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Existence and uniqueness of global solutions to fully nonlinear second order elliptic systems

Katzourakis, N. (2016) Existence and uniqueness of global solutions to fully nonlinear second order elliptic systems. Nonlinear differential equations and applications, 23 (3). 33. ISSN 1420-9004

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To link to this item DOI: 10.1007/s00030-016-0381-2

Abstract/Summary

We consider the problem of existence and uniqueness of strong a.e. solutions u:Rn⟶RNu:Rn⟶RN to the fully nonlinear PDE system F(⋅,D2u)=f, a.e. on Rn,(1) F(⋅,D2u)=f, a.e. on Rn,(1) when f∈L2(Rn)Nf∈L2(Rn)N and F is a Carathéodory map. (1) has not been considered before. The case of bounded domains has been studied by several authors, firstly by Campanato and under Campanato’s ellipticity condition on F. By introducing a new much weaker notion of ellipticity, we prove solvability of (1) in a tailored Sobolev “energy” space and a uniqueness estimate. The proof is based on the solvability of the linearised problem by Fourier transform methods, together with a “perturbation device” which allows to use Campanato’s near operators. We also discuss our hypothesis via counterexamples and give a stability theorem of strong global solutions for systems of the form (1).

Item Type:Article
Refereed:Yes
Divisions:Faculty of Science > School of Mathematical, Physical and Computational Sciences > Department of Mathematics and Statistics
ID Code:67061
Publisher:Springer

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