scholarly journals On the regularity of the total variation minimizers

2019 ◽  
Vol 23 (01) ◽  
pp. 1950082
Author(s):  
Alessio Porretta
Keyword(s):  
Image Processing ◽  
Total Variation ◽  
Source Term ◽  
Global Regularity ◽  
Regular Domain ◽  
Unique Minimizer ◽  
Convex Domains ◽  
Lipschitz Norm ◽  
Dimension Formula ◽  
Regularity Results

We prove regularity results for the unique minimizer of the total variation functional, currently used in image processing analysis since the work by Rudin, Osher and Fatemi. In particular, we show that if the source term [Formula: see text] is locally (respectively, globally) Lipschitz, then the solution has the same regularity with local (respectively, global) Lipschitz norm estimated accordingly. The result is proved in any dimension and for any (regular) domain. So far we extend a similar result proved earlier by Caselles, Chambolle and Novaga for dimension [Formula: see text] and (in case of the global regularity) for convex domains.

scholarly journals Global Schauder Estimates for the $$\mathbf {p}$$-Laplace System

2021 ◽  
Author(s):  
D. Breit ◽  
A. Cianchi ◽  
L. Diening ◽  
S. Schwarzacher
Keyword(s):  
Global Regularity ◽  
Regularity Theory ◽  
Divergence Form ◽  
Dirichlet Problems ◽  
Schauder Estimates ◽  
First Order ◽  
Right Hand ◽  
Regularity Results ◽  
The Right ◽  
Linear Setting

AbstractAn optimal first-order global regularity theory, in spaces of functions defined in terms of oscillations, is established for solutions to Dirichlet problems for the p-Laplace equation and system, with the right-hand side in divergence form. The exact mutual dependence among the regularity of the solution, of the datum on the right-hand side, and of the boundary of the domain in these spaces is exhibited. A comprehensive formulation of our results is given in terms of Campanato seminorms. New regularity results in customary function spaces, such as Hölder, $$\text {BMO}$$ BMO and $${{\,\mathrm{VMO}\,}}$$ VMO spaces, follow as a consequence. Importantly, the conclusions are new even in the linear case when $$p=2$$ p = 2 , and hence the differential operator is the plain Laplacian. Yet in this classical linear setting, our contribution completes and augments the celebrated Schauder theory in Hölder spaces. A distinctive trait of our results is their sharpness, which is demonstrated by a family of apropos examples.

A Weighted Difference of Anisotropic and Isotropic Total Variation Model for Image Processing

2015 ◽  
Vol 8 (3) ◽  
pp. 1798-1823 ◽  
Author(s):  
Yifei Lou ◽  
Tieyong Zeng ◽  
Stanley Osher ◽  
Jack Xin
Keyword(s):  
Image Processing ◽  
Total Variation ◽  
Total Variation Model ◽  
Variation Model

Regularity results for the 2D critical Oldroyd-B model in the corotational case

2019 ◽  
Vol 150 (4) ◽  
pp. 1871-1913
Author(s):  
Zhuan Ye
Keyword(s):  
Stress Tensor ◽  
Critical Case ◽  
Global Regularity ◽  
A Priori ◽  
Regularity Result ◽  
Difficult Problem ◽  
Classical Solutions ◽  
Two Dimensional ◽  
Time Singularities ◽  
Regularity Results

AbstractThis paper studies the regularity results of classical solutions to the two-dimensional critical Oldroyd-B model in the corotational case. The critical case refers to the full Laplacian dissipation in the velocity or the full Laplacian dissipation in the non-Newtonian part of the stress tensor. Whether or not their classical solutions develop finite time singularities is a difficult problem and remains open. The object of this paper is two-fold. Firstly, we establish the global regularity result to the case when the critical case occurs in the velocity and a logarithmic dissipation occurs in the non-Newtonian part of the stress tensor. Secondly, when the critical case occurs in the non-Newtonian part of the stress tensor, we first present many interesting global a priori bounds, then establish a conditional global regularity in terms of the non-Newtonian part of the stress tensor. This criterion comes naturally from our approach to obtain a global L∞-bound for the vorticity ω.

