Regularity of minimizers of the calculus of variations in Carnot groups via hypoellipticity of systems of Hörmander type
type: Item
instance of: scholarly article
title: Regularity of minimizers of the calculus of variations in Carnot groups via hypoellipticity of systems of Hörmander type
zbMATH Open document ID: 1064.49026
author: Luca Capogna
author: Nicola Garofalo
published in: Journal of the European Mathematical Society (JEMS)
publication date: 2003-08-20T00:00:00Z
review text: 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 …
reviewed by: Tilak Bhattacharya
Mathematics Subject Classification ID: 49N60
Mathematics Subject Classification ID: 35H10
Mathematics Subject Classification ID: 22E25
zbMATH DE Number: 1968514
zbMATH Keywords: Carnot groups
zbMATH Keywords: calculus of variations
zbMATH Keywords: minimizers
zbMATH Keywords: regularity
MaRDI profile type: Publication
full work available at URL: s100970200043
OpenAlex ID: W2088076367
cites work: Rectifiable sets in metric and Banach spaces
cites work: Currents in metric spaces
cites work: Q3941958
DOI: 10.1007/S100970200043
description: scientific article; zbMATH DE number 1968514