Related papers: Local regularity in non-linear generalized functio…
We study local regularity properties for solutions of linear, non-uniformly elliptic equations. Assuming certain integrability conditions on the coefficient field, we prove local boundedness and Harnack inequality. The assumed integrability…
We synthesize and unify notions of regularity, both of individual sets and of collections of sets, as they appear in the convergence theory of projection methods for consistent feasibility problems. Several new characterizations of…
Using similarity transformations we construct explicit nontrivial solutions of nonlinear Schr\"odinger equations with potentials and nonlinearities depending on time and on the spatial coordinates. We present the general theory and use it…
In this paper, locally Lipschitz functions acting between infinite dimensional normed spaces are considered. When the range is a dual space and satisfies the Radon--Nikod\'ym property, Clarke's generalized Jacobian will be extended to this…
This paper deals with a new kind of generalized functions, called "ultrafunctions" which have been introduced recently and developed in some previous works. Their peculiarity is that they are based on a Non-Archimedean field namely on a…
We study $H^1$ versus $C^1$ local minimizers for functionals defined on spaces of symmetric functions, namely functions that are invariant by the action of some subgroups of $\mathcal{O}(N)$. These functionals, in many cases, are associated…
Two classical results characterizing regularity of a convergence space in terms of continuous extensions of maps on one hand, and in terms of continuity of limits for the continuous convergence on the other, are extended to…
We characterize microlocal regularity of Colombeau generalized functions by an appropriate extension of the classical notion of micro-ellipticity to pseudodifferential operators with slow scale generalized symbols. Thus we obtain an…
We present numerical techniques based on generalized functions adapted to nonlinear calculations. They concern main numerical engineering problems ruled by-or issued from-nonlinear equations of continuum mechanics. The aim of this text is…
We solve here the so called division problem for wave equations with generic quadratic non-linearities in high dimensions. Specifically, we show that semilinear wave equations which can be written as systems involving quadratic derivative…
This paper investigates the localization properties of solutions to the semi-classical Schr\"odinger equation on closed Riemann surfaces. Unlike classical studies that assume a smooth potential, our work addresses the challenges arising…
We introduce and study some generalizations of regular spaces, which were motivated by studying continuity properties of functions between (regular) topological spaces. In particular, we prove that a first-countable Hausdorff topological…
Solutions to nonlinear integro-differential systems are regular outside a negligible closed subset whose Hausdorff dimension can be explicitly bounded from above. This subset can be characterized using quantitative, universal energy…
The main aim of the present work is to arrive at a mathematical theory close to the historically original conception of generalized functions, i.e. set theoretical functions defined on, and with values in, a suitable ring of scalars and…
In this paper, we study the connections between the normality, regularity, full regularity, and chain-complete property in partially ordered Banach spaces. Then, by applying these properties, we prove some fixed point theorems on partially…
We examine three primal space local Hoelder type regularity properties of finite collections of sets, namely, [q]-semiregularity, [q]-subregularity, and uniform [q]-regularity as well as their quantitative characterizations. Equivalent…
We extend the theory of regularity structures [Hai14] to allow processes belonging to locally $m$-convex topological algebras. This extension includes processes in the locally $C^{*}$-algebras of [CHP25] used to localise singular stochastic…
In this paper the necessary and sufficient conditions were given for Orlicz-Lorentz function space endowed with the Orlicz norm having non-squareness and local uniform non-squareness.
In these lecture notes we present an introduction to non-standard analysis especially written for the community of mathematicians, physicists and engineers who do research on J. F. Colombeau' theory of new generalized functions and its…
A previously established correspondence between definite-parity real functions and inner analytic functions is generalized to real functions without definite parity properties. The set of inner analytic functions that corresponds to the set…