Benjamin Foster
Published 2023-04-22Version 1
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 effective critical set 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.
Schwarz-Pick lemma for harmonic functions
Decomposition of $L^{2}$-vector fields on Lipschitz surfaces: characterization via null-spaces of the scalar potential
Isoperimetric inequalities and regularity of $A$-harmonic functions on surfaces
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