygmund theory to derive W^{2,p} bounds on u, leading to Hölder continuity and uniform lower bounds.
• Employs a cutoff and truncation argument to control the Lp norm of 1/u, leveraging the stability inequality and Sobolev embedding.
• Uses the decay estimate u(x) ≤ C(dist(x,A))^α for α < 1 to bound the integral of 1/u over cubes intersecting the singular set A.
• Applies geometric measure theory to show that the β-dimensional Hausdorff content of A ∩ B_{ρ/2} is uniformly bounded for β < n − 4 − 2√2.

Experimental results

Research questions

• RQ1 Can singular solutions to Δu = 1/u arise as limits of positive smooth solutions, and under what conditions are they stable?
• RQ2 What is the maximal Hausdorff dimension of the singular set A = {u = 0} for stable solutions of Δu = 1/u?
• RQ3 In which dimensions does the nonexistence of singular stable solutions hold?
• RQ4 How does the stability condition influence the regularity and lower bounds on u in the interior of the domain?

Key findings

• For dimensions 2 ≤ n ≤ 6, any stable solution u of Δu = 1/u with positive boundary data is uniformly bounded below by a positive constant δ > 0.
• In dimensions n ≤ 6, there are no singular stable solutions to Δu = 1/u, meaning u cannot vanish in the interior if it is stable.
• The Hausdorff dimension of the singular set A = {u = 0} for a limit of positive stable solutions satisfies dim_H(A) ≤ n − 4 − 2√2.
• Stable solutions are Hölder continuous with exponent α ∈ (0,1) on compact subsets of the domain, uniformly depending on n, M, and the domain.
• The Lp norm of 1/u is uniformly bounded on compact subsets for p > 1, independent of the solution, under stability and boundary data constraints.
• The lower bound on u implies that 1/u is Hölder continuous, enabling bootstrapping to C^∞ regularity in the interior for n ≤ 6.