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In this paper, the authors study the distributed optimal control of a system of three evolutionary equations involving fractional powers of three selfadjoint, monotone, unbounded linear operators having compact resolvents. The system is a generalization of a Cahn-Hilliard type phase field system modeling tumor growth that goes back to Hawkins-Daarud et al. (Int. J. Numer. Math. Biomed. Eng. 28 (2012), 3-24). The aim of the control process, which could be realized by either administering a drug or monitoring the nutrition, is to keep the tumor cell fraction under control while avoiding possible harm for the patient. In contrast to previous studies, in which the occurring unbounded operators governing the diffusional regimes were all given by the Laplacian with zero Neumann boundary conditions, the operators may in our case be different; more generally, we consider systems with fractional powers of the type that were studied in the recent work Adv. Math. Sci. Appl. 28 (2019), 343-375 (see arXiv:1906.10874), by the present authors. In the analysis, the Fr\'echet differentiability of the associated control-to-state operator is shown, by also establishing the existence of solutions to the associated adjoint system, and deriving the first-order necessary conditions of optimality for a cost functional of tracking type., Comment: 35 pages. Key words: Fractional operators, Cahn-Hilliard systems, well-posedness, regularity, optimal control, necessary optimality conditions

A distributed optimal control problem for an extended model of phase field type for tumor growth is addressed. In this model, the chemotaxis effects are also taken into account. The control is realized by two control variables that design the dispensation of some drugs to the patient. The cost functional is of tracking type, whereas the potential setting has been kept quite general in order to allow regular and singular potentials to be considered. In this direction, some relaxation terms have been introduced in the system. We show the well-posedness of the state system, the Fr\'echet differentiability of the control-to-state operator in a suitable functional analytic framework, and, lastly, we characterize the first-order necessary conditions of optimality in terms of a variational inequality involving the adjoint variables.

In this paper, we study a system of three evolutionary operator equations involving fractional powers of selfadjoint, monotone, unbounded, linear operators having compact resolvents. This system constitutes a generalization of a phase field system of Cahn-Hilliard type modelling tumor growth that has been proposed in Hawkins-Daarud et al. (Int. J. Numer. Math. Biomed. Eng. 28 (2012), 3-24) and investigated in recent papers co-authored by the present authors and E. Rocca. The model consists of a Cahn-Hilliard equation for the tumor cell fraction, coupled to a reaction-diffusion equation for a function S representing the nutrient-rich extracellular water volume fraction. Effects due to fluid motion are neglected. The generalization investigated in this paper is motivated by the possibility that the diffusional regimes governing the evolution of the different constituents of the model may be of different (e.g., fractional) type. Under rather general assumptions, well-posedness and regularity results are shown. In particular, by writing the equation governing the evolution of the chemical potential in the form of a general variational inequality, also singular or nonsmooth contributions of logarithmic or of double obstacle type to the energy density can be admitted., Comment: Key words: fractional operators, Cahn-Hilliard systems, well-posedness, regularity of solutions, tumor growth models

This paper is concerned with a boundary control problem for the Cahn--Hilliard equation coupled with dynamic boundary conditions. In order to handle the control problem, we restrict our analysis to the case of regular potentials defined on the whole real line, assuming the boundary potential to be dominant. The existence of optimal control, the Fr\'echet differentiability of the control-to-state operator between appropriate Banach spaces, and the first-order necessary conditions for optimality are addressed. In particular, the necessary condition for optimality is characterized by a variational inequality involving the adjoint variables.

In this contribution, we deal with the longtime behavior of the solutions to the fractional variant of the Cahn-Hilliard system, with possibly singular potentials, that we have recently investigated in the paper `Well-posedness and regularity for a generalized fractional Cahn-Hilliard system' (see arXiv:1804.11290). More precisely, we study the omega-limit of the phase parameter and characterize it completely. Our characterization depends on the first eigenvalue of one of the operators involved: if it is positive, then the chemical potential vanishes at infinity and every element of the omega-limit is a stationary solution to the phase equation; if, instead, the first eigenvalue is 0, then every element of the omega-limit satisfies a problem containing a real function related to the chemical potential. Such a function is nonunique and time dependent, in general, as we show by an example. However, we give sufficient conditions in order that this function be uniquely determined and constant., Comment: Key words: Fractional operators, Cahn-Hilliard systems, longtime behavior

