Parametrices and exact paralinearisation of semi-linear boundary problems

The subject is to establish solution formulae for elliptic (and parabolic) semi-linear boundary problems. The result should be new in at least two respects: the desired formulae result from a parametrix construction for semi-linear problems, using only parametrices from the linear theory and the mild assumption that the non-linearity may be decomposed into a suitable solution-dependent linear operator acting on the solution itself. Secondly non-linearities of so-called product type are shown to admit such decompositions via exact paralinearisation. The parametrices give regularity properties under rather weak conditions, with examples of properties that are unobtainable by boot-strap methods. Regularity improvements in submanifolds are deduced from the auxiliary result that operators of type 1,1 are pseudo-local on large parts of their domains. The framework is flexible, encompassing a broad class of boundary problems and Hölder and Sobolev spaces, or the more general Besov and Triebel-Lizorkin spaces. The example include the von Karman equation.