A degenerate operator in non divergence form

TL;DR

This paper investigates a fourth order degenerate differential operator in nondivergence form, establishing conditions under which it generates an analytic semigroup, and extends the results to higher order operators.

Contribution

It provides new conditions ensuring the generation of an analytic semigroup by degenerate nondivergence form operators and extends these results to higher order derivatives.

Findings

The operator generates an analytic semigroup under certain degeneracy conditions.

Results are extended to operators of order 2n.

Provides a framework for analyzing degenerate higher order operators.

Abstract

In this paper we consider a fourth order operator in nondivergence form $Au:= au''''$, where $a: [0,1] \rightarrow \mathcal R_+$ is a function that degenerates somewhere in the interval. We prove that the operator generates an analytic semigroup, under suitable assumptions on the function $a$. We extend these results to a general operator $A_nu := au^{(2n)}$.