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We introduce a definition of the fractional Laplacian (- Δ) s (·) with spatially variable order s : Ω → [ 0 , 1 ] and study the solvability of the associated Poisson problem on a bounded domain Ω . The initial motivation arises from the extension results of Caffarelli and Silvestre, and Stinga and Torrea; however the analytical tools and approaches developed here are new. For instance, in some cases we allow the variable order s (·) to attain the values 0 and 1 leading to a framework on weighted Sobolev spaces with non-Muckenhoupt weights. Initially, and under minimal assumptions, the operator (- Δ) s (·) is identified as the Lagrange multiplier corresponding to an optimization problem; and its domain is determined as a quotient space of weighted Sobolev spaces. The well-posedness of the associated Poisson problem is then obtained for data in the dual of this quotient space. Subsequently, two trace regularity results are established, allowing to partially characterize functions in the aforementioned quotient space whenever a Poincaré type inequality is available. Precise examples are provided where such inequality holds, and in this case the domain of the operator (- Δ) s (·) is identified with a subset of a weighted Sobolev space with spatially variant smoothness s (·) . The latter further allows to prove the well-posedness of the Poisson problem assuming functional regularity of the data. [ABSTRACT FROM AUTHOR]

This work is concerned with nonlinear feedback control design for the problem of fluid mixing via advection. The overall dynamics is governed by the transport and Stokes equations in an open bounded and connected domain Ω ⊂ R d , with d = 2 or d = 3. The feedback laws are constructed based on the ideas of instantaneous control as well as a direct approximation of the optimality system derived from an optimal open-loop control problem. It can be shown that under appropriate numerical discretization schemes, two approaches generate the same sub-optimal feedback law. On the other hand, different discretization schemes may result in feedback laws of different regularity, which determine different mixing results. The Sobolev norm of the dual space (H 1 (Ω)) ′ of H 1 (Ω) is used as the mix-norm to quantify mixing based on the known property of weak convergence. The major challenge is encountered in the analysis of the asymptotic behavior of the closed-loop systems due to the absence of diffusion in the transport equation together with its nonlinear coupling with the flow equations. To address these issues, we first establish the decay properties of the velocity, which in turn help obtain the estimates on scalar mixing and its long-time behavior. Finally, mixed continuous Galerkin (CG) and discontinuous Galerkin (DG) methods are employed to discretize the closed-loop system. Numerical experiments are conducted to demonstrate our ideas and compare the effectiveness of different feedback laws. [ABSTRACT FROM AUTHOR]

In this paper, we consider feedback stabilization for parabolic variational inequalities of obstacle type with time and space depending reaction and convection coefficients and show exponential stabilization to nonstationary trajectories. Based on a Moreau–Yosida approximation, a feedback operator is established using a finite (and uniform in the approximation index) number of actuators leading to exponential decay of given rate of the state variable. Several numerical examples are presented addressing smooth and nonsmooth obstacle functions. [ABSTRACT FROM AUTHOR]

This paper provides an extension to fractional order Sobolev spaces of the classical result of Murat and Brezis which states that the cone of positive elements in H -1(Ω) compactly embeds in W -1,q(Ω) for every q < 2 and for any open and bounded set Ω with Lipschitz boundary. In particular, our proof contains the classical result. Several new analysis tools are developed during the course of the proof to our main result which are of wider interest. Subsequently, we apply our results to the convergence of convex sets and establish a fractional version of the Mosco convergence result of Boccardo and Murat. We conclude with an application of this result to quasi-variational inequalities. [ABSTRACT FROM AUTHOR]

We propose a new variational model in weighted Sobolev spaces with nonstandard 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. [ABSTRACT FROM AUTHOR]

We address the problem of optimally placing sensor networks for convection-diffusion processes where the convective part is perturbed. The problem is formulated as an optimal control problem where the integral Riccati equation is a constraint and the design variables are sensor locations. The objective functional involves a term associated to the trace of the solution to the Riccati equation and a term given by a constrained optimization problem for the directional derivative of the previous quantity over a set of admissible perturbations. This paper addresses the existence of the derivative with respect to the convective part of the solution to the Riccati equation, the well-posedness of the optimization problem, and finalizes with a range of numerical tests. [ABSTRACT FROM AUTHOR]

Variable splitting schemes for the function space version of the image reconstruction problem with total variation regularization (TV-problem) in its primal and pre-dual formulations are considered. For the primal splitting formulation, while existence of a solution cannot be guaranteed, it is shown that quasi-minimizers of the penalized problem are asymptotically related to the solution of the original TV-problem. On the other hand, for the pre-dual formulation, a family of parametrized problems is introduced and a parameter dependent contraction of an associated fixed point iteration is established. Moreover, the theory is validated by numerical tests. Additionally, the augmented Lagrangian approach is studied, details on an implementation on a staggered grid are provided and numerical tests are shown. [ABSTRACT FROM AUTHOR]

