Parametrices of semi-linear elliptic boundary problems

As an example one can take the Laplace equation plus the square of the unknown, $-\Delta u + u2=f$ in an open set $\Omega$ in $R^n$, considered with a Dirichl\'et condition $u=g$ on the boundary. The purpose of the talk is to explain how one can obtain parametrices $P_N$ of this non-linear problem. The resulting parametrix formula $u=P_N(Rf+Kg)+ (RL)^Nu$ expresses a given solution $u$ via terms depending on the data $(f,g)$ \[$R$, $K$ are solution operators of the corresponding linear problem\] plus a remainder in $C^k$ for arbitrarily large $k$. The formula implies that solutions belong to the same spaces as in the linear case, under some mild assumptions allowing non-classical cases in which the solution `ends up' in a space on which the non-linear term $u2$ is ill-defined. The parametrix construction uses pseudo- and paradifferential techniques, and it extends to general semi-linear elliptic systems with non-linear terms of product type.