A boundary uniqueness theorem for Sobolev functions (Q1283233)

Uniqueness theorems for analytic functions having zero radial limits on a set of positive capacity are well known. The authors obtain a result of such a kind for continuous functions \(u\) in \(\mathbb{R}^n\) with a finite seminorm \(\int_{\mathcal{B}}| \nabla u| ^p dm\), \(\mathcal{B}\) being a bounded domain in \(\mathbb{R}^n\). This result is formulated in terms of the behaviour of the functional \[ I(\varepsilon)=\int_{x\in\mathcal{B}, | u(x)| < \varepsilon}| \nabla u| ^p dm \] as \(\varepsilon \to 0\).