Related papers: Characterization of Filippov representable maps an…
In this paper, locally Lipschitz, regular functions are utilized to identify and remove infeasible directions from set-valued maps that define differential inclusions. The resulting reduced set-valued map is point-wise smaller (in the sense…
A function $\mathfrak{F}$ with simple and nice algebraic properties is defined on a subset of the space of complex sequences. Some special functions are expressible in terms of $\mathfrak{F}$, first of all the Bessel functions of first…
Although the property of strong metric subregularity of set-valued mappings has been present in the literature under various names and with various definitions for more than two decades, it has attracted much less attention than its older…
Self-maps everywhere defined on the projective space $\P^N$ over a number field or a function field are the basic objects of study in the arithmetic of dynamical systems. One reason is a theorem of Fakkruddin \cite{Fakhruddin} (with…
We prove some regularity results for a priori bounded local minimizers of non-autonomous integral functionals of the form $$\mathcal{F}(v,\Omega)=\int_\Omega F(x,Dv)dx,$$ under the constraint $v \ge \psi$ a.e. in $\Omega$, where $\psi$ is a…
Using a geometric argument, we show that under a reasonable continuity condition, the Clarke subdifferential of a semi-algebraic (or more generally stratifiable) directionally Lipschitzian function admits a simple form: the normal cone to…
An equation $f(x)=a$, where $f$ is a complex meromorphic function and $a\in\mathbb{C}$ is a parameter, is solvable in elementary functions if the inverse map $x=f^{-1}(a)$ can be expressed as a finite composition of arithmetic operations…
We prove the local Lipschitz regularity of the local minimizers of scalar integral functionals of the form \begin{equation*} \mathcal{F}(v;\Omega)= \int_{\Omega} f (x, Dv) dx \end{equation*} under $(p,q)$-growth conditions. The main novelty…
Given a map with underlying graph $\mathcal{G}$, if the set of prime divisors of $|V(\mathcal{G}|$ is denoted by $\pi$, then we call the map a {\it $\pi$-map}. An orientably-regular (resp. A regular ) $\pi$-map is called {\it solvable} if…
The central theme in this paper is the Hopf-Laplace equation, which represents stationary solutions with respect to the inner variation of the Dirichlet integral. Among such solutions are harmonic maps. Nevertheless, minimization of the…
Functional equations satisfied by additive functions have a special interest not only in the theory of functional equations, but also in the theory of (commutative) algebra because the fundamental notions such as derivations and…
To minimize or upper-bound the value of a function "robustly", we might instead minimize or upper-bound the "epsilon-robust regularization", defined as the map from a point to the maximum value of the function within an epsilon-radius. This…
We prove local Lipschitz regularity for local minimiser of \[ W^{1,1}(\Omega)\ni v\mapsto \int_\Omega F(Dv)\, dx \] where $\Omega\subseteq {\mathbb R}^N$, $N\ge 2$ and $F:{\mathbb R}^N\to {\mathbb R}$ is a quasiuniformly convex integrand in…
We introduce real vector spaces composed of set-valued maps on an open set. They are also complete metric spaces, lattices, commutative rings. The set of differentiable functions is a dense subset of these spaces and the classical gradient…
We find sufficient conditions on a set $\mathscr{M}\subset\mathbf{R}^n\times\mathscr{L}(\mathbf{R}^n,\mathbf{R}^m)$ ensuring that the set of functions such that $(F(x),DF(x))\in\mathscr{M}$ is rectifiable. We also prove a more general…
In this paper we characterize sparse solutions for variational problems of the form $\min_{u\in X} \phi(u) + F(\mathcal{A} u)$, where $X$ is a locally convex space, $\mathcal{A}$ is a linear continuous operator that maps into a finite…
In the paper, we use the idea of normal family to find out the possible solution of the following special case of algebraic differential equation \[P_k\big(z,f,f^{(1)},\ldots, f^{(k)}\big)=f^{(1)}(f-\mathscr{L}_k(f))-\varphi (f-a)(f-b)=0,\]…
Recent years have witnessed the introduction and development of extremely fast rational function algorithms. Many ideas in this realm arose from polynomial-based linear-algebraic algorithms. However, polynomial approximation is occasionally…
We establish the higher differentiability of solutions to a class of obstacle problems for integral functionals where the convex integrand f satisfies p-growth conditions with respect to the gradient variable. We derive that the higher…
The main contribution of this paper is that every convex function with non-empty relative algebraic interior of its domain is Lipschitz and subdifferentiable in some algebraic sense without any additional topological constraints. The…