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## Über dieses Buch

This volume gathers contributions in the field of partial differential equations, with a focus on mathematical models in phase transitions, complex fluids and thermomechanics. These contributions are dedicated to Professor Gianni Gilardi on the occasion of his 70th birthday. It particularly develops the following thematic areas: nonlinear dynamic and stationary equations; well-posedness of initial and boundary value problems for systems of PDEs; regularity properties for the solutions; optimal control problems and optimality conditions; feedback stabilization and stability results. Most of the articles are presented in a self-contained manner, and describe new achievements and/or the state of the art in their line of research, providing interested readers with an overview of recent advances and future research directions in PDEs.

## Inhaltsverzeichnis

### Rate of Convergence for Eigenfunctions of Aharonov-Bohm Operators with a Moving Pole

Abstract
We study the behavior of eigenfunctions for magnetic Aharonov-Bohm operators with half-integer circulation and Dirichlet boundary conditions in a planar domain. We prove a sharp estimate for the rate of convergence of eigenfunctions as the pole moves in the interior of the domain.
Laura Abatangelo, Veronica Felli

### Nondecreasing Solutions to Doubly Nonlinear Equations

Abstract
Nondecreasing evolution is described via the coupling of an abstract doubly nonlinear differential inclusion and a constraint on the rate. The latter is formulated by imposing the monotonicity in time of the solution with respect to a given preorder in a Hilbert space. We discuss a solution notion for this problem and prove existence and long-time behavior.
Goro Akagi, Ulisse Stefanelli

### Identification Problems for Degenerate Integro-Differential Equations

Abstract
We are concerned with two different inverse problems for degenerate integro-differential equations in Banach spaces. In the first, we handle a strongly degenerate problem on a finite interval, while in the second we consider a related inverse problem for integro-differential equations studied by G. Da Prato and A. Lunardi in the regular case. All these results can be applied to inverse problems for equations from mathematical physics.
Mohammed Al Horani, Mauro Fabrizio, Angelo Favini, Hiroki Tanabe

### A Phase Transition Model Describing Auxetic Materials

Abstract
In this paper we introduce a new model describing the behavior of auxetic materials in terms of a phase-field PDE system. More precisely, the evolution equations are recovered by a generalization of the principle of virtual power in which microscopic motions and forces, responsible for the phase transitions, are included. The momentum balance is written in the setting of a second gradient theory, and it presents nonlinear contributions depending on the phases. The evolution of the phases is governed by variational inclusions with non-linear coupling terms. By use of a fixed point theorem and monotonicity arguments, we are able to show that the resulting initial and boundary value problem admits a weak solution.
Elena Bonetti, Mauro Fabrizio, Michel Frémond

### Global Well-Posedness for a Phase Transition Model with Irreversible Evolution and Acceleration Forces

Abstract
In this paper we investigate a nonlinear PDE system describing irreversible phase transition phenomena where inertial effects are also included. Its derivation comes from the modelling approach proposed by M. Frémond. We obtain a global in time existence and uniqueness result for the related initial and boundary value problem.
Giovanna Bonfanti, Fabio Luterotti

### Perimeter Symmetrization of Some Dynamic and Stationary Equations Involving the Monge-Ampère Operator

Abstract
We apply the perimeter symmetrization to a two-dimensional pseudo-parabolic dynamic problem associated to the Monge-Ampère operator as well as to the second order elliptic problem which arises after an implicit time discretization of the dynamical equation. Curiously, the dynamical problem corresponds to a third order operator but becomes a singular second order parabolic equation (involving the 3-Laplacian operator) in the class of radially symmetric convex functions. Using symmetrization techniques some quantitative comparison estimates and several qualitative properties of solutions are given.
Barbara Brandolini, Jesús Ildefonso Díaz

### Optimal Boundary Control of a Nonstandard Cahn–Hilliard System with Dynamic Boundary Condition and Double Obstacle Inclusions

Abstract
In this paper, we study an optimal boundary control problem for a model for phase separation taking place in a spatial domain that was introduced by P. Podio-Guidugli in Ric. Mat. 55 (2006), pp. 105–118. The model consists of a strongly coupled system of nonlinear parabolic differential inclusions, in which products between the unknown functions and their time derivatives occur that are difficult to handle analytically; the system is complemented by initial and boundary conditions. For the order parameter of the phase separation process, a dynamic boundary condition involving the Laplace–Beltrami operator is assumed, which models an additional nonconserving phase transition occurring on the surface of the domain. We complement in this paper results that were established in the recent contribution appeared in Evol. Equ. Control Theory 6 (2017), pp. 35–58, by the two authors and Gianni Gilardi. In contrast to that paper, in which differentiable potentials of logarithmic type were considered, we investigate here the (more difficult) case of nondifferentiable potentials of double obstacle type. For such nonlinearities, the standard techniques of optimal control theory to establish the existence of Lagrange multipliers for the state constraints are known to fail. To overcome these difficulties, we employ the following line of approach: we use the results contained in the preprint arXiv:1609.07046 [math.AP] (2016), pp. 1–30, for the case of (differentiable) logarithmic potentials and perform a so-called “deep quench limit”. Using compactness and monotonicity arguments, it is shown that this strategy leads to the desired first-order necessary optimality conditions for the case of (nondifferentiable) double obstacle potentials.
Pierluigi Colli, Jürgen Sprekels

