On multiwell Liouville theorems in higher dimension (Q2842996)

The authors consider certain subsets of the space of \(n\times n\) matrices of the form NEWLINE\[NEWLINEK= \bigcup^m_{i=1} \text{SO}(n) A_i,NEWLINE\]NEWLINE and show that for \(p>1\), \(q\geq 1\) and for connected \(\Omega'\subset\subset\Omega\subset \mathbb{R}^n\), there exists a positive constant \(a< 1\) depending on \(n\), \(p\), \(q\), \(\Omega\), \(\Omega'\) such that for NEWLINENEWLINE\[NEWLINE\varepsilon= \|\text{dist}(Du,K)\|^p_{L^p(\Omega)},NEWLINE\]NEWLINENEWLINE it follows that NEWLINENEWLINE\[NEWLINE\underset R\in {K}{\text{inf}}\| Du- R\|^p_{L^p(\Omega')}\leq M\varepsilon^{1/p}NEWLINE\]NEWLINE subject to the condition that for \(u\) the following inequality is satisfied NEWLINE\[NEWLINE\| D^2 u\|^q_{L^q(\Omega)}\leq a\varepsilon^{1-q}.NEWLINE\]NEWLINE The authors' result is valid in the case \(m=2\), and generically if \(m\leq n\) for every dimension greater than or equal to 3 if the wells \(\text{SO}(n)A_1,\dots, \text{SO}(n) A_m\) satisfy a connectivity condition. The results of this paper are known to some extent for \(n=2\). However, the results for \(n\geq 3\) are new. The proofs of the individual theorems in the present paper are precisely presented. The authors provide an essential contribution on multi-well Liouville theorems in higher dimension.