In this paper, we propose a new iteration method based on the hybrid steepest descent method and Ishikawa-type method for seeking a solution of a variational inequality involving a Lipschitz continuous and strongly monotone mapping on the set of common fixed points for a finite family of Lipschitz continuous and quasi-pseudocontractive mappings in a real Hilbert space. MSC: Primary 41A65; 47H17...

We prove convergence of the solutions Xn of semilinear stochastic evolution equations on a Banach space B, driven by a cylindrical Brownian motion in a Hilbert space H, dXn(t) = (AnX(t) + Fn(t,Xn(t))) dt+Gn(t,Xn(t)) dWH(t), Xn(0) = ξn, assuming that the operators An converge to A and the locally Lipschitz functions Fn and Gn converge to the locally Lipschitz functions F and G in an appropriate ...

Many classical problems in geometry and analysis involve the gluing together of local information to produce a coherent global picture. Inevitably, the difficulty of such a procedure lies at the local boundary, where overlapping views of the same locality must somehow be merged. It is therefore desirable that the boundaries be “smooth,” allowing a graceful transition from one viewpoint to the n...

In this paper we prove a new version of the Schoenflies extension theorem for collared domains Ω and Ω in Rn: for p ∈ [1, n), locally bi-Lipschitz homeomorphisms from Ω to Ω with locally p-integrable, second-order weak derivatives admit homeomorphic extensions of the same regularity. Moreover, the theorem is essentially sharp. The existence of exotic 7-spheres shows that such extension theorems...

A metric gauge on a set is a maximal collection of metrics on the set such that the identity map between any two metrics from the collection is locally bi-Lipschitz. We characterize metric gauges that are locally branched Euclidean and discuss an obstruction to removing the branching. Our characterization is a mixture of analysis, geometry, and topology with an argument of Yu. Reshetnyak to pro...

We characterize compact metric spaces whose locally flat Lipschitz functions separate points uniformly as exactly those that are purely 1-unrectifiable, resolving a problem of Weaver. subsequently use this geometric characterization to answer several questions in analysis. Notably, it follows the Lipschitz-free space $\mathcal{F}(M)$ over $M$ is dual if and only 1-unrectifiable. Furthermore, we...

We establish interior gradient bounds for functions u ∈ W 1 1,loc (Ω) which locally minimize the variational integral J[u, Ω] = ∫ Ω h (|∇u|) dx under the side condition u ≥ Ψ a.e. on Ω with obstacle Ψ being locally Lipschitz. Here h denotes a rather general N-function allowing (p, q)-ellipticity with arbitrary exponents 1 < p ≤ q < ∞. Our arguments are based on ideas developed in [BFM] combined...