Generalization of Rellich-Kondrachov theorem and trace compacteness in the framework of irregular and fractal boundaries

Anna Rozanova-Pierrat (1)

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

We present a survey of recent results of the functional analysis allowing 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 on the domains with the boundaries on which the measure is not necessarily Ahlfors regular d-measure. This gives a generalization of Rellich-Kondrachov theorem and the compactness of the trace operator, allowing to obtain, as for a regular classical case the unicity/existence of weak solutions of Poisson boundary valued problem with the Robin boundary condition and to obtain the usual properties of the associated spectral problem.

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

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

Anna Rozanova-Pierrat. Generalization of Rellich-Kondrachov theorem and trace compacteness in the framework of irregular and fractal boundaries. 2020. ⟨hal-02489325v1⟩