Related papers: Least gradient functions in metric random walk spa…
We study the two dimensional least gradient problem in a convex, but not necessary strictly convex region. We look for solutions in the space of $BV$ functions satisfying the boundary data $f$ in trace sense. We assume that $f$ is in $BV$…
We study the planar least gradient problem with respect to an anisotropic norm $\phi$ for continuous boundary data. We prove existence of minimizers for strictly convex domains $\Omega$. Furthermore, we inspect the issue of uniqueness and…
In this paper, we study some functional inequalities (such as Poincar\'e inequalities, logarithmic Sobolev inequalities, generalized Cheeger isoperimetric inequalities, transportation-information inequalities and transportation-entropy…
This paper is a follow up to an article by two of the authors dedicated to the study of Poincar\'e and logarithmic Sobolev inequalities for measures of the form $d\mu = e^{-U} d\nu$ where $e^{-U}$ is seen as a perturbation of $d\nu$.…
Given a nondegenerate harmonic structure, we prove a Poincar\'e-type inequality for functions in the domain of the Dirichlet form on nested fractals. We then study the Hajlasz-Sobolev spaces on nested fractals. In particular, we describe…
We prove two compactness results for function spaces with finite Dirichlet energy of half-space nonlocal gradients. In each of these results, we provide sufficient conditions on a sequence of kernel functions that guarantee the asymptotic…
Nonlinear Poincar\'e inequalities are indispensable tools in the study of dimension reduction and low-distortion embeddings of graphs into metric spaces, and have found remarkable algorithmic applications. A basic open problem, posed by Jon…
In the framework of the first-order differential structure introduced by Gigli, we obtain a Gauss-Green formula on regular bounded open sets of metric measure spaces, valid for BV functions and vector fields with integrable divergence.…
We compare several notion of weak (modulus of) gradient in metric measure spaces. Using tools from optimal transportation theory we prove density in energy of Lipschitz maps independenly of doubling and Poincar\'e assumptions on the metric…
We study the convergence of the gradient descent method for solving ill-posed problems where the solution is characterized as a global minimum of a differentiable functional in a Hilbert space. The classical least-squares functional for…
We show that quasi-minimizers of non-homogeneous energy functionals on metric measure spaces are locally H\"older continuous and satisfy the Harnack inequality. We assume that the spaces are doubling and support a Poincar\'e inequality. The…
This paper is concerned with the approximation of continuously differentiable functions with high-dimensional input by a composition of two functions: a feature map that extracts few features from the input space, and a profile function…
We prove local $L^p$-Poincar\'e inequalities, $ p\in[1,\infty]$, on quasiconvex sets in infinite graphs endowed with a family of locally doubling measures, and global $L^p$-Poincar\'e inequalities on connected sets for flow measures on…
We study functions of least gradient as well as related superminimizers and solutions of obstacle problems in metric spaces that are equipped with a doubling measure and support a Poincar\'e inequality. We show a standard weak Harnack…
In this paper, we study symmetry and existence of solutions of minimal gradient graph equations on punctured space $\mathbb R^n\setminus\{0\}$, which include the Monge-Amp\`ere equation, inverse harmonic Hessian equation and the special…
In this paper we study the $(BV,L^p)$-decomposition, $p=1,2$, of functions in metric random walk spaces, a general workspace that includes weighted graphs and nonlocal models used in image processing. We obtain the Euler-Lagrange equations…
Finding a local minimum or maximum of a function is often achieved through the gradient-descent optimization method. For a function in dimension d, the gradient requires to compute at each step d partial derivatives. This method is for…
The recent literature has intensively studied two classes of nonlocal variational problems, namely the ones related to the minimisation of energy functionals that act on functions in suitable Sobolev-Gagliardo spaces, and the ones related…
We show that gradient descent can converge to any local minimum of a smooth semi-algebraic function. This holds if the step sizes are nonsummable and sufficiently small. The same results hold for the subgradient method on locally Lipschitz…
We study a class of nonlocal partial differential equations presenting a tensor-mobility, in space, obtained asymptotically from nonlocal dynamics on localising infinite graphs. Our strategy relies on the variational structure of both…