I heard that De Giorgi-Nash-Moser type regularity arguments fail for elliptic systems, but do not know where to start looking for more substantial information. Why does the regularity fail? Is there some cases where the Moser iteration can be successfully applied to elliptic systems?



Moser iteration for elliptic systems
Hi.


The point is not the ellipticity. In fact the Argument of De Giorgi, Moser and Nash was designed for elliptic problems. The point is that solutions $u: \Omega\to\mathbb{R}^N$ of elliptic problems with $N>1$ just aren't $C^{1,\alpha}$ any more in general. This is no problem with the method, it's intrinsic. The famous counterexample is by De Giorgi himself.


See Giusti - The direct method of variational calculus for more details to this topic. The counterexample itself can be found as example 6.3 in this book. The paper from De Giorgi is "Un esempio di estremal discontinue per un problema variazionale di tipo ellittico", Boll. U.M.I., 4 (1968), 135-137


