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

Abstract

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 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. Mathematics Subject Classification (2000). Primary 35J47, 35B65; Secondary 35A01.

Everywhere 𝒞α-estimates for a class of nonlinear elliptic systems with critical growth

“…Moreover, for general elliptic systems which need not be derived from an Euler system and for which u is not a minimizer to (1.3), it is known by a counterexample of Struwe [39] that (1.13) does not imply Hölder continuity of bounded W 1;2 solutions if f has quadratic growth in ru. For further counterexamples to this subject see [1]. On the other hand it is shown in [42] that the so-called angle condition, which is a refinement of (1.13) implies the Hölder continuity of a weak solution.…”

Hölder continuity of bounded, weak solutions of a variational system in the critical case

“…Beck and Frehse [1,Thm. 1.4] demonstrated that, under zero-Dirchlet boundary conditions, there is at least one weak vector-valued solution u : Ω → R N that is locally Hölder continuous in the critical setting.…”

“…Let Ω ⊂ R n , n ≥ 2 be a bounded, Lipschitz domain. Beck and Frehse [1] considered elliptic systems of the type: div (a(x, u, Du)) = a 0 (x, u, Du) in Ω ⊂ R n .…”

On the H-systems in higher dimension

“…In this connection remark that (1.1) is a nonlinear degenerate elliptic system with critical growth nonlinearity H(u)J u on the right-hand side that is merely integrable for maps u of class W 1,n . It is well-known that such systems in general admit very singular solutions and that a general regularity theory is only possible provided the right-hand side nonlinearity has a special structure (see for instance [1]). As mentioned, this is fully exploited for the system in two dimensions n = 2 by Rivière's result, and the hope is that the structure of H(u)J u remains strong enough to ensure regularity also for n ≥ 3, at least when backed up by appropriate additional hypotheses on H. We mention the results of Mou and Yang [20] who proved that solutions of (1.1) are C 1 , provided that either H is constant or u is weakly conformal.…”