Let ~ u be a vector eld on a bounded Lipschitz domain in R 3 , and let ~ u together with its divergence and curl be square integrable. If either the normal or the tangential component of ~ u is square inte-grable over the boundary, then ~ u belongs to the Sobolev space H 1=2 on the domain. This result gives a simple explanation for known results on the compact embedding of the space of solution...

Control and state estimation of nonlinear systems satisfying a Lipschitz continuity condition have been important topics in nonlinear system theory for over three decades, resulting in a substantial amount of literature. The main criticism behind this approach, however, has been the restrictive nature of the Lipschitz continuity condition and the conservativeness of the related results. This wo...

where u : (0, T ]× Ω→ R is the unknown function, Ω ⊂ R is a bounded domain with a Lipschitz continuous boundary Γ, 0 < T <∞ and ∂νβ(u) denotes outside normal derivative. Function β : R → R is nondecreasing Lipschitz continuous function satisfying (2.1). Function f : (0, T ]×Ω×R→ R is Lipschitz continuous with Lipschitz constant Lf and satisfies (2.2). We denote g(t, x, s) : = γs+φ(t, x), where ...

This is one of a series of papers examining the interplay between differentiation theory for Lipschitz maps, X → V , and bi-Lipschitz nonembeddability, where X is a metric measure space and V is a Banach space. Here, we consider the case V = L, where differentiability fails. We establish another kind of differentiability for certain X , including R and H, the Heisenberg group with its Carnot-Ca...

We introduce the general and powerful scheme of predicting information re-use in optimization algorithms. This allows us to devise a computationally efficient algorithm for bandit convex optimization with new state-of-the-art guarantees for both Lipschitz loss functions and loss functions with Lipschitz gradients. This is the first algorithm admitting both a polynomial time complexity and a reg...

we show that the character space of the vector-valued lipschitz algebra $lip^{alpha}(x, e)$ of order $alpha$ is homeomorphic to the cartesian product $xtimes m_e$ in the product topology, where $x$ is a compact metric space and $e$ is a unital commutative banach algebra. we also characterize the form of each character on $lip^{alpha}(x, e)$. by appealing to the injective tensor product, we then...