Analysis Seminar: "Pseudo-holomorphic conjugate functions and the Dirichlet problem"

Speaker: Laurent Baratchart, INRIA

Abstract: On a simply connected rectifiable planar domain, we consider  pseudo holomorphic functions $w$ satisfying $\partial w=\alpha \bar{w}$, where $\alpha$ is a square summable complex valued function.

We introduce Hardy-Smirnov  classes of exponent $1<p<\infty$ together with their boundary values, and  discuss the M. Riesz problem: given a real valued $u$ in $L^p$ of the boundary,  to find a real valued $v$ such that $u+iv$ is the trace of a solution. For a  large class of rectifiable domains, we characterize solvability in terms of the   $A-p$ condition of the derivative of a conformal map onto the disk. This in turn impinges on the range of exponents for which the Dirichlet problem with $L^p$-boundary values is solvable for elliptic equations with sufficiently smooth (say Lipschitz) coefficients. This is joint work with E. Russ and E. Pozzi.

Host: Brett Wick