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?
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$\begingroup$ You made five new tags for this question. This seems a little excessive: in particular, at least one of the tags is supposed to be an arxiv tag. But this is so far from my expertise that I better not tamper with it. Can someone else make any constructive suggestions here? $\endgroup$– Pete L. ClarkCommented Sep 15, 2010 at 7:22
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1$\begingroup$ Note the correct spelling: Ennio De Giorgi. $\endgroup$– Pietro MajerCommented Sep 15, 2010 at 7:44
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2 Answers
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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
answered Sep 15, 2010 at 9:39
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$\begingroup$ Thanks a lot for your answer! It seems that one counterexample would not rule out the possibility that the argument can be applied to specific situations. Or this example just kills everything? Do you know a specific example where similar type of argument has been successfully applied? $\endgroup$– timurCommented Sep 15, 2010 at 15:52
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1$\begingroup$ You're right: There are cases where a De-Giorgi-type argument still works and this is currently an active field of research. I don't know a specific example. $\endgroup$ Commented Sep 15, 2010 at 17:18
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$\begingroup$ I don't know how much you want to be precise in imitating the arguments, but if you admit some sloppiness, then you'll be in trouble finding anything not using those methods. There is the "small energy implies regularity" result for systems (as in the treatment Evans in the answer of Hung Tran below) where the excess decay is proved much like in De Giorgi's method. A bootstrap like Moser is instead present in the very influential paper by Uhlenbeck form 1977 (Regularity for a class of non-linear elliptic systems. Acta Math. 138 (1977), no. 3-4, 219–240.). $\endgroup$– MirceaCommented Mar 22, 2012 at 19:00
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$\begingroup$ It is all about what definition you use for coercivity. Many problems are fine, of course, as they are just like the scalar ones, e.g. uniformly convex ones, diagonally dominant problems ... $\endgroup$– usernameCommented Oct 6, 2015 at 12:29
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Basically, the De-Giorgi - Moser - Nash regularity result fails for elliptic system. As Johannes pointed out, there is a counter-example in Giusti book.
For system, usually, one can get "partial regularity result". What I mean by that is: there exists $\Omega_0 \subset \Omega$ open such that $|\Omega\setminus \Omega_0|=0$ and $u \in C^{1,\alpha}(\Omega_0)$. Then with the smoothness of coefficients, you can have $u \in C^\infty(\Omega_0)$.
There are several references you can have a look at are:
M. GIAQUINTA, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Princeton U. Press, Princeton, 1983.
M. GIAQUINTA • E. GIUSTI, On the regularity of the minima of variational inte- grals, Acta. Math. 148 (1982), 31-46.
Evans, Lawrence C. Quasiconvexity and partial regularity in the calculus of variations. Arch. Rational Mech. Anal. 95 (1986), no. 3, 227--252.
In the third one, Evans proved the "partial regularity result" for a minimizer of certain energy functional. The most important thing is that the Lagrangian only need to be uniformly stricly-quasiconvex instead of uniformly convex. And it has some very important applications in elasticity. You can search for some papers of John Ball for this issue.
