Resumen de Unique continuation on convex domains

Sean McCurdy

• In this paper, we obtain estimates on the quantitative strata of the critical set of non-trivial harmonic functions u which vanish continuously on ⊂∂ΩV⊂∂Ω, a relatively open subset of the boundary of a convex domain Ω⊂Ω⊂Rn. In particular, these estimates improve dimensional estimates on {∣∇∣=0}{∣∇u∣=0} both in ⊂∂ΩV⊂∂Ω and as it approaches ∩Ω‾V∩Ω. These estimates are not obtainable by naively combining interior and boundary estimates, and represent a significant improvement upon existing results for boundary analytic continuation in the convex case.