Fourth order differential operators with interior degeneracy and generalized Wentzell boundary conditions

TL;DR

This paper investigates fourth order differential operators with interior degeneracy and generalized Wentzell boundary conditions, establishing their well-posedness via semigroup methods for associated parabolic problems.

Contribution

It introduces new analysis of degenerate fourth order operators with interior degeneracy and boundary conditions, proving generation and well-posedness results.

Findings

Operators generate analytic semigroups

Well-posedness of associated parabolic problems

Conditions ensuring generation property

Abstract

In this paper we consider the fourth order operators A1u := (au")" and A2u := au"" in divergence form and non divergence form, respectively, where a, defined in [0, 1] with values in R+, degenerates in an interior point of the interval. Using the semigroup technique, under suitable assumptions on a, we study the generation property of these operators associated to generalized Wentzell boundary conditions, proving the well posedness of the corresponding parabolic problems.