Regular and irregular solutions for a class of elliptic systems in the critical dimension

Abstract

We study regularity properties of weak solutions in the Sobolev space \({W^{1,n}_0}\) to inhomogeneous elliptic systems under a natural growth condition and on bounded Lipschitz domains in \({\mathbb{R}^n}\) , i. e. we investigate weak solutions in the limiting situation of the Sobolev embedding. Several counterexamples of irregular solutions are constructed in cases, where additional structure conditions might have led to regularity. Among others we present both bounded irregular and unbounded weak solutions to elliptic systems obeying a one-sided condition, and we further construct unbounded extremals of two-dimensional variational problems. These counterexamples do not exclude the existence of a regular solution. In fact, we establish the existence of regular solutions—under standard assumptions on the principal part and the aforementioned one-sided condition on the inhomogeneity. This extends previous works for n = 2 to more general cases, including arbitrary dimensions. Moreover, this result is achieved by a simplified proof invoking modern techniques.