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On the best possible character of the \(L^Q\) norm in some a priori estimates for non-divergence form equations in Carnot groups. (English) Zbl 1036.35047

Let \(G\) be a group of Heisenberg type with homogeneous dimension \(Q.\) The authors construct for every \(0<\varepsilon<Q\) a non-divergence form operator \(L^\varepsilon\) and a nontrivial solution \(u^\varepsilon\in {\mathcal L}^{2,Q-\varepsilon}(\Omega)\cap C(\overline\Omega)\) to the Dirichlet problem: \(Lu=0\) in \(\Omega,\) \(u=0\) on \(\partial\Omega.\) This non-uniqueness result shows the impossibility of controlling the maximum of \(u\) with an \(L^p\) norm of \(L u\) when \(p<Q.\) Another consequence is the impossibility of an Alexandrov-Bakelman type estimates such as \[ \sup_\Omega\leq C \left(\int_\Omega | \operatorname {det} (u_{,ij})| \,dg \right)^{1/m}, \] where \(m\) is the dimension of the horizontal layer of the Lie algebra and \((u_{,ij})\) is the symmetrized horizontal Hessian of \(u.\)

MSC:

35B50 Maximum principles in context of PDEs
22E30 Analysis on real and complex Lie groups
52A30 Variants of convex sets (star-shaped, (\(m, n\))-convex, etc.)
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