Topics in Applied Analysis and Optimisation

We consider state-of-the-art methods, theoretical limitations, and open problems in elliptic Quasi-Variational Inequalities (QVIs). This involves the development of solution algorithms in function space, existence theory, and the study of optimization problems with QVI constraints. We address the range of applicability and theoretical limitations of fixed point and other popular solution algorithms, also based on the nature of the constraint, e.g., obstacle and gradient-type. For optimization problems with QVI constraints, we study novel formulations that capture the multivalued nature of the solution mapping to the QVI, and generalized differentiability concepts appropriate for such problems.

The mathematical modelling in mechanics has a long-standing history as related to geometry, and significant progresses have often been achieved by the invention of new geometrical tools. Also, it happened that the elucidation of practical issues led to the invention of new scientific concepts, and possibly new paradigms, with potential impact far beyond. One such example is Riemann’s intrinsic view in geometry, that offered a radically new insight in the Physics of the early 20th century. On the other hand, the rather recent intrinsic approaches in elasticity and elasto-plasticity also share this philosophical standpoint of looking from inside, i.e., from the “manifold” point of view. Of course, this approach requires smoothness, and is thus incomplete for an analyst. Nevertheless, its first aim is to highlight the concepts of metric, curvature and torsion; these notions are addressed in the first part of this survey paper. In a second part, they are given a precise functional meaning and their properties are studied systematically. Further, a novel approach to elasto-plasticity constructed upon a model of incompatible elasticity is designed, carrying this intrinsic spirit. The main mathematical object in this theory is the incompatibility operator, i.e., a linearized version of Riemann’s curvature tensor. So far, this route not only has led the authors to a new model with a solid functional foundation and proof of existence results, but also to a framework with a minimal amount of ad-hoc assumptions, and complying with both the basic principles of thermodynamics and invariance principles of Physics. The questions arising from this novel approach are complex and intriguing, but we believe that the model is now sufficiently well posed to be studied simultaneously as a problem of mathematics and of mechanics. Most of the research programme remains to be done, and this survey paper is written to present our model, with a particular care to put this approach into a historical perspective.

A nonlocal phase field model of viscous Cahn–Hilliard type is considered. This model constitutes a nonlocal version of a model for two-species phase segregation on an atomic lattice under the presence of diffusion that has been studied in a series of papers by P. Podio-Guidugli and the present authors. The resulting system of differential equations consists of a highly nonlinear parabolic equation coupled to a nonlocal ordinary differential equation, which has singular terms that render the analysis difficult. Some results are presented on the well-posedness and stability of the system as well as on the distributed optimal control problem.

Statistical solutions for differential equations, as opposed to pointwise classical ones, are solutions where the initial data is taken in a measure one set in some probability space. The statistical approach to differential equations was partly motivated by A.N. Kolmogorov’s ideas on turbulence ( [28]), a subject where the interest relies on computation of mean quantities, and is suitable for situations where initial conditions are not precisely known. The use of probability theory to describe the concept of ensemble average, as far as Hydrodynamical equations are concerned, can be traced back to the work of E. Hopf [23]. Later on statistical solutions were introduced by C. Foias [20, 21] and studied also by M.I. Vishik and A.V. Fursikov [33].

In this set of notes, we present some recent developments on the fractional \( {(-\Delta)^{s}} u = u - {{u}^{3}}\) Allen-Cahn equation with special attention to -convergence results, energy and density estimates, convergence of level sets, Hamiltonian estimates, rigidity and symmetry results.

This work addresses an issue of statistical inference for the datasets lacking underlying linear structure, which makes impossible the direct application of standard inference techniques and requires a development of a new tool-box taking into account properties of the underlying space.We present an approach based on optimal transportation theory that is a convenient instrument for the analysis of complex data sets. The theory originates from seminal works of a french mathematician Gaspard Monge published at the end of 18th century. This chapter recalls the basics on optimal transportations theory, explains the ideas behind statistical inference on non-linear manifolds, and as an illustrative example presents a novel approach of construction of non asymptotic confidence sets for so calledWasserstein barycenter, a generalized analogous of Euclidean mean to the case of non-linear space endowed with a particular distance belonging to a class of Earth-Mover distances that it is a main object of study in optimal transportation theory. The chapter is based on the paper [18].

