##
**Convex integration for Lipschitz mappings and counterexamples to regularity.**
*(English)*
Zbl 1083.35032

In this very interesting paper the authors solve a long–standing problem in the regularity theory for elliptic systems. Let \(\Omega\) be an open disk of \({\mathbb R}^2\) and let \(u\) denote a map \(u:\Omega\to {\mathbb R}^2\). The authors show the existence of a variational integral
\[
I (u)\;:=\; \int_\Omega F( \nabla u) \tag{1}
\]
such that

(a) \(F\) is smooth, strongly quasiconvex and has bounded second derivatives;

(b) The Euler–Lagrange equation of \(I\) has a large class of Lipschitz weak solutions which are nowhere \(C^1\).

This is in contrast with the well–known partial regularity of minimizers of \(I\), proved by L. C. Evans in [Arch. Ration. Mech. Anal. 95, 227–252 (1986; Zbl 0627.49006)]. The result is achieved with a new original approach, quite different from those of the classical counterexamples to regularity for elliptic systems. The authors reduce the task of finding solutions of (1) to that of solving a related partial differential inclusion of type \[ \nabla v \in K \qquad {\text{ for a.e. }} x\in \Omega,\tag{2} \] where \(v\) is a Lipschitz map from \(\Omega\) into \({\mathbb R}^4\). Then they construct rough solutions of (2) by combining an extension of Gromov’s theory of partial differential relations with a clever use of certain particular configurations of \(4\times 2\) matrices, called \({\mathbb T}_4\). A \({\mathbb T}_4\) configuration of \(m\times n\) matrices is a set four matrices \(M:=\{M_1, M_2, M_3, M_4\}\) such that \(\text{ rank} (M_i-M_j)> 1\) for every \(j\neq i\) but the rank–one convex hull of \(M\) is quite large. The existence of such configurations was discovered independently by many authors in the last twenty years; in a context closely related to this paper it was first pointed out by L. Tartar.

The approach of Müller and Sverak has been recently improved by L. Szekelyhidi in [Arch. Ration. Mech. Anal. 172, No. 1, 133–152 (2004; Zbl 1049.49017)] where the author shows the same pathology for some polyconvex integrands \(F\). The connection between nonlinear PDEs and questions about hulls of sets of matrices has been fruitfully used in many other contexts; the interested reader is referred to [B. Kirchheim, S. Müller and V. Sverak, ”Studying nonlinear PDE by geometry in matrix space. Hildebrandt, Stefan (ed.) et al., Geometric analysis and nonlinear partial differential equations.” Berlin: Springer, 347–395 (2003; Zbl 1290.35097)]. Finally, the authors mention that very recently they became aware of some important partial results on the regularity problem for elliptic systems obtained by V. Scheffer in [Regularity and irregularity of solutions to nonlinear second order elliptic systems of partial differential equations and inequalities, Dissertation, Princeton University (1974, unpublished)] using a very similar point of view. It seems that this work was never published in a journal and did not receive the attention it deserves.

(a) \(F\) is smooth, strongly quasiconvex and has bounded second derivatives;

(b) The Euler–Lagrange equation of \(I\) has a large class of Lipschitz weak solutions which are nowhere \(C^1\).

This is in contrast with the well–known partial regularity of minimizers of \(I\), proved by L. C. Evans in [Arch. Ration. Mech. Anal. 95, 227–252 (1986; Zbl 0627.49006)]. The result is achieved with a new original approach, quite different from those of the classical counterexamples to regularity for elliptic systems. The authors reduce the task of finding solutions of (1) to that of solving a related partial differential inclusion of type \[ \nabla v \in K \qquad {\text{ for a.e. }} x\in \Omega,\tag{2} \] where \(v\) is a Lipschitz map from \(\Omega\) into \({\mathbb R}^4\). Then they construct rough solutions of (2) by combining an extension of Gromov’s theory of partial differential relations with a clever use of certain particular configurations of \(4\times 2\) matrices, called \({\mathbb T}_4\). A \({\mathbb T}_4\) configuration of \(m\times n\) matrices is a set four matrices \(M:=\{M_1, M_2, M_3, M_4\}\) such that \(\text{ rank} (M_i-M_j)> 1\) for every \(j\neq i\) but the rank–one convex hull of \(M\) is quite large. The existence of such configurations was discovered independently by many authors in the last twenty years; in a context closely related to this paper it was first pointed out by L. Tartar.

The approach of Müller and Sverak has been recently improved by L. Szekelyhidi in [Arch. Ration. Mech. Anal. 172, No. 1, 133–152 (2004; Zbl 1049.49017)] where the author shows the same pathology for some polyconvex integrands \(F\). The connection between nonlinear PDEs and questions about hulls of sets of matrices has been fruitfully used in many other contexts; the interested reader is referred to [B. Kirchheim, S. Müller and V. Sverak, ”Studying nonlinear PDE by geometry in matrix space. Hildebrandt, Stefan (ed.) et al., Geometric analysis and nonlinear partial differential equations.” Berlin: Springer, 347–395 (2003; Zbl 1290.35097)]. Finally, the authors mention that very recently they became aware of some important partial results on the regularity problem for elliptic systems obtained by V. Scheffer in [Regularity and irregularity of solutions to nonlinear second order elliptic systems of partial differential equations and inequalities, Dissertation, Princeton University (1974, unpublished)] using a very similar point of view. It seems that this work was never published in a journal and did not receive the attention it deserves.

Reviewer: Camillo De Lellis (Zürich)