If u 7→ A(u) is a C1,α-mapping having as values unbounded selfadjoint operators with compact resolvents and common domain of definition, parametrized by u in an (even infinite dimensional) space then any continuous arrangement of the eigenvalues u 7→ λi(u) is C0,1 in u. If u 7→ A(u) is C0,1, then the eigenvalues may be chosen C0,1/N (even C0,1 if N = 2), locally in u, where N is locally the max...

In a recent paper by H. X. Cao, J. H. Zhang and Z. B. Xu an α-Lipschitz operator from a compact metric space into a Banach space A is defined and characterized in a natural way in the sence that F : K → A is a α-Lipschitz operator if and only if for each σ ∈ X∗ the mapping σ ◦ F is a α-Lipschitz function. The Lipschitz operators algebras Lα(K,A) and lα(K,A) are developed here further, and we st...

In a recent paper by H.X. Cao, J.H. Zhang and Z.B. Xu a α-Lipschitz operator from a compact metric space into a Banach space A is defined and characterized in a natural way in the sence that F : K → A is a α-Lipschitz operator if and only if for each σ ∈ X∗ the mapping σoF is a α-Lipschitz function. The Lipschitz operators algebras L(K,A) and l(K,A) are developed here further, and we study thei...

We show that the Zariski canonical stratification of complex hypersurfaces is locally bi-Lipschitz trivial along strata codimension two. More precisely, we study equisingular families surface, not necessarily isolated, singularities in ${\\mathbb{C}^3}$. a natural such family, given by singular set and generic family polar curves, provides Lipschitz sense Mostowski. In particular are trivial, w...

3. The Implicit Function Theorem and its special case, the Inverse Function Theorem. Relation to the Contraction Mapping Fixed Point Theorem. Computation of several iterates of the approximating sequence to the implicit function. Continuity and differentiability of the implicit function. Application to differentiable variation of the fixed point in the Contraction Mapping Fixed Point Theorem. 4...

Traditional finite-time convergence theory for numerical methods applied to stochastic differential equations (SDEs) requires a global Lipschitz assumption on the drift and diffusion coefficients. In practice, many important SDE models satisfy only a local Lipschitz property and, since Brownian paths can make arbitrarily large excursions, the global Lipschitz-based theory is not directly releva...

We study p-harmonic functions on metric measure spaces, which are formulated as minimizers to certain energy functionals. For spaces supporting a p-Poincaré inequality, we show that such functions satisfy an infinitesmal Lipschitz condition almost everywhere. This result is essentially sharp, since there are examples of metric spaces and p-harmonic functions that fail to be locally Lipschitz co...

Following classical work by Freidlin [Trans. Amer. Math. Soc. (1988) 305 665–657] and subsequent works by Sowers [Ann. Probab. (1992) 20 504–537] and Peszat [Probab. Theory Related Fields (1994) 98 113–136], we prove large deviation estimates for the small noise limit of systems of stochastic reaction–diffusion equations with globally Lipschitz but unbounded diffusion coefficients, however, ass...

We consider a nonlinear (possibly) degenerate elliptic operator Lv = − div a(∇v) + b(x, v) where the functions a and b are, unnecessarly strictly, monotonic. For a given boundary datum φ we prove the existence of the maximum and the minimum of the solutions and formulate a Haar-Rado type result, namely a continuity property for these solutions that may follow from the continuity of φ. In the ho...

We extend the proof of Reifenberg’s Topological Disk Theorem to allow the case of sets with holes, and give sufficient conditions on a set E for the existence of a bi-Lipschitz parameterization of E by a d-dimensional plane or smooth manifold. Such a condition is expressed in terms of square summability for the P. Jones numbers β1(x, r). In particular, it applies in the locally Ahlfors-regular ...