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Daniel Azagra

Publications and source records attributed to Daniel Azagra.

34 records · Page 2Linked to original sources
arXiv2016

Whitney Extension Theorems for convex functions of the classes $C^1$ and $C^{1,ω}$

Let $C$ be a subset of $\mathbb{R}^n$ (not necessarily convex), $f:C\to\mathbb{R}$ be a function, and $G:C\to\mathbb{R}^n$ be a uniformly continuous function, with modulus of continuity $ω$. We provide a necessary and sufficient condition on $f$, $G$ for the existence of a convex function $F\in C^{1, ω}(\mathbb{R}^n)$ such that $F=f$ on $C$ and $\nabla F=G$ on $C$, with a good control of the modulus of continuity of $\nabla F$ in terms of that of $G$. On the other hand, assuming that $C$ is compact, we also solve a similar problem for the class of $C^1$ convex functions on $\mathbb{R}^n$, with a good control of the Lipschitz constants of the extensions (namely, $\textrm{Lip}(F)\lesssim \|G\|_{\infty}$). Finally, we give a geometrical application concerning interpolation of compact subsets $K$ of $\mathbb{R}^n$ by boundaries of $C^1$ or $C^{1,1}$ convex bodies with prescribed outer normals on $K$.

math.CA↗
arXiv2016

An Extension Theorem for convex functions of class $C^{1,1}$ on Hilbert spaces

Let $\mathbb{H}$ be a Hilbert space, $E \subset \mathbb{H}$ be an arbitrary subset and $f: E \rightarrow \mathbb{R}, \: G: E \rightarrow \mathbb{H}$ be two functions. We give a necessary and sufficient condition on the pair $(f,G)$ for the existence of a \textit{convex} function $F\in C^{1,1}(\mathbb{H})$ such that $F=f$ and $\nabla F =G$ on $E$. We also show that, if this condition is met, $F$ can be taken so that $\textrm{Lip}(\nabla F) = \textrm{Lip}(G)$. We give a geometrical application of this result, concerning interpolation of sets by boundaries of $C^{1,1}$ convex bodies in $\mathbb{H}$. Finally, we give a counterexample to a related question concerning smooth convex extensions of smooth convex functions with derivatives which are not uniformly continuous.

math.FA↗
arXiv2014

Global approximation of convex functions by differentiable convex functions on Banach spaces

We show that if $X$ is a Banach space whose dual $X^{*}$ has an equivalent locally uniformly rotund (LUR) norm, then for every open convex $U\subseteq X$, for every $\varepsilon >0$, and for every continuous and convex function $f:U \rightarrow \mathbb{R}$ (not necessarily bounded on bounded sets) there exists a convex function $g:X \rightarrow \mathbb{R}$ of class $C^1(U)$ such that $f-\varepsilon\leq g\leq f$ on $U.$ We also show how the problem of global approximation of continuous (not necessarily bounded on bounded sets) and convex functions by $C^k$ smooth convex functions can be reduced to the problem of global approximation of Lipschitz convex functions by $C^k$ smooth convex functions.

math.FA↗
arXiv2014

Global and fine approximation of convex functions

Let $U\subseteq\mathbb{R}^d$ be open and convex. We prove that every (not necessarily Lipschitz or strongly) convex function $f:U\to\mathbb{R}$ can be approximated by real analytic convex functions, uniformly on all of $U$. We also show that $C^0$-fine approximation of convex functions by smooth (or real analytic) convex functions on $\mathbb{R}^d$ is possible in general if and only if $d=1$. Nevertheless, for $d\geq 2$ we give a characterization of the class of convex functions on $\mathbb{R}^d$ which can be approximated by real analytic (or just smoother) convex functions in the $C^0$-fine topology. It turns out that the possibility of performing this kind of approximation is not determined by the degree of local convexity or smoothness of the given function, but by its global geometrical behaviour. We also show that every $C^{1}$ convex and proper function on $U$ can be approximated by $C^{\infty}$ convex functions in the $C^{1}$-fine topology, and we provide some applications of these results, concerning prescription of (sub-)differential boundary data to convex real analytic functions, and smooth surgery of convex bodies.

