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 Hölder 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.

57 pages, 3 figures. Accepted version. Appeared online in Communications of partial differential equations on 4 December 2008; available at Taylor and Francis: http://www.tandfonline.com/doi/full/10.1080/03605300802289188

References in corpus (5)

• Pointwise multiplication of Besov and Triebel--Lizorkin spaces
• A direct proof of Sobolev embeddings for quasi-homogeneous Lizorkin--Triebel spaces with mixed norms
• Domains of pseudo-differential operators: a case for the Triebel--Lizorkin spaces
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Cited by in corpus (2)

• $L_p$-Theory of Type $1,1$-Operators
• Fundamental Results for Pseudo-Differential Operators of Type $1,1$