In this work we establish log type stability estimates for the inverse potential and conductivity problems with partial Dirichlet-to-Neumann map, where the Dirichlet data is homogeneous on the inaccessible part. This result, to some extent, improves our former result on the partial data problem [HW06] in which log-log type estimates were derived.

In this paper we prove a quantitative form of Landis' conjecture in the plane. Precisely, let W (z) be a measurable real vector-valued function and V (z) ≥ 0 be a real measurable scalar function, satisfyingC depends on C 0 . In addition to the case of the whole plane, we also establish a quantitative form of Landis' conjecture defined in an exterior domain.where W (x, y) = (W 1 (x, y), W 2 (x, y)) and V (x, y) are measurable, real-valued, and V (x, y) ≥ 0 a.e. In view of the scaling argument in [BK05], Landis' conjecture is an easy consequence *

In this work we study the inverse boundary value problem of determining the refractive index in the acoustic equation. It is known that this inverse problem is ill-posed. Nonetheless, we show that the ill-posedness decreases when we increase the frequency and the stability estimate changes from logarithmic type for low frequencies to a Lipschitz estimate for large frequencies.

ABSTRACT. In this work, we study the Landis conjecture for second-order elliptic equations in the plane. Precisely, assume that V ≥ 0 is a measurable real-valued function satisfying ||V || L ∞ (R 2 ) ≤ 1. Let u be a real solution to div (A∇u) −Vu = 0 in R 2 . Assume that |u (z)| ≤ exp (c 0 |z|) and u (0) = 1. Then, for any R sufficiently large,In addition to equations with electric potentials, we also derive similar estimates for equations with magnetic potentials. The proofs rely on transforming the equations to Beltrami systems and Hadamard's three-quasicircle theorem.

In this paper we study the local behavior of a solution to the Stokes system with singular coefficients. One of the main results is the bound on the vanishing order of a nontrivial solution to the Stokes system, which is a quantitative version of the strong unique continuation property. Our proof relies on some delicate Carleman-type estimates. We first use these estimates to derive crucial optimal three-ball inequalities. Taking advantage of the optimality, we then derive an upper bound on the vanishing order of any nontrivial solution to the Stokes system from those three-ball inequalities.