TITLE

# Hausdorff measure of critical sets of solutions to magnetic schrÃ¶dinger equations

AUTHOR(S)
Liu, Dan
PUB. DATE
May 2011
SOURCE
Calculus of Variations & Partial Differential Equations;May2011, Vol. 41 Issue 1/2, p179
SOURCE TYPE
DOC. TYPE
Article
ABSTRACT
In this paper, we study the critical set of a complex-valued solution to a SchrÃ¶dinger equation involving the magnetic field and with a nonlinear term, where the critical set is $${\{x\in\Omega:~\psi(x)=0, ~\nabla\psi(x)=0\}}$$ . We consider this equation in a bounded domain of $${\mathbb{R}^3}$$ with the boundary condition: $${\nabla _{\mathbf{A}}\psi\cdot \nu=0}$$ , and we establish a global 1-dimensional Hausdorff measure estimate for the critical sets. From the proof of global estimates, we find that our methods work as well for more general equations with a magnetic potential.
ACCESSION #
59439120

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