math.DG↗
arXiv2014

Regularization by sup-inf convolutions on Riemannian manifolds: an extension of Lasry-Lions theorem to manifolds of bounded curvature

We show how Lasry-Lions's result on regularization of functions defined on $\mathbb{R}^n$ or on Hilbert spaces by sup-inf convolutions with squares of distances can be extended to (finite or infinite dimensional) Riemannian manifolds $M$ of bounded sectional curvature. More specifically, among other things we show that if the sectional curvature $K$ of $M$ satisfies $-K_0\leq K\leq K_0$ on $M$ for some $K_0>0$, and if the injectivity and convexity radii of $M$ are strictly positive, then every bounded, uniformly continuous function $f:M\to\mathbb{R}$ can be uniformly approximated by globally $C^{1,1}$ functions defined by $$ (f_λ)^μ=\sup_{z\in M}\inf_{y\in M}\{f(y)+\frac{1}{2λ} d(z,y)^{2}-\frac{1}{2μ}d(x,z)^2\} $$ as $λ, μ\to 0^{+}$, with $0<μ<λ/2$. Our definition of (global) $C^{1,1}$ smoothness is intrinsic and natural, and it reduces to the usual one in flat spaces, but we warn the reader that, in the noncompact case, this definition differs from other notions of (rather local) $C^{1,1}$ smoothness that have been recently used, for instance, by A. Fathi and P. Bernard (based on charts). The importance of this regularization method lies (rather than on the degree of smoothness obtained) on the fact that the correspondence $f\mapsto (f_λ)^μ$ is explicit and preserves many significant geometrical properties that the given functions $f$ may have, such as invariance by a set of isometries, infima, sets of minimizers, ordering, local or global Lipschitzness, and (only when one additionally assumes that $K\leq 0$) local or global convexity. We also give two examples showing that this result completely fails, even for (nonflat) Cartan-Hadamard manifolds, whenever $f$ or $K$ are not bounded.

math.DG↗
arXiv2012

On $C^{0}$-fine approximation of convex functions by real analytic convex functions

We show that $C^0$-fine approximation of convex functions by smooth (or real analytic) convex functions on $\R^d$ is possible in general if and only if $d=1$. Nevertheless, for $d\geq 2$ we give a characterization of the class of convex functions on $\R^d$ which can be approximated by real analytic (or just smoother) convex functions in the $C^0$-fine topology. It turns out that the possibility of performing this kind of approximation is not determined by the degree of local convexity or smoothness of the given function, but by its global geometrical behavior. We give some applications concerning prescription of (sub-)differential boundary data to convex real analytic functions, and smooth surgery of convex bodies.

math.CA↗
arXiv2010

Concentration of symmetric eigenfunctions

In this article we examine the concentration and oscillation effects developed by high-frequency eigenfunctions of the Laplace operator in a compact Riemannian manifold. More precisely, we are interested in the structure of the possible invariant semiclassical measures obtained as limits of Wigner measures corresponding to eigenfunctions. These measures describe simultaneously the concentration and oscillation effects developed by a sequence of eigenfunctions. We present some results showing how to obtain invariant semiclassical measures from eigenfunctions with prescribed symmetries. As an application of these results, we give a simple proof of the fact that in a manifold of constant positive sectional curvature, every measure which is invariant by the geodesic flow is an invariant semiclassical measure.

math.AP↗
arXiv2008

Viscosity solutions to second order partial differential equations on Riemannian manifolds

We prove comparison, uniqueness and existence results for viscosity solutions to a wide class of fully nonlinear second order partial differential equations $F(x, u, du, d^{2}u)=0$ defined on a finite-dimensional Riemannian manifold $M$. Finest results (with hypothesis that require the function $F$ to be degenerate elliptic, that is nonincreasing in the second order derivative variable, and uniformly continuous with respect to the variable $x$) are obtained under the assumption that $M$ has nonnegative sectional curvature, while, if one additionally requires $F$ to depend on $d^{2}u$ in a uniformly continuous manner, then comparison results are established with no restrictive assumptions on curvature.

math.AP↗
arXiv2004

Proximal calculus on Riemannian manifolds, with applications to fixed point theory

