Related papers: Weakly differentiable functions on varifolds
Local solutions for variational and quasi-variational inequalities are usually the best type of solutions that could practically be obtained when in case of lack of convexity or else when available numerical techniques are too limited for…
We introduce the concept of functions of locally bounded variation on abstract Wiener spaces and study their properties. Some nontrivial examples and applications to stochastic analysis are also discussed.
This work provides a systematic study of the variational properties of decomposable functions which are compositions of an outer support function and an inner smooth mapping under certain constraint qualifications. A particular focus is put…
We define the notion of colocally weakly differentiable maps from a manifold $M$ to a manifold $N$. If $p \ge 1$ and $M$ and $N$ are endowed with a Riemannian metric, this allows us to define intrinsically the homogeneous Sobolev space…
The goal of the paper is four-fold. In the setting of non-smooth spaces with Ricci curvature lower bounds (more precisely RCD(K,N) metric measure spaces): - we develop an intrinsic theory of Laplacian bounds in viscosity sense and in a…
We study minimization of a structured objective function, being the sum of a smooth function and a composition of a weakly convex function with a linear operator. Applications include image reconstruction problems with regularizers that…
We represent minimal upper gradients of Newtonian functions, in the range $1\le p<\infty$, by maximal directional derivatives along "generic" curves passing through a given point, using plan-modulus duality and disintegration techniques. As…
The diffuse domain method for partial differential equations on complicated geometries recently received strong attention in particular from practitioners, but many fundamental issues in the analysis are still widely open. In this paper we…
In this paper we study the shape differentiability properties of a class of boundary integral operators and of potentials with weakly singular pseudo-homogeneous kernels acting between classical Sobolev spaces, with respect to smooth…
We establish a novel convergent iteration framework for a weak approximation of general switching diffusion. The key theoretical basis of the proposed approach is a restriction of the maximum number of switching so as to untangle and…
We obtain a sharp lower bound on the isoperimetric deficit of a general polygon in terms of the variance of its side lengths, the variance of its radii, and its deviation from being convex. Our technique involves a functional minimization…
We study fine potential theory and in particular partitions of unity in quasiopen sets in the case $p=1$. Using these, we develop an analog of the discrete convolution technique in quasiopen (instead of open) sets. We apply this technique…
We adapt previous research on category theory and topological unsupervised learning to develop a functorial perspective on manifold learning, also known as nonlinear dimensionality reduction. We first characterize manifold learning…
Evolution PDEs for dispersive waves are considered in both linear and nonlinear integrable cases, and initial-boundary value problems associated with them are formulated in spectral space. A method of solution is presented, which is based…
The purpose of this short article is to prove some potential estimates that naturally arise in the study of subelliptic Sobolev inequalites for functions. This will allow us to prove a local subelliptic Sobolev inequality with the optimal…
In this document I develop a weight function theory of positive order basis function interpolants and smoothers. **In Chapter 1 the basis functions and data spaces are defined directly using weight functions. The data spaces are used to…
We study boundary values of harmonic functions in spaces of quasianalytic functionals and spaces of ultradistributions of non-quasianalytic type. As an application, we provide a new approach to H\"ormander's support theorem for…
We introduce a theoretical framework for differentiable surface evolution that allows discrete topology changes through the use of topological derivatives for variational optimization of image functionals. While prior methods for inverse…
The aim of the present paper is to define a notion of weakly differentiable cochain in the generality of metric measure spaces and to study basic properties of such cochains. Our cochains are (sub-)linear functionals on a subspace of…
We prove that any weakly differentiable function with square integrable gradient can be extended to a capacitary boundary of any simply connected plane domain $\Omega\ne\mathbb R^2$ except a set of a conformal capacity zero. For locally…