### Nontrivial Solutions of Quasilinear Elliptic Equations with Natural Growth Term

Abstract
We prove the existence of multiple solutions for a quasilinear elliptic equation containing a term with natural growth, under assumptions that are invariant by diffeomorphism. To this purpose we develop an adaptation of degree theory.
Marco Degiovanni, Alessandra Pluda

### On a Diffuse Interface Model for Tumour Growth with Non-local Interactions and Degenerate Mobilities

Abstract
We study a non-local variant of a diffuse interface model proposed by Hawkins–Daarud et al. (Int. J. Numer. Methods Biomed. Eng. 28:3–24, 2012) for tumour growth in the presence of a chemical species acting as nutrient. The system consists of a Cahn–Hilliard equation coupled to a reaction-diffusion equation. For non-degenerate mobilities and smooth potentials, we derive well-posedness results, which are the non-local analogue of those obtained in Frigeri et al. (European J. Appl. Math. 2015). Furthermore, we establish existence of weak solutions for the case of degenerate mobilities and singular potentials, which serves to confine the order parameter to its physically relevant interval. Due to the non-local nature of the equations, under additional assumptions continuous dependence on initial data can also be shown.
Sergio Frigeri, Kei Fong Lam, Elisabetta Rocca

### A Boundary Control Problem for the Equation and Dynamic Boundary Condition of Cahn–Hilliard Type

Abstract
A dynamic boundary condition is a type of partial differential equation that describes the dynamics of a system on the boundary. Combining with the heat equation in a smooth-bounded domain, the characteristic structure of “total mass conservation” appears, namely, the volume in the bulk plus the volume on the boundary is conserved. Based on this interesting structure, an equation and dynamic boundary condition of Cahn–Hilliard type was introduced by Goldstein–Miranville–Schimperna. In this paper, based on the previous work of Colli–Gilardi–Sprekels, a boundary control problem for the equation and dynamic boundary condition of Cahn–Hilliard type is considered. The optimal boundary control that realizes the minimal cost under a control constraint is determined, and a necessary optimality condition is obtained.
Takeshi Fukao, Noriaki Yamazaki

### New Class of Doubly Nonlinear Evolution Equations Governed by Time-Dependent Subdifferentials

Abstract
We discuss a new class of doubly nonlinear evolution equations governed by time-dependent subdifferentials in uniformly convex Banach spaces, and establish an abstract existence result of solutions. Also, we show non-uniqueness of solution, giving some examples. Moreover, we treat a quasi-variational doubly nonlinear evolution equation by applying this result extensively, and give some applications to nonlinear PDEs with gradient constraint for time-derivatives.
Nobuyuki Kenmochi, Ken Shirakawa, Noriaki Yamazaki

### Boundedness of Solutions to a Degenerate Diffusion Equation

Abstract
The diffusion equation with a bounded saturation range under the time derivative and with Robin boundary conditions is shown to admit a regular bounded solution provided that the saturation function and the permeability coefficient have controlled decay at infinity. The result remains valid even if Preisach hysteresis is present in the pressure-saturation relation. The method of proof is based on a Moser-Alikakos iteration scheme which is compatible with a generalized Preisach energy dissipation mechanism.
Pavel Krejčí

### Optimal A Priori Error Estimates of Parabolic Optimal Control Problems with a Moving Point Control

Abstract
In this paper we consider a parabolic optimal control problem with a Dirac type control with moving point source in two space dimensions. We discretize the problem with piecewise constant functions in time and continuous piecewise linear finite elements in space. For this discretization we show optimal order of convergence with respect to the time and the space discretization parameters modulo some logarithmic terms. Error analysis for the same problem was carried out in the recent paper (Gong and Yan, SIAM J Numer Anal 54:1229–1262, 2016), however, the analysis there contains a serious flaw. One of the main goals of this paper is to provide the correct proof. The main ingredients of our analysis are the global and local error estimates on a curve, that have an independent interest.
Dmitriy Leykekhman, Boris Vexler