We formulate and study two mathematical models of a thermoforming process involving a membrane and a mould as implicit obstacle problems. In particular, the membrane-mould coupling is determined by the thermal displacement of the mould that depends in turn on the membrane through the contact region. The two models considered are a stationary (or elliptic) model and an evolutionary (or quasistatic) one. For the first model, we prove the existence of weak solutions by solving an elliptic quasi-variational inequality coupled to elliptic equations. By exploring the fine properties of the variation of the contact set under non-degenerate data, we give sufficient conditions for the existence of regular solutions, and under certain contraction conditions, also a uniqueness result. We apply these results to a series of semi-discretised problems that arise as approximations of regular solutions for the evolutionary or quasistatic problem. Here, under certain conditions, we are able to prove existence for the evolutionary problem and for a special case, also the uniqueness of time-dependent solutions. [ABSTRACT FROM AUTHOR]

This paper considers a class of nonlinear evolution quasi-variational inequality (QVI) problems with pointwise gradient constraints in vector-valued function spaces. The existence and approximation of solutions is addressed based on a combination of C0-semigroup methods, Mosco convergence, and monotone operator techniques developed by Brézis. An algorithm based on semidiscretization in time is proposed and its numerical implementation based on a penalty approach and semismooth Newton methods is studied. This paper ends with a report on numerical tests which involve the p-Laplacian and several types of nonlinear constraints. [ABSTRACT FROM AUTHOR]

A class of nonlinear elliptic quasi-variational inequality (QVI) problems with gradient constraints in function space is considered. Problems of this type arise, for instance, in the mathematical description of the magnetization of superconductors, in problems in elastoplasticity, or in electrostatics as well as in game theory. The paper addresses the iterative solution of the QVIs by a sequential minimization technique relying on the repeated solution of variational inequality--type problems. A monotone operator theoretic approach is developed which does not resort to Mosco convergence as is often done in connection with existence analysis for QVIs. For the numerical solution of the QVIs a penalty approach combined with a semismooth Newton iteration is proposed. The paper ends with a report on numerical tests involving the p-Laplace operator and various types of nonlinear constraints. [ABSTRACT FROM AUTHOR]

Total Generalized Variation (TGV) regularization in image reconstruction relies on an infimal convolution type combination of generalized first- and second-order derivatives. This helps to avoid the staircasing effect of Total Variation (TV) regularization, while still preserving sharp contrasts in images. The associated regularization effect crucially hinges on two parameters whose proper adjustment represents a challenging task. In this work, a bilevel optimization framework with a suitable statistics-based upper level objective is proposed in order to automatically select these parameters. The framework allows for spatially varying parameters, thus enabling better recovery in high-detail image areas. A rigorous dualization framework is established, and for the numerical solution, a Newton type method for the solution of the lower level problem, i.e. the image reconstruction problem, and a bilevel TGV algorithm are introduced. Denoising tests confirm that automatically selected distributed regularization parameters lead in general to improved reconstructions when compared to results for scalar parameters. [ABSTRACT FROM AUTHOR]

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 nonstandard 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. [ABSTRACT FROM AUTHOR]

In this paper we study the structure of solutions of the one dimensional weighted total variation regularisation problem, motivated by its application in signal recovery tasks. We study in depth the relationship between the weight function and the creation of new discontinuities in the solution. A partial semigroup property relating the weight function and the solution is shown and analytic solutions for simply data functions are computed. We prove that the weighted total variation minimisation problem is well-posed even in the case of vanishing weight function, despite the lack of coercivity. This is based on the fact that the total variation of the solution is bounded by the total variation of the data, a result that it also shown here. Finally the relationship to the corresponding weighted fidelity problem is explored, showing that the two problems can produce completely different solutions even for very simple data functions. [ABSTRACT FROM AUTHOR]

We consider the problem of the optimal shape of a two dimensional duct which contains a fluid governed by the Stokes equations with mixed boundary conditions. The conduit domain is assumed to be non-smooth and perturbations are allowed only on the walls, while the objective functional aims at minimizing the head loss and to enforce a uniform velocity profile at the outlet. We show existence of solutions to the shape optimization problem and determine the existence of the shape derivative. [ABSTRACT FROM AUTHOR]

We study a Boussinesq system in a bounded domain with an outlet boundary portion where fluid can leave or re-enter. On this boundary part, we consider a do-nothing condition for the fluid flow, and a new artificial condition for the heat transfer that couples nonlinearly the fluid velocity and temperature. The latter can be further adjusted if convective or conductive phenomena are dominant. We prove existence and, in some cases, uniqueness of weak solutions to stationary and evolutionary problems by a fixed point strategy under suitable assumptions on the data. A variety of numerical tests shows the improved performance of the new artificial condition with respect to other standard choices in the literature. [ABSTRACT FROM AUTHOR]