scholarly journals Fréchet differentiability of mild solutions to SPDEs with respect to the initial datum

2019 ◽  
Vol 20 (3) ◽  
pp. 1093-1130
Author(s):  
Carlo Marinelli ◽  
Luca Scarpa
Keyword(s):  
Mild Solutions ◽  
Random Measure ◽  
Classical Solutions ◽  
Noise Sources ◽  
Lipschitz Continuous ◽  
Fréchet Differentiability ◽  
Cylindrical Wiener Process ◽  
Non Local ◽  
Frechet Differentiability ◽  
Derivatives Of

Abstract We establish n-th-order Fréchet differentiability with respect to the initial datum of mild solutions to a class of jump diffusions in Hilbert spaces. In particular, the coefficients are Lipschitz-continuous, but their derivatives of order higher than one can grow polynomially, and the (multiplicative) noise sources are a cylindrical Wiener process and a quasi-left-continuous integer-valued random measure. As preliminary steps, we prove well-posedness in the mild sense for this class of equations, as well as first-order Gâteaux differentiability of their solutions with respect to the initial datum, extending previous results by Marinelli, Prévôt, and Röckner in several ways. The differentiability results obtained here are a fundamental step to construct classical solutions to non-local Kolmogorov equations with sufficiently regular coefficients by probabilistic means.

scholarly journals Bifurcation at isolated singular points of the Hadamard derivative

2014 ◽  
Vol 144 (5) ◽  
pp. 1027-1065 ◽  
Author(s):  
C. A. Stuart
Keyword(s):  
Banach Spaces ◽  
Elliptic Equations ◽  
Nonlinear Elliptic Equations ◽  
Open Neighbourhood ◽  
Lipschitz Continuous ◽  
Nonlinear Elliptic ◽  
Fréchet Differentiability ◽  
Frechet Differentiability ◽  
Hadamard Derivative ◽  
Shed Light

For Banach spaces X and Y, we consider bifurcation from the line of trivial solutions for the equation F (λ, u) = 0, where F : ℝ × X → Y with F (λ, 0) = 0 for all λ ∈ ℝ. The focus is on the situation where F (λ, ·) is only Hadamard differentiable at 0 and Lipschitz continuous on some open neighbourhood of 0, without requiring any Fréchet differentiability. Applications of the results obtained here to some problems involving nonlinear elliptic equations on ℝN, where Fréchet differentiability is not available, are presented in some related papers, which shed light on the relevance of our hypotheses.

Fréchet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces (AM-179)

2012 ◽  
Author(s):  
Joram Lindenstrauss ◽  
David Preiss ◽  
Jaroslav Tier
Keyword(s):  
Banach Spaces ◽  
Set Theory ◽  
Lipschitz Functions ◽  
Descriptive Set Theory ◽  
Fréchet Differentiability ◽  
Frechet Differentiability ◽  
Porous Sets ◽  
Descriptive Set ◽  
Vector Valued ◽  
Derivatives Of

This book makes a significant inroad into the unexpectedly difficult question of existence of Fréchet derivatives of Lipschitz maps of Banach spaces into higher dimensional spaces. Because the question turns out to be closely related to porous sets in Banach spaces, it provides a bridge between descriptive set theory and the classical topic of existence of derivatives of vector-valued Lipschitz functions. The topic is relevant to classical analysis and descriptive set theory on Banach spaces. The book opens several new research directions in this area of geometric nonlinear functional analysis. The new methods developed here include a game approach to perturbational variational principles that is of independent interest. Detailed explanation of the underlying ideas and motivation behind the proofs of the new results on Fréchet differentiability of vector-valued functions should make these arguments accessible to a wider audience. The most important special case of the differentiability results, that Lipschitz mappings from a Hilbert space into the plane have points of Fréchet differentiability, is given its own chapter with a proof that is independent of much of the work done to prove more general results. The book raises several open questions concerning its two main topics.