### A Note on the Feedback Stabilization of a Cahn–Hilliard Type System with a Singular Logarithmic Potential

Abstract
This article deals with the internal feedback stabilization of a phase field system of Cahn–Hilliard type involving a logarithmic potential F, and extends the recent results provided in Barbu et al. (J Differ Equ 262:2286–2334, 2017) for the double-well potential. The stabilization is searched around a stationary solution, by a feedback controller with support in a subset ω of the domain. The controller stabilizing the linearized system is constructed as a finite combination of the unstable modes of the operator acting in the linear system and it is further provided in a feedback form by solving a certain minimization problem. Finally, it is proved that this feedback form stabilizes the nonlinear system too, if the stationary solution has not large variations. All these results are provided in the three-dimensional case for a regularization of the singular potential F, and allow the same conclusion for the singular logarithmic potential in the one-dimensional case.
Gabriela Marinoschi

### Mathematical Analysis of a Parabolic-Elliptic Model for Brain Lactate Kinetics

Abstract
Our aim in this paper is to study properties of a parabolic-elliptic system related with brain lactate kinetics. These equations are obtained from a reaction-diffusion system, when a small parameter vanishes. In particular, we prove the existence and uniqueness of nonnegative solutions and obtain error estimates on the difference of the solutions to the initial reaction-diffusion system and those to the limit one, on bounded time intervals. We also study the linear stability of the unique spatially homogeneous equilibrium.
Alain Miranville

### Weak Formulation for Singular Diffusion Equation with Dynamic Boundary Condition

Abstract
In this paper, we propose a weak formulation of the singular diffusion equation subject to the dynamic boundary condition. The weak formulation is based on a reformulation method by an evolution equation including the subdifferential of a governing convex energy. Under suitable assumptions, the principal results of this study are stated in forms of Main Theorems A and B, which are respectively to verify: the adequacy of the weak formulation; the common property between the weak solutions and those in regular problems of standard PDEs.
Ryota Nakayashiki, Ken Shirakawa

### Smooth and Broken Minimizers of Some Free Discontinuity Problems

Abstract
We show that minimizers of free discontinuity problems with energy dependent on jump integrals and Dirichlet boundary conditions are smooth provided a smallness condition is imposed on data. We examine in detail two examples: the elastic-plastic beam and the elastic-plastic plate with free yield lines. In both examples there is a gap between the condition for solvability (safe load condition) and this smallness condition (load regularity condition) which imply regularity and uniqueness of minimizers. Such gap allows the existence of damaged/creased minimizers. Eventually we produce explicit examples of irregular solutions when the load is in the gap.
Danilo Percivale, Franco Tomarelli

### Stability Results for Abstract Evolution Equations with Intermittent Time-Delay Feedback

Abstract
We consider abstract evolution equations with on–off time delay feedback. Without the time delay term, the model is described by an exponentially stable semigroup. We show that, under appropriate conditions involving the delay term, the system remains asymptotically stable. Under additional assumptions exponential stability results are also obtained. Concrete examples illustrating the abstract results are finally given.
Cristina Pignotti

### From Visco-Energetic to Energetic and Balanced Viscosity Solutions of Rate-Independent Systems

Abstract
This paper focuses on weak solvability concepts for rate-independent systems in a metric setting. Visco-Energetic solutions have been recently obtained by passing to the time-continuous limit in a time-incremental scheme, akin to that for Energetic solutions, but perturbed by a ‘viscous’ correction term, as in the case of Balanced Viscosity solutions. However, for Visco-Energetic solutions this viscous correction is tuned by a fixed parameter μ. The resulting solution notion is characterized by a stability condition and an energy balance analogous to those for Energetic solutions, but, in addition, it provides a fine description of the system behavior at jumps as Balanced Viscosity solutions do. Visco-Energetic evolution can be thus thought as ‘in-between’ Energetic and Balanced Viscosity evolution. Here we aim to formalize this intermediate character of Visco-Energetic solutions by studying their singular limits as μ ↓ 0 and μ ↑∞. We shall prove convergence to Energetic solutions in the former case, and to Balanced Viscosity solutions in the latter situation.
Riccarda Rossi, Giuseppe Savaré

### A Duality Approach in Some Boundary Value Problems

Abstract
We describe several results from the literature concerning approximation procedures for variational boundary value problems, via duality techniques. Applications in shape optimization are also indicated. Some properties are quite unexpected and this is an argument that the present duality approach may be of interest in a large class of problems.
Dan Tiba

### On the Structural Properties of Nonlinear Flows

Abstract
This work deals with the structural stability of quasilinear first-order flows w.r.t. arbitrary perturbations not only of data but also of operators. This rests upon a variational formulation based on works of Brezis, Ekeland, Nayroles and Fitzpatrick, and on the use of evolutionary Γ-convergence w.r.t. a nonlinear topology of weak type. This approach is extended to flows of a class of nonmonotone operators. A theory in progress is outlined, and is also used to prove the structural compactness and stability of doubly-nonlinear parabolic flows of the form
$$\displaystyle\begin{array}{rcl} \alpha (D_{t}u) + \partial \gamma (u) \ni h,& & {}\\ \end{array}$$
α being a maximal monotone operator, and γ a lower semicontinuous convex function on a Hilbert space.
Augusto Visintin
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