scholarly journals Singularity Analysis for a Class of Porous Medium Equation with Time-Dependent Coefficients

2016 ◽  
Vol 2016 ◽  
pp. 1-5 ◽  
Author(s):  
Anyin Xia ◽  
Xianxiang Pu ◽  
Shan Li
Keyword(s):  
Porous Medium ◽  
Global Regularity ◽  
Blow Up ◽  
Porous Medium Equation ◽  
Singularity Analysis ◽  
Dirichlet Boundary Conditions ◽  
Time Dependent ◽  
Medium Equation ◽  
Regularity Results ◽  
Homogeneous Dirichlet Boundary Conditions

This paper concerns the singularity and global regularity for the porous medium equation with time-dependent coefficients under homogeneous Dirichlet boundary conditions. Firstly, some global regularity results are established. Furthermore, we investigate the blow-up solution to the boundary value problem. The upper and lower estimates to the lifespan of the singular solution are also obtained here.

scholarly journals Regularity results for singular elliptic problems

2006 ◽  
Vol 4 (3) ◽  
pp. 243-259 ◽  
Author(s):  
Loredana Caso
Keyword(s):  
Boundary Conditions ◽  
Weight Function ◽  
Elliptic Equations ◽  
Global Regularity ◽  
Elliptic Problems ◽  
The Other ◽  
Weighted Spaces ◽  
Regularity Results

Some local and global regularity results for solutions of linear elliptic equations in weighted spaces are proved. Here the leading coefficients are VMO functions, while the hypotheses on the other coefficients and the boundary conditions involve a suitable weight function.

W1,p REGULARITY IN TRANSPORT THEORY

1991 ◽  
Vol 01 (04) ◽  
pp. 477-499 ◽  
Author(s):  
MUSTAPHA MOKHTAR-KHARROUBI
Keyword(s):  
Fourier Analysis ◽  
Singular Integral ◽  
General Class ◽  
Transport Theory ◽  
Neutron Transport ◽  
Convex Domains ◽  
Integral Analysis ◽  
Neutron Transport Theory ◽  
Type C ◽  
Regularity Results

Let T be the streaming operator arising in neutron transport theory, T=−ν(∂/∂x)−σ(ν), with non-incoming boundary conditions in convex domains. The purpose of this paper is to give a general class [Formula: see text] of pair (C1, C2) of collision operators such that C1(λ−T)−1C2 maps continuously Lp into W1,p(1<p<+∞). The result is based on a singular integral analysis of Calderon-Zygmund type. In the case p=2 we obtain, by Fourier analysis, the same result under weaker assumptions. We also derive W1,p regularity results for operators of type C1(λ−T)−1.

Global solvability and regularity for ∂¯ on an annulus between two weakly convex domains which satisfy property (P)

2019 ◽  
Vol 12 (03) ◽  
pp. 1950041
Author(s):  
Sayed Saber
Keyword(s):  
Neumann Problem ◽  
Complex Manifold ◽  
Global Solvability ◽  
Closed Formula ◽  
Convex Domains ◽  
Weakly Convex ◽  
Canonical Solution ◽  
Dimension Formula ◽  
Satisfy Property

Let [Formula: see text] be a complex manifold of dimension [Formula: see text] and let [Formula: see text]. Let [Formula: see text] be a weakly [Formula: see text]-convex and [Formula: see text] be a weakly [Formula: see text]-convex in [Formula: see text] with smooth boundaries such that [Formula: see text]. Assume that [Formula: see text] and [Formula: see text] satisfy property [Formula: see text]. Then the compactness estimate for [Formula: see text]-forms [Formula: see text] holds for the [Formula: see text]-Neumann problem on the annulus domain [Formula: see text]. Furthermore, if [Formula: see text] is [Formula: see text]-closed [Formula: see text]-form, which is [Formula: see text] on [Formula: see text] and which is cohomologous to zero on [Formula: see text], the canonical solution [Formula: see text] of the equation [Formula: see text] is smooth on [Formula: see text].

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