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The question of unique continuation of harmonic functions in a domain$\Omega$ $\subset$ R d with boundary $\partial$$\Omega$, satisfying Dirichletboundary conditions and with normal derivatives vanishing on a subset $\omega$of the boundary is a classical problem. When $\omega$ contains an open subsetof the boundary it is a consequence of Carleman estimates (uniqueness forsecond order elliptic operators across an hypersurface). The case where$\omega$ is a set of positive (d -- 1) dimensional measure has attracted a lotof attention, see e.g. [10, 3, 15], where qualitative results have beenobtained in various situations. The main open questions (about uniqueness)concern now Lipschitz domains and variable coefficients. Here, using results byLogunov and Malinnikova [13, 14], we consider the simpler case of W 2,$\infty$domains but prove quantitative uniqueness both for Dirichlet and Neumannboundary conditions. As an application, we deduce quantitative estimates forthe Dirichlet and Neumann Laplace eigenfunctions on a W 2,$\infty$ domain withboundary.

A remark on quantitative unique continuation from subsets of the boundary of positive measure