On nonexistence of Feller semigroups in the nontransversal case

TL;DR

This paper presents examples of elliptic operators with nonlocal boundary conditions that, despite having a closure in continuous functions, do not generate Feller semigroups, highlighting limitations in the theory of such operators.

Contribution

It provides explicit examples demonstrating the nonexistence of Feller semigroups for certain elliptic operators with nonlocal boundary conditions.

Findings

Examples of operators with nonlocal boundary conditions that do not generate Feller semigroups

Illustration of limitations in the theory of Feller semigroups for elliptic operators

Insights into the boundary conditions affecting semigroup generation

Abstract

We give three examples of second-order elliptic operators with nonlocal boundary conditions of the Ventsel type that admit a closure in the space of continuous functions, but do not generate a Feller semigroup (i.e., a strongly continuous contractive nonnegative semigroup).