We derive new representations for the generalised Jacobian of a locally Lipschitz map between finite dimensional real Euclidean spaces as lower limit (i.e., inferior) classical derivative where it exists. The lead to significantly shorter proofs basic properties subgradient and including chain rule. establish that sequence maps converges given in L-topology—that is, weakest refinement sup norm ...

Let X, Y be complete metric spaces and E, F be Banach spaces. A bijective linear operator from a space of E-valued functions on X to a space of F -valued functions on Y is said to be biseparating if f and g are disjoint if and only if Tf and Tg are disjoint. We introduce the class of generalized Lipschitz spaces, which includes as special cases the classes of Lipschitz, little Lipschitz and uni...

We show that if K is a compact metric space then C(K) is a 2-absolute Lipschitz retract. We then study the best Lipschitz extension constants for maps into C(K) from a given metric space M , extending recent results of Lancien and Randrianantoanina. They showed that a finitedimensional normed space which is polyhedral has the isometric extension property for C(K)-spaces; here we show that the s...

We show, as our main theorem, that if a Lipschitz map from compact Riemannian manifold $M$ to connected $N$, where $\dim M \geq \dim N$, has no singular points on in the sense of Clarke, then admits smooth approximation via Ehresmann fibrations. also show Reeb sphere theorem for functions, i.e., closed function with exactly two is homeomorphic sphere.

for x 6= x′. A set Y ⊂ Rn is a net if there is an R such that d(x, Y ) < R for every x ∈ Rn; it is separated if there is an ǫ > 0 such that |y − y′| > ǫ > 0 for every pair y 6= y′ in Y . History. In 1965, J. Moser showed that any two positive, C∞ volume forms on a compact manifold with the same total mass are related by a diffeomorphism [Mos]. Extensions of this result to other smoothness class...

A metric space (X, d) is called c-monotone if there is a linear order < on X and c > 0 such that d(x, y) 6 c d(x, z) for all x < y < z in X. A brief account of investigation of monotone spaces including applications is presented. 1 Monotone and σ-Monotone Spaces. In [6] I investigated existence of sets in Euclidean spaces that have large Hausdorff dimension and yet host no continuous finite Bor...

We study Robin-to-Robin maps, and Krein-type resolvent formulas for Schrödinger operators on bounded Lipschitz domains in R, n > 2, with generalized Robin boundary conditions.

We consider metric measure spaces satisfing a doubling condition and a Poincaré inequality in the upper gradient sense. We show that the results of [Che99] on differentiability of real valued Lipschitz functions and the resulting bi-Lipschitz nonembedding theorems for finite dimensional vector space targets extend to Banach space targets having what we term a good finite dimensional approximati...

We study geometric and topological properties of the image of a smooth submanifold of Runder a bi-Lipschitz map to R. In particular, we characterize how the dimension, diameter,volume, and reach of the embedded manifold relate to the original. Our main result establishesa lower bound on the reach of the embedded manifold in the case where m ≤ n and the bi-Lipschitz map is li...

We show the existence of a global unique and analytic solution for the mean curvature flow and the Willmore flow of entire graphs for Lipschitz initial data with small Lipschitz norm. We also show the existence of a global unique and analytic solution to the Ricci-DeTurck flow on euclidean space for bounded initial metrics which are close to the euclidean metric in L∞ and to the harmonic map fl...