I recently encountered a paper by Protter ("Unique continuation of elliptic equations") that starts out by saying "any solution of an elliptic equation that is defined on a domain $D$ must vanish on all of $D$ if it vanishes on an open set in $D$" and calls this the unique continuation principle. The problem is that this statement is so general that I'm afraid to use it. Does anyone know the rigorous statement of the unique continuation principle? In particular I wonder what are the conditions on the elliptic equation and the solution involved.
I think this is the basic reference:
A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of second order by N Aronszajn - J. Math. pur. appl., IX. Sér., 1957
Aronszajn considers second order elliptic equations AND inequalities, but his theorem is even more general. You only need that the difference of two solutions vanish at all orders at some point and already the solutions must be identical. Actually, its better than that, but I forget the details.
Besides the Carleman type estimate in Aronszajn work, Nicola Garofalo and Fang-Hua Lin, Monotonicity properties of variational integrals, Ap weights and unique continuation, using Almgren frequency monotonicity, also proved the unique continuation result.
For a linear elliptic operator, the Lipschitz continuity of the coefficient is sufficient. And as I remember, if the coefficients are Holder continuous, there are counterexamples.
See "Critical sets of solutions to elliptic equations", by R. Hardt, M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, and N. Nadirashvili, J. Differential Geom. Volume 51, Number 2 (1999), 359-373.
There is also an earlier paper by Lipman Bers, "Local behaviour of solutions of general linear elliptic equations", Comm. Pure Appl. Math. 8 (1955) 473-496.
Unique continuation fails on sets where two solutions agree, so their difference vanishes. So we can look for nonvanishing results on nonzero solutions. Some quantitative control on vanishing of the gradient of a nonzero solution, stronger than but implying an estimate on the Hausdorff dimension of the zero locus of the gradient of any nonzero solution, was proven by
Cheeger, Naber and Valorta, Critical Sets of Elliptic Equations, Communications in Pure and Applied Mathematics, 68 (2015), no. 2, 173–209.
On the other hand, there are counterexamples to various forms of unique continuation, for linear hyperbolic equations in
S. Alinhac and M. S. Baouendi, A nonuniqueness result for operators of principal type, Math. Z. 220 (1995), no. 4, 561–568. 85
and for nonlinear in
G. Métivier, Counterexamples to Hölmgren’s uniqueness for analytic nonlinear Cauchy problems, Invent. Math. 112 (1993), no. 1, 217–222.