We give a new self-contained proof of Bronshtein’s theorem, that any continuous root of a Cn−1,1-family of monic hyperbolic polynomials of degree n is locally Lipschitz, and obtain explicit bounds for the Lipschitz constant of the root in terms of the coefficients. As a by-product we reprove the recent result of Colombini, Orrú, and Pernazza, that a C-curve of hyperbolic polynomials of degree n...

Let Ω be a bounded domain in R, with surface measure dσ and outward unit normal n. Throughout the paper we shall assume that Ω is Lipschitz (i.e., ∂Ω can be locally described by means of graphs of Lipschitz continuous functions in appropriate systems of coordinates) and that ∂Ω is connected. Also, we set Ω+ := Ω and Ω− := R \ Ω̄. The classical Faraday-Amperé-Maxwell equations in Ω+ or Ω− read { ...

In this paper, we introduce a new resolvent operator and we call it relaxed resolvent operator. We prove that relaxed resolvent operator is single-valued and Lipschitz continuous and finally we approximate the solution of a variational inclusion problem in Hilbert spaces by defining an iterative algorithm based on relaxed resolvent operator. A few concepts like Lipschitz continuity and strong m...

We consider the area functional for t-graphs in the sub-Riemannian Heisenberg group and study minimizers of the associated Dirichlet problem. We prove that, under a bounded slope condition on the boundary datum, there exists a unique minimizer and that this minimizer is Lipschitz continuous. We also provide an example showing that, in the first Heisenberg group, Lipschitz regularity is sharp ev...