Complex variables are not dead

Complex variables are not dead

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Leon Ehrenpreis, Temple University
Fine Hall 110

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.