Regularity versus singularity for elliptic problems in two dimensions (Q2861482)

For a second-order elliptic system in divergence form NEWLINE\[NEWLINE-\mathrm{div }a(\cdot,u,Du)=b(\cdot,u,Du) \;\text{ in }\Omega,NEWLINE\]NEWLINE where \(\Omega\) is a bounded domain in \(\mathbb R^2\) and the coefficients \(a:\Omega \times \mathbb R^N\times \mathbb R^{2N}\rightarrow \mathbb R^{2N}\) satisfy standard growth, ellipticity and regularity assumptions, the author, using similar arguments as in [\textit{J. Kristensen} and \textit{G. Mingione}, Arch. Ration. Mech. Anal. 180, No. 3, 331--398 (2006; Zbl 1116.49010)], proves that a \(W^{1,p}(\Omega,\mathbb R^N)\) \((p>1)\) solution \(u\) belongs to \(C^{1,\alpha}_{\mathrm{loc}}(\Omega)\) with optimal exponent \(\alpha\), provided that one of following conditions holds: 1) the a priori integrability exponent \(p\) is greater than an exponent \(p_1\in (1,2)\) which depends on the constants involved in the ellipticity, regularity and growth conditions on the coefficients; 2) the coefficients are independent of \(u\); 3) the solution \(u\) is locally continuous. NEWLINENEWLINENEWLINEThe main objective of this paper is then giving an answer to the following problem: Keeping the same assumptions on the coefficients as above, is \(Du\) locally Hölder continuous for every \(W^{1,p}(\Omega,\mathbb R^N)\)-solution \(u\) of the system? The author gives a negative answer to this problem by exhibiting a non-smooth function defined in the unit ball of \(\mathbb R^2\) which is a solution of the above system for suitable coefficients satisfying the standard growth, ellipticity and regularity conditions.