Estimates of the Green function and the initial-Dirichlet problem for the heat equation in sub-Riemannian spaces

2016/03/09 by Nicola Garofalo, Garofalo, Nicola, Isidro H. Munive +1
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP

paper · pdf · doi:10.48550/arxiv.1603.02988

arXiv admin note: text overlap with arXiv:0803.0784

arxiv created 2016/03/09 · arxiv updated 2016/03/10

Abstract

In a cylinder D_T = Ω× (0,T), where Ω⊂ ℝⁿ, we examine the relation between the L-caloric measure, dω^(x,t), where L is the heat operator associated with a system of vector fields of Hörmander type, and the measure dσ_X× dt, where dσ_X is the intrinsic X-perimeter measure. The latter constitutes the appropriate replacement for the standard surface measure on the boundary and plays a central role in sub-Riemannian geometric measure theory. Under suitable assumptions on the domain Ω we establish the mutual absolute continuity of dω^(x,t) and dσ_X× dt. We also derive the solvability of the initial-Dirichlet problem for L with boundary data in appropriate L^p spaces, for every p>1.

Citations

• Mutual Absolute Continuity of Harmonic and Surface Measures for Hormander Type Operators

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