unique continuation

An analytic function has a rigid kind of memory: if you know it on any tiny patch, you know it everywhere — its values on a small piece determine the whole. Solutions of elliptic equations inherit a version of this rigidity. Unique continuation says that a solution which vanishes on any small open set (or, in a stronger form, vanishes to infinite order at a single point) must vanish throughout the connected domain. You cannot have a solution that is exactly zero in one region and nonzero in a neighbouring one.

Precisely: if u solves an elliptic equation on a connected domain and u is identically zero on some open subset, then u is zero on the whole domain. Equivalently, two solutions that agree on a small open set agree everywhere. The phrase strong unique continuation strengthens open set to a single point: if u vanishes faster than every power of the distance at one point (vanishes to infinite order there), it is identically zero. Harmonic functions get this for free from being real-analytic; for general elliptic operators with non-analytic coefficients it is a genuine theorem, proved with delicate Carleman estimates (weighted energy inequalities) rather than power series. There are parabolic versions too, reflecting diffusion's smoothing, while hyperbolic unique continuation is more subtle and tied to the geometry of characteristics.

Why care? Unique continuation is the abstract heart of why elliptic and parabolic solutions are so rigidly determined — it is the qualitative reason behind uniqueness for inverse problems and control problems, where you try to recover or steer a solution from partial information. If a quantity you can observe is zero, unique continuation often forces the hidden state to be zero too. Honesty note: it is a hallmark of elliptic/parabolic rigidity and can FAIL for general or non-smooth equations — there are famous counterexamples (Plis, and others) of elliptic-type equations with smooth coefficients whose nonzero solutions vanish on an open set, so unique continuation is a theorem with hypotheses, not a free lunch.