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

Abstract

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.

References in corpus (1)

• Characterization of traces of smooth functions on Ahlfors regular sets

Cited by in corpus (3)

• Steklov eigenvalues for the Lamé operator in linear elasticity
• Mixed boundary valued problem for linear and nonlinear wave equations in domains with fractal boundaries
• Optimal absorption of acoustical waves by a boundary