Lecture notes on quantitative unique continuation for solutions of second order elliptic equations

Abstract

In these lectures we present some useful techniques to study quantitative properties of solutions of elliptic PDEs. Our aim is to outline the proof of a recent result on propagation of smallness. The ideas are also useful in the study of the zero sets of eigenfunctions of the Laplace-Beltrami operator. Some basic facts about second order elliptic PDEs in divergent form are collected in the Appendix at the end of the notes.

An elliptic adaptation of ideas of Carleman and Domar from complex analysis related to Levinson's log log theorem

“…We note that (3) holds for solutions of second order equations in divergence form in R n : div(A∇u) = 0, where the matrix A is assumed to be uniformly elliptic and its coefficients are Lipschitz in Ω. The version of the three balls theorem for wild sets is non-trivial when the coefficients are assumed to be just smooth (but not real-analytic), and it is related to a recently solved conjecture of Landis ( [12], p.169) and a conjecture of Donnelly and Fefferman [6], see [15,16] for the proofs.…”

Nodal set of monochromatic waves satisfying the Random Wave Model

“…Finally, to study the nodal domains of f , we will to use the doubling index to control the growth of f in sets which might not be balls. That is, we will need the following lemma [40]: Lemma 2.7 (Remez type inequality). Let B be the unit ball in R m and suppose that h : B → R ia an harmonic function.…”

Optimal unique continuation for periodic elliptic equations on large scales

“…Both of these inequalities obviously imply, on a qualitative level, the unique continuation property: a harmonic function cannot vanish in a ball without vanishing identically. They also contain quantitative information regarding the growth of harmonic functions which have important applications, including to spectral properties of the operator, the behavior of nodal sets of eigenfunctions, and inverse problems (see for instance [11,7,13,1]).…”

“…Since the explicit dependence of the doubling ratio on the Lipschitz constant is required, it is necessary to inspect the argument of [6] and track this dependence line-by-line. This is routine and has essentially already been done 1 in [13], so we omit the details. Since Corollary 1.3 was proved already in [2] by a simple argument which is similar to the one implied here, it has also been left to the reader.…”