Sobolev spaces with non-Muckenhoupt weights, fractional elliptic operators, and applications

We propose a new variational model in weighted Sobolev spaces with non-standard weights and applications to image processing. We show that these weights are, in general, not of Muckenhoupt type and therefore the classical analysis tools may not apply. For special cases of the weights, the resulting variational problem is known to be equivalent to the fractional Poisson problem. The trace space for the weighted Sobolev space is identified to be embedded in a weighted $L^2$ space. We propose a finite element scheme to solve the Euler-Lagrange equations, and for the image denoising application we propose an algorithm to identify the unknown weights. The approach is illustrated on several test problems and it yields better results when compared to the existing total variation techniques

Existence, iteration procedures and directional differentiability for parabolic QVIs

We study parabolic quasi-variational inequalities (QVIs) of obstacle type. Under appropriate assumptions on the obstacle mapping, we prove the existence of solutions of such QVIs by two methods: one by time discretisation through elliptic QVIs and the second by iteration through parabolic variational inequalities (VIs). Using these results, we show the directional differentiability (in a certain sense) of the solution map which takes the source term of a parabolic QVI into the set of solutions, and we relate this result to the contingent derivative of the aforementioned map. We finish with an example where the obstacle mapping is given by the inverse of a parabolic differential operator.Comment: 41 page

Stability of the solution set of quasi-variational inequalities and optimal control

For a class of quasi-variational inequalities (QVIs) of obstacle-type the stability of its solution set and associated optimal control problems are considered. These optimal control problems are non-standard in the sense that they involve an objective with set-valued arguments. The approach to study the solution stability is based on perturbations of minimal and maximal elements of the solution set of the QVI with respect to {monotone} perturbations of the forcing term. It is shown that different assumptions are required for studying decreasing and increasing perturbations and that the optimization problem of interest is well-posed.Comment: 29 page

The Boussinesq system with mixed non-smooth boundary conditions and do-nothing boundary flow

A @stationary Boussinesq system for an incompressible viscous fluid in a bounded domain with a nontrivial condition at an open boundary is studied. We consider a novel non-smooth boundary condition associated to the heat transfer on the open boundary that involves the temperature at the boundary, the velocity of the fluid, and the outside temperature. We show that this condition is compatible with two approaches at dealing with the do-nothing boundary condition for the fluid: 1) the directional do-nothing condition and 2) the do-nothing condition together with an integral bound for the backflow. Well-posedness of variational formulations is proved for each problem

Sobolev spaces with non-Muckenhoupt weights, fractional elliptic operators, and applications

We propose a new variational model in weighted Sobolev spaces with non-standard weights and applications to image processing. We show that these weights are, in general, not of Muckenhoupt type and therefore the classical analysis tools may not apply. For special cases of the weights, the resulting variational problem is known to be equivalent to the fractional Poisson problem. The trace space for the weighted Sobolev space is identified to be embedded in a weighted L2 space. We propose a finite element scheme to solve the Euler-Lagrange equations, and for the image denoising application we propose an algorithm to identify the unknown weights. The approach is illustrated on several test problems and it yields better results when compared to the existing total variation techniques

Optimal selection of the regularization function in a generalized total variation model. Part I: Modelling and theory

A generalized total variation model with a spatially varying regularization weight is considered. Existence of a solution is shown, and the associated Fenchel-predual problem is derived. For automatically selecting the regularization function, a bilevel optimization framework is proposed. In this context, the lower-level problem, which is parameterized by the regularization weight, is the Fenchel predual of the generalized total variation model and the upper-level objective penalizes violations of a variance corridor. The latter object relies on a localization of the image residual as well as on lower and upper bounds inspired by the statistics of the extremes

The Boussinesq system with mixed non-smooth boundary conditions and ``do-nothing'' boundary flow

A stationary Boussinesq system for an incompressible viscous fluid in a bounded domain with a nontrivial condition at an open boundary is studied. We consider a novel non-smooth boundary condition associated to the heat transfer on the open boundary that involves the temperature at the boundary, the velocity of the fluid, and the outside temperature. We show that this condition is compatible with two approaches at dealing with the do-nothing boundary condition for the fluid: 1) the directional do-nothing condition and 2) the do-nothing condition together with an integral bound for the backflow. Well-posedness of variational formulations is proved for each problem

Convergence aspects for sets of measures with divergences and boundary conditions

In this paper we study set convergence aspects for Banach spaces of vector-valued measures with divergences (represented by measures or by functions) and applications. We consider a form of normal trace characterization to establish subspaces of measures that directionally vanish in parts of the boundary, and present examples constructed with binary trees. Subsequently we study convex sets with total variation bounds and their convergence properties together with applications to the stability of optimization problems

On the uniqueness and numerical approximation of solutions to certain parabolic quasi-variational inequalities

We provide a series of rigidity results for a nonlocal phase transition equation. The results that we obtain are an improvement of flatness theorem and a series of theorems concerning the one-dimensional symmetry for monotone and minimal solutions, in the research line dictaded by a classical conjecture of E. De Giorgi. Here, we collect a series of pivotal results, of geometric type, which are exploited in the proofs of the main results in a companion paper