Abstract or Additional Information
Among different systems of elliptic PDE's of particular interest are those with the so-called critical growth, i.e. systems with (some of the) nonlinear terms a priori only integrable. For such systems, standard constructions of test functions fail and the applicable toolbox of regularity theory for PDE's is very limited. Important examples include systems describing harmonic maps between manifolds, parametrizations of surfaces with prescribed mean curvature and their numerous generalizations.
A tool that has proved very beneficial in the last 2 decades is known as Uhlenbeck-Riviere decomposition theorem. Originally proved by Karen Ulenbeck as a result in Yang-Mills theory on existence of a gauge, in which the connection form has particularly simple form, it has been reformulated by Riviere and used in his breakthrough proof of Heinz-Hildebrandt conjecture. Since then, different versions of this theorem have been proved and applied to many geometrically motivated problems.
In a joint work with Anna Zatorska-Goldstein we seek a general form of Uhlenbeck-Riviere decomposition theorem and study possible generalizations.