Preliminaries to Main Results

2012 ◽  
Author(s):  
Joram Lindenstrauss ◽  
David Preiss ◽  
Tiˇser Jaroslav
Keyword(s):  
Flat Surface ◽  
Tensor Products ◽  
Linear Operators ◽  
Euclidean Spaces ◽  
Lipschitz Maps ◽  
Fréchet Differentiability ◽  
Frechet Differentiability ◽  
Fréchet Differentiable ◽  
Derivatives Of ◽  
Surface Passing

This chapter presents a number of results and notions that will be used in subsequent chapters. In particular, it considers the concept of regular differentiability and the lemma on deformation of n-dimensional surfaces. The idea is to deform a flat surface passing through a point x (along which we imagine that a certain function f is almost affine) to a surface passing through a point witnessing that f is not Fréchet differentiable at x. This is done in such a way that certain “energy” associated to surfaces increases less than the “energy” of the functionf along the surface. The chapter also discusses linear operators and tensor products, various notions and notation related to Fréchet differentiability, and deformation of surfaces controlled by ω‎ⁿ. Finally, it proves some integral estimates of derivatives of Lipschitz maps between Euclidean spaces (not necessarily of the same dimension).

Unavoidable Porous Sets and Nondifferentiable Maps

2012 ◽  
Author(s):  
Joram Lindenstrauss ◽  
David Preiss ◽  
Tiˇser Jaroslav
Keyword(s):  
Banach Space ◽  
Mean Value ◽  
Porous Set ◽  
Lipschitz Maps ◽  
Fréchet Differentiability ◽  
Frechet Derivatives ◽  
Fréchet Derivatives ◽  
Frechet Differentiability ◽  
Porous Sets ◽  
Derivatives Of

This chapter discusses Γ‎ₙ-nullness of sets porous “¹at infinity” and/or existence of many points of Fréchet differentiability of Lipschitz maps into n-dimensional spaces. The results reveal a σ‎-porous set whose complement is null on all n-dimensional surfaces and the multidimensional mean value estimates fail even for ε‎-Fréchet derivatives. Previous chapters have established conditions on a Banach space X under which porous sets in X are Γ‎ₙ-null and/or the the multidimensional mean value estimates for Fréchet derivatives of Lipschitz maps into n-dimensional spaces hold. This chapter investigates in what sense the assumptions of these main results are close to being optimal.

Fr ´Echet Differentiability Except For Γ‎-Null Sets

2012 ◽  
Author(s):  
Joram Lindenstrauss ◽  
David Preiss ◽  
Tiˇser Jaroslav
Keyword(s):  
Banach Spaces ◽  
Common Point ◽  
Lipschitz Functions ◽  
Infinite Dimensional ◽  
Lipschitz Maps ◽  
Almost Everywhere ◽  
Fréchet Differentiability ◽  
Gateaux Derivatives ◽  
Frechet Differentiability ◽  
Porous Sets

This chapter gives an account of the known genuinely infinite dimensional results proving Fréchet differentiability almost everywhere except for Γ‎-null sets. Γ‎-null sets provide the only notion of negligible sets with which a Fréchet differentiability result is known. Porous sets appear as sets at which Gâteaux derivatives can behave irregularly, and they turn out to be the only obstacle to validity of a Fréchet differentiability result Γ‎-almost everywhere. Furthermore, geometry of the space may (or may not) guarantee that porous sets are Γ‎-null. The chapter also shows that on some infinite dimensional Banach spaces countable collections of real-valued Lipschitz functions, and even of fairly general Lipschitz maps to infinite dimensional spaces, have a common point of Fréchet differentiability.

Ε‎-Fr ´Echet Differentiability

2012 ◽  
Author(s):  
Joram Lindenstrauss ◽  
David Preiss ◽  
Tiˇser Jaroslav
Keyword(s):  
Banach Space ◽  
Convex Hull ◽  
Simple Proof ◽  
Common Point ◽  
Lipschitz Functions ◽  
Closed Convex Hull ◽  
Uniformly Smooth ◽  
Fréchet Differentiability ◽  
Finite Set ◽  
Frechet Differentiability

This chapter treats results on ε‎-Fréchet differentiability of Lipschitz functions in asymptotically smooth spaces. These results are highly exceptional in the sense that they prove almost Frechet differentiability in some situations when we know that the closed convex hull of all (even almost) Fréchet derivatives may be strictly smaller than the closed convex hull of the Gâteaux derivatives. The chapter first presents a simple proof of an almost differentiability result for Lipschitz functions in asymptotically uniformly smooth spaces before discussing the notion of asymptotic uniform smoothness. It then proves that in an asymptotically smooth Banach space X, any finite set of real-valued Lipschitz functions on X has, for every ε‎ > 0, a common point of ε‎-Fréchet differentiability.

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