Related papers: A two spaces extension of Cauchy-Lipschitz Theorem
Convexity is an important notion in non linear optimization theory as well as in infinite dimensional functional analysis. As will be seen below, very simple and powerful tools will be derived from elementary duality arguments (which are…
The question is addressed of when a Sobolev type space, built upon a general rearrangement-invariant norm, on an $n$-dimensional domain, is a Banach algebra under pointwise multiplication of functions. A sharp balance condition among the…
The dynamics of defect excitations in crystalline solids is necessary to understand the macroscopic low-energy properties of elastic media. We use fracton-elasticity duality to systematically study the defect dynamics and interactions in…
We present an elementary Functional Analytic proof of the roughness of Exponential Dichotomy of Ordinary Differential Equations (with exponential growth) on an arbitrary Banach Space.
Theorem: Let X and Y be two Banach spaces, Phi: X to Y a continuous, linear, surjective operator, and Psi: X to Y a completely continuous operator with bounded range. Then, one has dim{x in X : Phi(x)=Psi(x)} >= dim(Phi^{-1}(0)). Here dim…
The study of the 2D Euler equation with non Lipschitzian velocity was initiated by Yudovich in [19] where a result of global well-posedness for essentially bounded vorticity is proved. A lot of works have been since dedicated to the…
In this paper, several differentiability criteria for real functions of multiple variables in n-dimensional Euclidean space are considered. Simple and easy-to-use Cauchy-like criterion is formulated and proven. Relaxed sufficient conditions…
We study translation-invariant integrodifferential operators that generate L\'{e}vy processes. First, we investigate different notions of what a solution to a nonlocal Dirichlet problem is and we provide the classical representation formula…
We consider a Cauchy Dirichlet problem for a quasilinear second order parabolic equation with lower order term driven by a singular coefficient. We establish an existence result to such a problem and we describe the time behavior of the…
In the paper is considered two problems on extension of operators whose range space for the first problem (or domain space for the second one) belongs to the fixed class of finite equivalence, which is generated by a given Banach space $X$.…
Riccati's differential equation is formulated as abstract equation in finite or infinite dimensional Banach spaces. Since the Riccati's differential equation with the Cole-Hopf transform shows a relation between the first order evolution…
We introduce and study the notion of a locally proper map between topological spaces. We show that fundamental constructions of sheaf theory, more precisely proper base change, projection formula, and Verdier duality, can be extended from…
We establish pointwise estimates expressed in terms of a nonlinear potential of a generalized Wolff type for $A$-superharmonic functions with nonlinear operator $A:\Omega\times\mathbb{R}^n\to\mathbb{R}^n$ having measurable dependence on the…
We discuss an alternative approach to Fr\'echet derivatives on Banach spaces inspired by a characterisation of derivatives due to Carath\'eodory. The approach allows us to reduce many questions of differentiability to a question of…
We study the Cauchy problem for a class of third order linear anisotropic evolution equations with complex valued lower order terms depending both on time and space variables. Under suitable decay assumptions for $|x| \to \infty$ on these…
This paper is concerned with the study of a class of nonlinear nonlocal functional evolution problems defined in an abstract Banach algebra. We introduce an abstract functional setting that encompasses a wide range of structured population…
In this paper, we study a second-order, nonlinear evolution equation with damping arising in elastodynamics. The nonlinear term is monotone and possesses a convex potential but exhibits anisotropic and nonpolynomial growth. The appropriate…
We address the Cauchy problem for a nonlinear Schr{\"o}dinger equation where the dispersion is modulated by a deterministic noise. The noise is understood as the derivative of a self-affine function of order H $\in$ (0, 1). Due to the…
We consider parabolic problems with non-Lipschitz nonlinearity in the different scales of Banach spaces and prove local-in-time existence theorem. New class of parabolic equations that have analytic solutions is obtained.
This thesis explores two important areas in the mathematical analysis of nonlinear partial differential equations: Generalized gradient flows and vector valued Orlicz spaces. The first part deals with the existence of strong solutions for…