ABSTRACT. We prove quantitative unique continuation results for solutions of −∆u + W · ∇u + Vu = λ u, where λ ∈ C and V and W are complex-valued decaying potentials that satisfy, we show that if the solution u is non-zero, bounded, andwhere β 0 = max 2 − 2P, 4 − 2N 3 ,1 . Under certain conditions on N, P and λ , we construct examples (some of which are in the style of Meshkov) to prove that this estimate for M (R) is sharp. That is, we construct functions u,V and W such that −∆u +W · ∇u +Vu = λ u, |V (x)| x −N , |W (x)| x −P and |u(x)| exp −c|x| β 0 (log |x|) C .

In this article, we study some quantitative unique continuation properties of solutions to second order elliptic equations with singular lower order terms. First, we quantify the strong unique continuation property by estimating the maximal vanishing order of solutions. That is, when u is a non-trivial solution to ∆u + W · ∇u + V u = 0 in some open, connected subset of R n , where n ≥ 3, we characterize the vanishing order of solutions in terms of the norms of V and W in their respective Lebesgue spaces. Then, using these maximal order of vanishing estimates, we establish quantitative unique continuation at infinity results for solutions to ∆u + W · ∇u + V u = 0 in R n . The main tools in our work are new versions of L p → L q Carleman estimates for a range of p-and q-values.

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.

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In this paper, we establish existence, uniqueness, and scale-invariant estimates for fundamental solutions of non-homogeneous second order elliptic systems with bounded measurable coefficients in R n and for the corresponding Green functions in arbitrary open sets. We impose certain non-homogeneous versions of de Giorgi-Nash-Moser bounds on the weak solutions and investigate in detail the assumptions on the lower order terms sufficient to guarantee such conditions. CONTENTSV 44 Appendix D. Smoothing and Approximations 46 References 48 for some 0 < λ , Λ < ∞ and for all x in the domain. Note that, in particular, equations with complex bounded measurable coefficients fit into this scheme. We establish existence, uniqueness, as well as global scale-invariant estimates for the fundamental solution in R n and for the Dirichlet Green function in any connected, open set Ω ⊂ R n , where n ≥ 3.The key difficulty in our work is the lack of homogeneity of the system since this typically results in a lack of scale-invariant bounds. Here, the existence of solutions relies on a coercivity assumption, which controls the lower-order terms, and the validity of the Caccioppoli inequality. Furthermore, following many predecessors (see, e.g., [HK07], [KK10]), we require certain quantitative versions of the local boundedness of solutions. This turns out to be a delicate game, however, to impose local conditions which are sufficient for the construction of fundamental solutions and necessary for most prominent examples. Indeed, they have not been completely wellunderstood even in the case of real equations, due to the same type of difficulties: Solutions to nonhomogeneous equations can grow exponentially with the growth of the domain in the absence of a suitable control on the potential V, even if b = d = 0. This affects the construction of the fundamental solution. Let us discuss the details. The fundamental solutions and Green functions for homogeneous second order elliptic systems are fairly well-understood by now. We do not aim to review the vast literature addressing various situations with additional smoothness assumptions on the coefficients of the operator and/or the domain, and will rather comment on those works that are most closely related to ours. The analysis of Green functions for operators with bounded measurable coefficients goes back to the early 80's, [GW82] (see also [LSW63] for symmetric operators), in the case of homogeneous equations with real coefficients (N = 1). The case of homogeneous systems, and, respectively, equations with complex coefficients, has been treated much more recently in [HK07] and [KK10] under the assumptions of local boundedness and Hölder continuity of solutions, the so-called de Giorgi-Nash-Moser estimates. Later on, in [Ros13], the fundamental solution in R n was constructed using only the assumption of local boundedness, that is, without the requirement of Hölder continuity. In [Bar14], Barton constructed fundamental solutions, also in R n only, in the full generality of homogeneous elli...