Integrating graphene with plasmonic nanostructures results in multifunctional hybrid systems with enhanced performance for numerous applications. In this work, we take advantage of the remarkable mechanical properties of graphene to combine it with scalable 3D plasmonic nanostructured silicon substrates, which enhance the interaction of graphene with electromagnetic radiation. Large areas of femtosecond laser-structured arrays of silicon nanopillars, decorated with gold nanoparticles, are integrated with graphene, which conforms to the substrate nanotopography. We obtain Raman spectra at 488, 514, 633, and 785 nm excitation wavelengths, spanning the entire visible range. For all excitation wavelengths, the Raman signal of graphene is enhanced by 2-3 orders of magnitude, similarly to the highest enhancements measured to date, concerning surface-enhanced Raman Spectroscopy (SERS) of graphene on plasmonic substrates. Moreover, in contrast to traditional deposition and lithographic methods, the fabrication method employed here relies on single-step, maskless, cost-effective, rapid laser processing of silicon in water, amenable to large-scale fabrication. Finite-difference time-domain simulations elucidate the advantages of the 3D topography of the substrate. Conformation of graphene to the Au-decorated silicon nanopillars enables graphene to sample near fields from an increased number of nanoparticles. Due to synergistic effects with the nanopillars, different nanoparticles become more active for different wavelengths and locations on the pillars, providing broadband enhancement. Nanostructured plasmonic silicon is a promising platform for integration with graphene and other 2D materials, for next-generation applications of large-area hybrid nanomaterials in the fields of sensing, photonics, optoelectronics, and medical diagnostics.

This paper is concerned with a thermomechanical model describing phase separation phenomena in terms of the entropy balance and equilibrium equations for the microforces. The related system is highly nonlinear and admits singular potentials in the phase equation. Both the viscous and the non-viscous cases are considered in the Cahn--Hilliard relations characterizing the phase dynamics. The entropy balance is written in terms of the absolute temperature and of its logarithm, appearing under time derivative. The initial and boundary value problem is considered for the system of partial differential equations. The existence of a global solution is proved via some approximations involving Yosida regularizations and a suitable time discretization.

The paper arXiv:1804.11290 contains well-posedness and regularity results for a system of evolutionary operator equations having the structure of a Cahn-Hilliard system. The operators appearing in the system equations were fractional versions in the spectral sense of general linear operators A and B having compact resolvents and are densely defined, unbounded, selfadjoint, and monotone in a Hilbert space of functions defined in a smooth domain. The associated double-well potentials driving the phase separation process modeled by the Cahn-Hilliard system could be of a very general type that includes standard physically meaningful cases such as polynomial, logarithmic, and double obstacle nonlinearities. In the subsequent paper arXiv:1807.03218, an analysis of distributed optimal control problems was performed for such evolutionary systems, where only the differentiable case of certain polynomial and logarithmic double-well potentials could be admitted. Results concerning existence of optimizers and first-order necessary optimality conditions were derived. In the present paper, we complement these results by studying a distributed control problem for such evolutionary systems in the case of nondifferentiable nonlinearities of double obstacle type. For such nonlinearities, it is well known that the standard constraint qualifications cannot be applied to construct appropriate Lagrange multipliers. To overcome this difficulty, we follow here the so-called "deep quench" method. We first give a general convergence analysis of the deep quench approximation that includes an error estimate and then demonstrate that its use leads in the double obstacle case to appropriate first-order necessary optimality conditions in terms of a variational inequality and the associated adjoint state system., Comment: Key words: Fractional operators, Cahn-Hilliard systems, optimal control, double obstacles, necessary optimality conditions

This work represents a first contribution on the problem of boundary stabilization for the phase field system of Cahn-Hilliard type, which models the phase separation in a binary mixture. The feedback controller we design here is with actuation only on the temperature flow of the system, on one part of the boundary only. Moreover, it is of proportional type, given in an explicit form, expressed only in terms of the eigenfunctions of the Laplace operator, being easy to manipulate from the computational point of view. Furthermore, it ensures that the closed loop nonlinear system exponentially reaches the prescribed stationary solution provided that the initial datum is close enough to it., Comment: Keywords: Cahn-Hilliard system, Feedback boundary control, Eigenfunctions, Closed loop system, Stabilization

