Boundary unique continuation on $C^1$-Dini domains and the size of the singular set

Abstract

Let $u$ be a harmonic function in a $C^1$-Dini domain $D$ such that $u$ vanishes on a boundary surface ball $\partial D\cap B_{5R}(0)$. We consider an effective version of its singular set (up to boundary) $\mathcal{S}(u):=\{X\in \overline{D}:u(X)=|\nabla u(X)|=0\}$ and give an estimate of its $(d-2)$-dimensional Minkowski content, which only depends on the upper bound of some modified frequency function of $u$ centered at $0$. Such results are already known in the interior and at the boundary of convex domains, when the standard frequency function is monotone at every point. The novelty of our work on Dini domains is how to compensate for the lack of such monotone quantities at boundary as well as interior points.

arXiv comment78 pages

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Cited by in corpus (4)

• Size of the zero set of solutions of elliptic PDEs near the boundary of Lipschitz domains with small Lipschitz constant
• Unique continuation at the boundary for harmonic functions in $C^1$ domains and Lipschitz domains with small constant
• Łojasiewicz Inequalities and Generic Smoothness of Nodal Sets of Solutions to Elliptic PDE
• Expansion of harmonic functions near the boundary of Dini domains