Nonsmooth solutions for a class of fully nonlinear PDE’s on Lie groups

Existence and uniqueness results for linear second-order equations in the Heisenberg group

“…In spite of the fact that equation (1.1) is a relatively simple model, to the best of our knowledge, there are no comparison and uniqueness results that may be applied to the full form (1.1) (compare with the recent works [22] and [21]). Indeed, for the available comparison principles for second-order elliptic equations F (x, u, ∇ H u, ∇ 2 H 0 u) = 0, the following assumptions appear, in general, as essential tools:…”

“…Observe that equation (1.1) does not depend on the variable u explicitly. This is because the u-dependence has been already treated in the literature [20,22,21].…”

Nonsmooth viscosity solutions of elementary symmetric functions of the complex Hessian

“…. The proof of this theorem uses Pogorelov's counterexamples (see [6,Section 5.5]) and its extensions developed in [16] , [7], [12].…”

A study of comparison, existence and regularity of viscosity and weak solutions for quasilinear equations in the Heisenberg group

“…where Ω is a John domain, A : Ω × R 2 → R 2 and f : Ω × R → R. We refer the reader to the Appendix for the proofs of existence and uniqueness of weak solutions of equations like (30). We highlight the parallelism between the assumptions on A and f in (30) to obtain the mentioned results to the corresponding well-known conditions in the Euclidean scenario. Moreover, even do one can not apply directly the Euclidean results in the Heisenberg framework, the use of slightly modifications of Euclidean techniques works in the sub-Riemannian setting.…”

“…Towards getting comparison principles, it is usual to assume that a given second order operator F = F(p, u, ∇ 1 u, ∇ 2, * 0 u) does not depend on the spatial variable p, or the first-derivative variable ∇ 1 u, and also assume that F has bounded away from zero derivatives ∂F/∂u (strict monotonicity in u). For examples of these results we quote [28], [31,Proposition 4.1], [30,Theorem 2.1] (here the authors remove the strictly increasing assumptions but they assume a sign condition). At this point we would like to quote the works [1] and [29] where the authors propose various form of partial nondegeneracy to weaken the uniform ellipticity assumption and apply their results to some sub-elliptic second order equations.…”