Essentially smooth Lipschitz functions
The authors observe fundamental questions regarding the differentiability structure of locally Lipschitz functions defined on Banach spaces. So the following 3 significant properties are discussed: (P1) \(D\)-representability, which means Gâteaux-differentiability on a dense subset \(D\) of the domain and the description of the Clarke subdifferential mapping by the gradients on \(D\); (P2) Integrability, which means the unique determination of the function (up to an additive constant) by means of its Clarke subdifferential mapping; (P3) Differentiability properties similar to those enjoyed by continuous convex functions. The aim of the paper is the examination of the relationships between these properties and the construction of a family of functions which satisfy all the three properties. After a short survey of uscos (compact-valued upper semicontinuous mappings), cuscos (convex-compact-valued upper semicontinuous mappings), and especially of minimal uscos and minimal cuscos, it is pointed out that the \(D\)-representability of a function is closely connected with the minimality of the Clarke subdifferential mapping. Moreover, it is shown that for locally Lipschitz functions which are densely Gâteaux differentiable (this is fulfilled, e.g., in separable Banach spaces) both notions are equivalent. In the main section of the paper the authors observe the class of the so-called essentially smooth functions, i.e., the class of functions which are strictly differentiable everywhere except possibly on a Haar-null set. It is shown that (in separable Banach spaces) these functions satisfy all the above mentioned properties. Moreover, this class of functions possesses very strong closure properties (e.g., regarding addition, subtraction, multiplication, division, and lattice operations) and contains many other important subclasses of Lipschitz functions (e.g., pseudo-regular and semi-smooth functions). The results are applied especially to perturbation functions and to distance functions. At the end of the paper, some examples are provided to show that in general \(D\)-representability and integrability are not comparable.
compact-valued upper semicontinuous mappings
convex-compact-valued upper semicontinuous mappings
scientific article; zbMATH DE number 1078818
scientific article