The Aronsson Equation for Absolute Minimizers of Supremal Functionals in Carnot-Carath\'eodory Spaces

Andrea Pinamonti; Simone Verzellesi; Changyou Wang

The Aronsson Equation for Absolute Minimizers of Supremal Functionals in Carnot-Carath\'eodory Spaces

Analysis of PDEs 2022-03-01 v1

Authors: Andrea Pinamonti , Simone Verzellesi , Changyou Wang

Abstract

Given a $C^2$ family of vector fields $X_1,...,X_m$ which induces a continuous Carnot-Carath\'eodory distance, we show that any absolute minimizer of a supremal functional defined by a $C^2$ quasiconvex Hamiltonian $f(x, z, p)$ , allowing $z$ -variable dependence, is a viscosity solution to the Aronsson equation $-\langle X(f(x, u(x), Xu(x))), D_p f(x, u(x), Xu(x))\rangle = 0.$

Keywords

aronsson equation absolute minimizer supremal functional carnot-caratheodory space viscosity solution quasiconvex hamiltonian

Cite

@article{arxiv.2202.14008,
  title  = {The Aronsson Equation for Absolute Minimizers of Supremal Functionals in Carnot-Carath\'eodory Spaces},
  author = {Andrea Pinamonti and Simone Verzellesi and Changyou Wang},
  journal= {arXiv preprint arXiv:2202.14008},
  year   = {2022}
}

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