This paper deals with the nonlinear phase field system \begin{equation*} \begin{cases} \partial_t (\theta +\ell \varphi) - \Delta\theta = f & \mbox{in}\ \Omega\times(0, T), \\[1mm] \partial_t \varphi - \Delta\varphi + \xi + \pi(\varphi) = \ell \theta,\ \xi\in\beta(\varphi) & \mbox{in}\ \Omega\times(0, T) \end{cases} \end{equation*} in a general domain $\Omega\subseteq\mathbb{R}^N$. Here $N \in \mathbb{N}$, $T>0$, $\ell>0$, $f$ is a source term, $\beta$ is a maximal monotone graph and $\pi$ is a Lipschitz continuous function. We note that in the above system the nonlinearity $\beta+\pi$ replaces the derivative of a potential of double well type. Thus it turns out that the system is a generalization of the Caginalp phase field model and it has been studied by many authors in the case that $\Omega$ is a bounded domain. However, for unbounded domains the analysis of the system seems to be at an early stage. In this paper we study the existence of solutions by employing a time discretization scheme and passing to the limit as the time step $h$ goes to $0$. In the limit procedure we face with the difficulty that the embedding $H^1(\Omega) \hookrightarrow L^2(\Omega)$ is not compact in the case of unbounded domains. Moreover, we can prove an interesting error estimate of order $h^{1/2}$ for the difference between continuous and discrete solutions.

In the recent paper `Well-posedness and regularity for a generalized fractional Cahn-Hilliard system' (arXiv:1804.11290) by the same authors, general well-posedness results have been established for a a class of evolutionary systems of two equations having the structure of a viscous Cahn-Hilliard system, in which nonlinearities of double-well type occur. The operators appearing in the system equations are fractional versions in the spectral sense of general linear operators A,B having compact resolvents, which are densely defined, unbounded, selfadjoint, and monotone in a Hilbert space of functions defined in a smooth domain. In this work we complement the results given in arXiv:1804.11290 by studying a distributed control problem for this evolutionary system. The main difficulty in the analysis is to establish a rigorous Frechet differentiability result for the associated control-to-state mapping. This seems only to be possible if the state stays bounded, which, in turn, makes it necessary to postulate an additional global boundedness assumption. One typical situation, in which this assumption is satisfied, arises when B is the negative Laplacian with zero Dirichlet boundary conditions and the nonlinearity is smooth with polynomial growth of at most order four. Also a case with logarithmic nonlinearity can be handled. Under the global boundedness assumption, we establish existence and first-order necessary optimality conditions for the optimal control problem in terms of a variational inequality and the associated adjoint state system., Comment: Key words: fractional operators, Cahn-Hilliard systems, optimal control, necessary optimality conditions

This paper is concerned with a non-conserved phase field system of Caginalp type in which the main operators are fractional versions of two fixed linear operators $A$ and $B$. The operators $A$ and $B$ are supposed to be densely defined, unbounded, self-adjoint, monotone in the Hilbert space $L^2(\Omega)$, for some bounded and smooth domain $\Omega$, and have compact resolvents. Our definition of the fractional powers of operators uses the approach via spectral theory. A nonlinearity of double-well type occurs in the phase equation and either a regular or logarithmic potential, as well as a non-differentiable potential involving an indicator function, is admitted in our approach. We show general well-posedness and regularity results, extending the corresponding results that are known for the non-fractional elliptic operators with zero Neumann conditions or other boundary conditions like Dirichlet or Robin ones. Then, we investigate the longtime behavior of the system, by fully characterizing every element of the $\omega$-limit as a stationary solution. In the final part of the paper we study the asymptotic behavior of the system as the parameter $\sigma$ appearing in the operator $B^{2\sigma}$ that plays in the phase equation decreasingly tends to zero. We can prove convergence to a phase relaxation problem at the limit, in which an additional term containing the projection of the phase variable on the kernel of $B$ appears., Comment: This paper is dedicated to Prof. Dr. Juergen Sprekels on the occasion of his 70th birthday, with best wishes from the authors. Key words: Fractional operators, Allen-Cahn equations, phase field system, well-posedness, regularity, asymptotics

