Asymptotics of Harmonic Functions in the Absence of Monotonicity Formulas

In this article, we study the asymptotics of harmonic functions. In literature, one typical method is by proving the monotonicity of a version of rescaled Dirichlet energies, and use it to study the renormalized solution—the Almgren’s blowup. However, such monotonicity formulas require strong smoothness assumptions on domains and operators. We are interested in the cases when monotonicity formulas are not available, including variable coefficient equations with unbounded lower order terms, Dirichlet problems on rough (non- \(C^1\) ) domains, and Robin problems with rough Robin potentials.