We introduce a proximal subdifferential and develop a calculus for nonsmooth functions defined on any Riemannian manifold $M$. We give several applications of this theory, concerning: 1) differentiability and geometrical properties of the distance function to a closed subset $C$ of $M$; 2) solvability and implicit function theorems for nonsmooth functions on $M$; 3) conditions on the existence of a circumcenter for three different points of $M$; and especially 4) fixed point theorems for expansive and nonexpansive mappings and certain perturbations of such mappings defined on $M$.

math.DG↗
arXiv2003

Nonsmooth analysis and Hamilton-Jacobi equations on Riemannian manifolds

We establish some perturbed minimization principles, and we develop a theory of subdifferential calculus, for functions defined on Riemannian manifolds. Then we apply these results to show existence and uniqueness of viscosity solutions to Hamilton-Jacobi equations defined on Riemannian manifolds.

math.DG↗
arXiv2002

Uniform approximation of continuous mappings by smooth mappings with no critical points on Hilbert manifolds

We prove that every continuous mapping from a separable infinite-dimensional Hilbert space $X$ into $\mathbb{R}^{m}$ can be uniformly approximated by $C^\infty$ smooth mappings {\em with no critical points}. This kind of result can be regarded as a sort of very strong approximate version of the Morse-Sard theorem. Some consequences of the main theorem are as follows. Every two disjoint closed subsets of $X$ can be separated by a one-codimensional smooth manifold which is a level set of a smooth function with no critical points; this fact may be viewed as a nonlinear analogue of the geometrical version of the Hahn-Banach theorem. In particular, every closed set in $X$ can be uniformly approximated by open sets whose boundaries are $C^\infty$ smooth one-codimensional submanifolds of $X$. Finally, since every Hilbert manifold is diffeomorphic to an open subset of the Hilbert space, all of these results still hold if one replaces the Hilbert space $X$ with any smooth manifold $M$ modelled on $X$.

math.DG↗
arXiv2001

Uniform approximation of continuous functions by smooth functions with no critical points on Hilbert manifolds

We prove that every continuous function on a separable infinite-dimensional Hilbert space X can be uniformly approximated by smooth functions with no critical points. This kind of result can be regarded as a sort of very strong approximate version of the Morse-Sard theorem. Some consequences of the main theorem are as follows. Every two disjoint closed subsets of X can be separated by a one-codimensional smooth manifold which is a level set of a smooth function with no critical points; this fact may be viewed as a nonlinear analogue of the geometrical version of the Hahn-Banach theorem. In particular, every closed set in X can be uniformly approximated by open sets whose boundaries are smooth one-codimensional submanifolds of X. Finally, since every Hilbert manifold is diffeomorphic to an open subset of the Hilbert space, all of these results still hold if one replaces the Hilbert space X with any smooth manifold M modelled on X.

math.DG↗
arXiv2000

Rolle's theorem is either false or trivial in infinite-dimensional Banach spaces

We prove the following new characterization of $C^p$ (Lipschitz) smoothness in Banach spaces. An infinite-dimensional Banach space $X$ has a $C^p$ smooth (Lipschitz) bump function if and only if it has another $C^p$ smooth (Lipschitz) bump function $f$ such that $f'(x)\neq 0$ for every point $x$ in the interior of the support of $f$ (that is, $f$ does not satisfy Rolle's theorem). Moreover, the support of this bump can be assumed to be a smooth starlike body. As a by-product of the proof of this result we also obtain other useful characterizations of $C^p$ smoothness related to the existence of a certain kind of deleting diffeomorphisms, as well as to the failure of Brouwer's fixed point theorem even for smooth self-mappings of starlike bodies in all infinite-dimensional spaces. Finally, we study the structure of the set of gradients of bump functions in the Hilbert space $\ell_2$, and as a consequence of the failure of Rolle's theorem in infinite dimensions we get the following result. The usual norm of the Hilbert space $\ell_2$ can be uniformly approximated by $C^1$ smooth Lipschiz functions $ψ$ so that the cones generated by the sets of derivatives $ψ'(\ell_{2})$ have empty interior. This implies that there are $C^1$ smooth Lipschitz bumps in $\ell_{2}$ so that the cones generated by their sets of gradients have empty interior.

math.FA↗