Let $M$ be a smooth compact manifold. If $M$ is closed, we have that the interpolation space $$(L^2(M), H^1(M))_{\frac 12}=H^{\frac 12}(M)$$ (see Taylor's book on PDE for example). Suppose $M$ has a boundary. Then what is $$(L^2(M), H^1_0(M))_{\frac 12}?$$ Is it some sort of Lions-Magenes space with a nice characterisation? A reference would be gratefully appreciated.