Parametrices and exact paralinearisation of semi-linear boundary problems

Abstract

The subject is parametrices for semi-linear problems, based on parametrices for linear boundary problems and on non-linearities that decompose into solution-dependent linear operators acting on the solutions. Non-linearities of product type are shown to admit this via exact paralinearisation. The parametrices give regularity properties under weak conditions; improvements in subdomains result from pseudo-locality of type $1,1$-operators. The framework encompasses a broad class of boundary problems in H\"older and $L_p$-Sobolev spaces (and also Besov and Lizorkin--Triebel spaces). The Besov analyses of homogeneous distributions, tensor products and halfspace extensions have been revised. Examples include the von Karman equation.