In this paper, we established a quantitative unique continuation results for a coupled heat equations, with the homogeneous Dirichlet boundary condition, on a bounded convex domain Ω of [Formula: see text] with smooth boundary ∂Ω. Our result shows that the value of the solutions can be determined uniquely by its value on an arbitrary open subset ω of Ω at any given positive time T.

We consider the vanishing viscosity limit of the Navier-Stokes equations in a half space, with Dirichlet boundary conditions. We prove that the inviscid limit holds in the energy norm if the product of the components of the Navier-Stokes solutions are equicontinuous at x2 = 0. A sufficient condition for this to hold is that the tangential Navier-Stokes velocity remains uniformly bounded and has...

Several abstract model problems of elliptic and parabolic type with inhomogeneous initial and boundary data are discussed. By means of a variant of the Dore-Venni theorem, real and complex interpolation, and trace theorems, optimal Lp-regularity is shown. By means of this purely operator theoretic approach, classical results on Lp-regularity of the diffusion equation with inhomogeneous Dirichle...

We show that on an open bounded smooth strongly pseudoconvex subset of C, there exists a Kähler-Einstein metric with positive Einstein constant, such that the metric restricted to the Levi distribution of the boundary is conformal to the Levi form. To achieve this, we solve an associated complex Monge-Ampère equation with Dirichlet boundary condition. We also prove uniqueness under some more as...

We consider asymptotically future de Sitter spacetimes endowed with an eternal observatory. In the conventional descriptions, the conformal metric at the future boundary I+ is deformed by the flux of gravitational radiation. We however impose an unconventional future “Dirichlet” boundary condition requiring that the conformal metric is flat everywhere except at the conformal point where the obs...