For harmonic functions in R^n, ε-smallness of the gradient on a set of Hausdorff dimension n-2+δ (any δ > 0) forces a C ε^α bound on the half-ball, reaching the sharp threshold and answering the Logunov-Malinnikova conjecture.
Results on gradients of harmonic functions on Lipschitz surfaces
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We study various properties of the gradients of solutions to harmonic functions on Lipschitz surfaces. We improve an exponential bound of Naber and Valtorta on the size of the superlevel sets for the frequency function to a sharp quadratic bound in this setting using complex analytic tools. We also develop a propagation of smallness for gradients of harmonic functions, settling an open question in this setting. Finally, we extend the estimate on superlevel sets of the frequency to more general divergence-form elliptic PDEs with bounded drift terms at the cost of a subpolynomial factor.
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Propagation of smallness near codimension two for gradients of harmonic functions
For harmonic functions in R^n, ε-smallness of the gradient on a set of Hausdorff dimension n-2+δ (any δ > 0) forces a C ε^α bound on the half-ball, reaching the sharp threshold and answering the Logunov-Malinnikova conjecture.