Generalization of Rellich-Kondrachov theorem and trace compacteness for fractal boundaries

Anna Rozanova-Pierrat (1)

1 MICS - Mathématiques et Informatique pour la Complexité et les Systèmes

We present a survey of recent results from the functional analysis that allow to solve PDEs in a large class of domains with irregular boundaries. We extend the previously introduced concept of admissible domains with a $d$-set boundary to domains with boundaries carrying measures that are not necessarily Ahlfors $d$-regular. We prove generalizations of the Rellich-Kondrachov theorem and the compactness of the trace operator and obtain uniqueness and existence results for weak solutions to Poisson boundary value problems with Robin boundary conditions. We observe the usual properties of the associated spectral problem.

Mots clés: Rellich-Kondrachov theorem; d-set; trace and extension operators; compact operators; fractal boundaries

Domaines: Analyse fonctionnelle [math.FA]; Equations aux dérivées partielles [math.AP]

Anna Rozanova-Pierrat. Generalization of Rellich-Kondrachov theorem and trace compacteness for fractal boundaries. Maria Rosaria Lancia and Anna Rozanova-Pierrat. Fractals in engineering: Theoretical aspects and Numerical approximations, 2020, ICIAM 2019 - SEMA SIMAI SPRINGER SERIES PUBLICATIONS. ⟨hal-02489325v2⟩