Regularity of minimizers of the calculus of variations in Carnot groups via hypoellipticity of systems of Hörmander type
The authors in this work attempt to develop the regularity theory for local minimizers of non-convex variational functionals and for weak solutions of quasilinear systems in Carnot groups of step 2. A Carnot group of step \(r\geq 1\) is a simply connected Lie group \(G\) whose Lie algebra admits a vector space decomposition in \(r\) layers which are stratified and \(r\)-nilpotent. For Carnot groups of step 2 with Lie algebra \(g=V^1\oplus V^2\), fix an orthonormal basis \(X=\{X_1,X_2,...,X_m\}\) of the first layer \(V^1\) of \(g\). The work first studies the partial regularity of local minimizers of functionals on bounded open sets \(\Omega\subset G\), \[ F(u,\Omega)=\int_{\Omega}F(Xu)dp,\tag{1} \] where \(F:\mathbb R^{mN}\rightarrow \mathbb R^+\) is a \(C^2\) function satisfying (i) \(F(\eta)\leq C(1+| \eta| ^q)\), for some \(2\leq q<\infty\), and (ii) \(\sum_{\alpha,\beta=1}^{N}\sum_{i,j=1}^{m} \partial_{\eta_i^{\alpha}}\partial_{\eta_i^{\alpha}}F(\eta)\xi_i^{\alpha}\xi_j^{\beta}\geq \lambda(1+| \eta| ^{q-2})| \xi| ^2.\) Note that while the above functional is strictly convex in \(Xu\) (the horizontal Jacobian matrix of \(u\)) it may not be so in the full Riemannian gradient \(Du\). The authors also study the weak solutions of \[ -\sum_{\beta=1}^N\sum_{i,j=1}^mX_i(A_{ij}^{\alpha\beta}(x,u)X_ju^{\beta})=f^{\alpha} (x,u,Xu),\; \alpha=1,\dots,N, \tag{2} \] where \(A_{ij}^{\alpha\beta}(x,u)\) are Hölder continuous, and for every \(M>0\) there exist \(\lambda(M)>0\) and \(a(M)>0\), such that for \(x\in \Omega\) and \(| u| \leq M\), \[ \sum_{\alpha,\beta=1}^N\sum_{i,j=1}^m A_{ij}^{\alpha\beta}(x,u)\eta^{\alpha}_i \eta^{\beta}_j \geq \lambda(M)| \eta| ^2,\;\;| f(x,u,\eta)|\leq a(M)| \eta| ^2+b. \tag{3} \] The authors show that (A) if \(u\) is a local minimizer of (1) then \(Xu\) is Hölder continuous in \(\Omega\setminus S\), where \(S\) has measure zero and in particular, \(H^{Q-2+\sigma}(S)=0\) for \(q=2\) and every \(\sigma>0\). Here \(Q\) is the homogeneous dimension of \(G\). In order to estimate the size of the singular set, they also show that if \(q=2\) then the Riemannian gradient has square integrable horizontal derivatives. The second set of results are (B) if \(u\) is a weak solution of (2), satisfying (3) and \(a(M)M\leq \lambda(M)/2\), then there is a closed set \(S\subset \Omega\) such that \(Xu\) is Hölder continuous in \(\Omega\setminus S\) and \(H^{Q-2+\sigma}(S)=0\), for every \(\sigma>0\). The authors also consider constant coefficient systems in (2) with \(C^{\infty}\) righthand sides, they show that such systems are then hypoelliptic, i.e., solutions are \(C^\infty\). The major contribution of this paper is to develop direct proofs since \(W^{2,2}\) estimates for constant coefficient systems of Hörmander type are lacking. They also adapt their methods to the non-abelian setting.
scientific article; zbMATH DE number 1968514