Schauder estimates at the boundary for sub-laplacians in Carnot groups

Abstract

In this paper we prove Schauder estimates at the boundary for sub-Laplacian type operators in Carnot groups. While internal Schauder estimates have been deeply studied, up to now subriemannian estimates at the boundary are known only in the Heisenberg groups. The proof of these estimates in the Heisenberg setting, due to Jerison ([32]), is based on the Fourier transform technique and can not be repeated in general Lie groups. After the result of Jerison no new contribution to the boundary problem has been provided. In this paper we introduce a new approach, which allows to build a Poisson kernel starting from the fundamental solution, from which we deduce the Schauder estimates at non characteristic boundary points.1991 Mathematics Subject Classification. 35R03, 35B65, 35J25.

Schauder estimates up to the boundary on H-type groups: an approach via the double layer potential

“…Recently Baldi, Citti and Cupini [3], assuming a geometric hypothesis on the boundary, obtained Schauder estimates at the boundary in neighborhood of noncharacteristic points for the problem ∆ G u = f in Ω with Dirichlet boundary condition u = g on ∂Ω. Here ∆ G = group G with distribution V 1 generated by the vector fields X 1 , .…”

“…, X k and Ω is an bounded open set of G. Here f and g belongs to the suitable classes of hölderian functions. The additional hypothesis that Baldi, Citti and Cupini [3] assume is that the induced distribution on the boundary, generated by the vector fields tangent to the boundary that belongs to the distribution V 1 , verifies the Hörmander condition. However even in the simplest case of the Heisenberg group H 1 this hypothesis is not verified.…”

Borderline gradient estimates at the boundary in Carnot groups

“…Very recently in [3], by suitably adapting the Levi's method of parametrix, A. Baldi, G. Citti and G. Cupini established Γ 2,α type Schauder estimate for nondivergence form operators upto the non-characteristic portion of a C ∞ boundary in more general Carnot groups. Subsequently in [4], by employing an alternate approach based on geometric compactness arguments, the authors showed the validity of Γ 1,α boundary Schauder estimate for divergence form operators as in (1.1) when boundary is C 1,α regular and a ij , f i ∈ Γ 0,α , h ∈ Γ 1,α , g ∈ L ∞ .…”

“…(a) In view of (3.102), it is clear that B(p, δ(p)) ⊂ Ω. Now, let us consider the function v := u − L p0 , where p 0 ∈ S 1/2 is the point corresponding to p discussed above and L p0 is the polynomial from step-(3). Again it is easy to see that v satisfies an equation of the type (3.97) in B(p, δ(p)) ⊂ Ω.…”