Our lecture will focus on two problems in pde which are solvable by ideas in holomorphic functions of complex variables. The first problem is called the strip theorem. Let $f$ be a function defined in the strip in the complex plane $|Im z| \leq 1$. Suppose $f$ agrees on the boundary of each unit circle centered on the real axis, radius $1$, with the solution (depending on the circle) of a suitable elliptic pde, the agreement being to order one greater than the order of the Dirichlet data. Then $f$ satisfies this pde. If the equation is the Cauchy-Riemann equation then equality suffices. The second type of problems we discuss are the Phragmen-Lindelof theorem for pde and a form of the Heisenberg uncertainty for pde. These were introduced in Kenig's lecture at Fefferman's birthday bash. We shall put them in a general framework.