Liquid electrolytes are fluidic mixtures containing electrically charged ions. Electrochemical energy conversion systems like fuel cells and batteries contain liquid electrolytes. In biological tissues, nanoscale pores in the cell membranes separate different types of ions inside the cell from those in the intercellular space. Nanopores between electrolyte reservoirs can be used for analytical applications in medicine. Water purification technologies like electrodialysis rely on the electrolytic flow properties. This short and by far not exhaustive list of occurrences of electrolytic flow processes shows the importance of correct modeling of electrolyte flows. Due to the complex physical interactions present in this type of flows, in many case numerical simulation techniques are required to facilitate a deeper understanding of the flow behavior..

Assuming a pipe-wise constant structure of the friction coefficient in the modeling of natural gas transport through a passive network of pipes via semilinear systems of balance laws with associated linear coupling and boundary conditions, uncertainty in this parameter is quantified by a Markov chain Monte Carlo method. Information on the prior distribution is obtained from practitioners. The results are applied to the problem of validating technical feasibility under random exit demand in gas transport networks. The impact of quantified uncertainty to the probability level of technical feasible exit demand situations is studied by two example networks of small and medium size. The gas transport of the network is modeled by stationary solutions that are steady states of the time dependent semilinear problems.

We present and comment some recent modeling and results from stochastic geometry about the functionality of spatial communication networks with a multihop system of message transmissions. Our novel approaches concern connectivity on random street systems, frustration probabilities for service quality under constraints with regard to interference and capacity, and a new model of random message routing. Our main focus is on the description of the influence of spatial aspects, predominantly the locations of all the users. As a leading mathematical tool, we introduce the probabilistic theory of large deviations to the study of such systems in a high-density situation.

AbstractWe discuss recent progress in the mathematical modeling of semiconductor devices. The central result of this paper is a combined quantum-classical model that self-consistently couples van Roosbroeck’s drift-diffusion system for classical charge transport with a Lindblad-type quantum master equation. The coupling is shown to obey fundamental principles of non-equilibrium thermodynamics. The appealing thermodynamic properties are shown to arise from the underlying mathematical structure of a damped Hamitlonian system, which is an isothermal version of socalled GENERIC systems. The evolution is governed by a Hamiltonian part and a gradient part involving a Poisson operator and an Onsager operator as geoemtric structures, respectively. Both parts are driven by the conjugate forces given in terms of the derivatives of a suitable free energy.

In this work we investigate a two-phase model for concentrated suspensions. We construct a PDE formulation using a gradient flow structure featuring dissipative coupling between fluid and solid phase as well as different driving forces. Our construction is based on the concept of flow maps that also allows it to account for flows in moving domains with free boundaries. The major difference compared to similar existing approaches is the incorporation of a non-smooth two-homogeneous term to the dissipation potential, which creates a normal pressure even for pure shear flows.

This survey on stationary and evolutionary problems with gradient constraints is based on developments of monotonicity and compactness methods applied to large classes of scalar and vectorial solutions to variational and quasi-variational inequalities. Motivated by models for critical state problems and applications to free boundary problems in Mechanics and in Physics, in this work several known properties are collected and presented and a few novel results and examples are found.

Several variants of models of damage in viscoelastic continua under small strains in the Kelvin-Voigt rheology are presented and analyzed by using the Galerkin method. The particular case, known as a phase-field fracture approximation of cracks, is discussed in detail. All these models are dynamic (i.e. involve inertia to model vibrations or waves possibly emitted during fast damage/fracture or induced by fast varying forcing) and consider viscosity which is also damageable. Then various options for time discretisation are devised. Eventually, extensions to more complex rheologies or a modification for large strains are briefly exposed, too.