| In this thesis we deduce some interesting facts about PDE in mathematical physics. In the first half, we establish a loglog-type estimate which shows that in dimension n ≥ 3 the magnetic field and the electric potential of the magnetic Schrodinger equation depends stably on the Dirichlet to Neumann (DN) map even when the boundary measurement is taken only on a subset that is slightly larger than half of the boundary ∂O. Furthermore, we prove that in the case when the measurement is taken on all of ∂O one can establish a better estimate that is of log-type. The proofs involve the use of the complex geometric optics (CGO) solutions of the magnetic Schrodinger equation constructed in [25] then follow a similar line of argument as in [1]. In the partial data estimate we follow the general strategy of [22] by using the Carleman estimate established in [14] and a continuous dependence result for analytic continuation developed in [32].; In the second half, a new variational principles were introduced based on the concept of anti-selfdual (ASD) Lagrangians. We continue here the program of using these Lagrangians to provide variational formulations and resolutions to various basic equations and evolutions which do not normally fit in the Euler-Lagrange framework. In particular, we consider stationary equations of the form -Au ∈ ∂ϕ(u) as well as i dissipative evolutions of the form -u˙( t) - Atu(t) + o u(t) ∂ϕ(t, u( t)) were ϕ is a convex potential on an infinite dimensional space. |
