Quantitative propagation of smallness for solutions of elliptic equations

2017/11/28 by Logunov, Alexander, Malinnikova, Eugenia · 6 citations
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1711.10076

Abstract

Let u be a solution to an elliptic equation div(A∇ u)=0 with Lipschitz coefficients in ℝⁿ. Assume |u| is bounded by 1 in the ball B=\|x|≤ 1\. We show that if |u| < ε on a set E ⊂ (1)/(2) B with positive n-dimensional Hausdorf measure, then |u|≤ Cε^γ on (1)/(2)B, where C>0, γ∈ (0,1) do not depend on u and depend only on A and the measure of E. We specify the dependence on the measure of E in the form of the Remez type inequality. Similar estimate holds for sets E with Hausdorff dimension bigger than n-1. For the gradients of the solutions we show that a similar propagation of smallness holds for sets of Hausdorff dimension bigger than n-1-c, where c>0 is a small numerical constant depending on the dimension only.

Cited by

• Global controllability of the Cahn-Hilliard equation
• Fractal Remez inequality on the sphere and observability of the heat equation
• Propagation of smallness near codimension two for gradients of harmonic functions
• Elliptic unique continuation below the Lipschitz threshold
• Spectral inequalities for Schrödinger equations and quantitative propagation of smallness in the plane
• Capacity estimates and improved lower bounds for the inner radius of nodal domains

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