In this paper, we investigate a rather general system of two operator equations that has the structure of a viscous or nonviscous Cahn--Hilliard system in which nonlinearities of double-well type occur. Standard cases like regular or logarithmic potentials, as well as non-differentiable potentials involving indicator functions, are admitted. The operators appearing in the system equations are fractional versions of general linear operators $A$ and $B$, where the latter are densely defined, unbounded, self-adjoint and monotone in a Hilbert space of functions defined in a smooth domain and have compact resolvents. We remark that our definition of the fractional power of operators uses the approach via spectral theory. Typical cases are given by standard second-order elliptic operators (e.g., the Laplacian) with zero Dirichlet or Neumann boundary conditions, but also other cases like fourth-order systems or systems involving the Stokes operator are covered by the theory. We derive general well-posedness and regularity results that extend corresponding results which are known for either the non-fractional Laplacian with zero Neumann boundary condition or the fractional Laplacian with zero Dirichlet condition. It turns out that the first eigenvalue $\lambda_1$ of $A$ plays an important und not entirely obvious role: if $\lambda_1$ is positive, then the operators $\,A\,$ and $\,B\,$ may be completely unrelated; if, however, $\lambda_1=0$, then it must be simple and the corresponding one-dimensional eigenspace has to consist of the constant functions and to be a subset of the domain of definition of a certain fractional power of $B$. We are able to show general existence, uniqueness, and regularity results for both these cases, as well as for both the viscous and the nonviscous system., Comment: 36 pages. Key words: fractional operators, Cahn-Hilliard systems, well-posedness, regularity of solutions

We study a coupled bulk-surface Allen-Cahn system with an affine linear transmission condition, that is, the trace values of the bulk variable and the values of the surface variable are connected via an affine relation, and this serves to generalize the usual dynamic boundary conditions. We tackle the problem of well-posedness via a penalization method using Robin boundary conditions. In particular, for the relaxation problem, the strong well-posedness and long-time behavior of solutions can be shown for more general and possibly nonlinear relations. New difficulties arise since the surface variable is no longer the trace of the bulk variable, and uniform estimates in the relaxation parameter are scarce. Nevertheless, weak convergence to the original problem can be shown. Using the approach of Colli and Fukao (Math. Models Appl. Sci. 2015), we show strong existence to the original problem with affine linear relations, and derive an error estimate between solutions to the relaxed and original problems., Comment: 34 pages

In this paper, we study the longtime asymptotic behavior of a phase separation process occurring in a three-dimensional domain containing a fluid flow of given velocity. This process is modeled by a viscous convective Cahn-Hilliard system, which consists of two nonlinearly coupled second-order partial differential equations for the unknown quantities, the chemical potential and an order parameter representing the scaled density of one of the phases. In contrast to other contributions, in which zero Neumann boundary conditions were are assumed for both the chemical potential and the order parameter, we consider the case of dynamic boundary conditions, which model the situation when another phase transition takes place on the boundary. The phase transition processes in the bulk and on the boundary are driven by free energies functionals that may be nondifferentiable and have derivatives only in the sense of (possibly set-valued) subdifferentials. For the resulting initial-boundary value system of Cahn-Hilliard type, general well-posedness results have been established in a recent contribution by the same authors. In the present paper, we investigate the asymptotic behavior of the solutions as times approaches infinity. More precisely, we study the $\omega$-limit (in a suitable topology) of every solution trajectory. Under the assumptions that the viscosity coefficients are strictly positive and that at least one of the underlying free energies is differentiable, we prove that the $\omega$-limit is meaningful and that all of its elements are solutions to the corresponding stationary system, where the component representing the chemical potential is a constant., Comment: Key words: Cahn-Hilliard systems, convection, dynamic boundary conditions, well-posedness, asymptotic behavior, omega-limit. arXiv admin note: text overlap with arXiv:1704.05337

The well-posedness for a system of partial differential equations and dynamic boundary conditions is discussed. This system is a sort of transmission problem between the dynamics in the bulk $\Omega $ and on the boundary $\Gamma$. The Poisson equation for the chemical potential, the Allen-Cahn equation for the order parameter in the bulk $\Omega$ are considered as auxiliary conditions for solving the Cahn-Hilliard equation on the boundary $\Gamma$. Recently the well-posedness for the equation and dynamic boundary condition, both of Cahn-Hilliard type, was discussed. Based on this result, the existence of the solution and its continuous dependence on the data are proved., Comment: Key words: Cahn-Hilliard equation, bulk condition, dynamic boundary condition, well-posedness. The interested reader can also see the related preprint arXiv:1502.05159 whose results are recalled and used for the analysis carried out in this paper