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Pierluigi Colli

Publications and source records attributed to Pierluigi Colli.

At least 109 records · Page 6Linked to original sources
arXiv2015

Asymptotic analyses and error estimates for a Cahn-Hilliard type phase field system modelling tumor growth

This paper is concerned with a phase field system of Cahn-Hilliard type that is related to a tumor growth model and consists of three equations in terms of the variables order parameter, chemical potential and nutrient concentration. This system has been investigated in the recent contributions arXiv:1401.5943 [math.AP] and arXiv:1501.07057 [math.AP] from the viewpoint of well-posedness, long time behavior and asymptotic convergence as two positive viscosity coefficients tend to zero at the same time. Here, we continue the analysis performed in arXiv:1501.07057 [math.AP] by showing two independent sets of results as just one of the coefficients tends to zero, the other remaining fixed. We prove convergence results, uniqueness of solutions to the two resulting limit problems, and suitable error estimates.

math.AP↗
arXiv2015

Equation and dynamic boundary condition of Cahn-Hilliard type with singular potentials

The well-posedness of a system of partial differential equations and dynamic boundary conditions, both of Cahn-Hilliard type, is discussed. The existence of a weak solution and its continuous dependence on the data are proved using a suitable setting for the conservation of a total mass in the bulk plus the boundary. A very general class of double-well like potentials is allowed. Moreover, some further regularity is obtained to guarantee the strong solution.

math.AP↗
arXiv2015

Vanishing viscosities and error estimate for a Cahn-Hilliard type phase field system related to tumor growth

In this paper we perform an asymptotic analysis for two different vanishing viscosity coefficients occurring in a phase field system of Cahn-Hilliard type that was recently introduced in order to approximate a tumor growth model. In particular, we extend some recent results obtained in the preprint arXiv:1401.5943, letting the two positive viscosity parameters tend to zero independently from each other and weakening the conditions on the initial data in such a way as to maintain the nonlinearities of the PDE system as general as possible. Finally, under proper growth conditions on the interaction potential, we prove an error estimate leading also to the uniqueness result for the limit system.

math.AP↗
arXiv2015

Singular limit of an integrodifferential system related to the entropy balance

A thermodynamic model describing phase transitions with thermal memory, in terms of an entropy equation and a momentum balance for the microforces, is adressed. Convergence results and error estimates are proved for the related integrodifferential system of PDE as the sequence of memory kernels converges to a multiple of a Dirac delta, in a suitable sense.

math.AP↗
arXiv2014

Cahn-Hilliard equation with dynamic boundary conditions and mass constraint on the boundary

The well-known Cahn-Hilliard equation entails mass conservation if a suitable boundary condition is prescribed. In the case when the equation is also coupled with a dynamic boundary condition, including the Laplace-Beltrami operator on the boundary, the total mass on the inside of the domain and its trace on the boundary should be conserved. The new issue of this paper is the setting of a mass constraint on the boundary. The effect of this additional constraint is the appearance of a Lagrange multiplier; in fact, two Lagrange multipliers arise, one for the bulk, the other for the boundary. The well-posedness of the resulting Cahn-Hilliard system with dynamic boundary condition and mass constraint on the boundary is obtained. The theory of evolution equations governed by subdifferentials is exploited and a complete characterization of the solution is given.

math.AP↗
arXiv2014

Optimal control of an Allen-Cahn equation with singular potentials and dynamic boundary condition

In this paper, we investigate optimal control problems for Allen-Cahn equations with singular nonlinearities and a dynamic boundary condition involving singular nonlinearities and the Laplace-Beltrami operator. The approach covers both the cases of distributed controls and of boundary controls. The cost functional is of standard tracking type, and box constraints for the controls are prescribed. Parabolic problems with nonlinear dynamic boundary conditions involving the Laplace-Beltrami operation have recently drawn increasing attention due to their importance in applications, while their optimal control was apparently never studied before. In this paper, we first extend known well-posedness and regularity results for the state equation and then show the existence of optimal controls and that the control-to-state mapping is twice continuously Fréchet differentiable between appropriate function spaces. Based on these results, we establish the first-order necessary optimality conditions in terms of a variational inequality and the adjoint state equation, and we prove second-order sufficient optimality conditions.

math.AP↗
arXiv2014

Optimal control for a phase field system with a possibly singular potential

In this paper we study a distributed control problem for a phase field system of Caginalp type with logarithmic potential. The main aim of this work would be to force the location of the diffuse interface to be as close as possible to a prescribed set. However, due to the discontinuous character of the cost functional, we have to approximate it by a regular one and, in this case, we solve the associated control problem and derive the related first order necessary optimality conditions.

math.AP↗
arXiv2014

Optimal boundary control of a viscous Cahn-Hilliard system with dynamic boundary condition and double obstacle potentials

In this paper, we investigate optimal boundary control problems for Cahn-Hilliard variational inequalities with a dynamic boundary condition involving double obstacle potentials and the Laplace-Beltrami operator. The cost functional is of standard tracking type, and box constraints for the controls are prescribed. We prove existence of optimal controls and derive first-order necessary conditions of optimality. The general strategy, which follows the lines of the recent approach by Colli, Farshbaf-Shaker, Sprekels (see the preprint arXiv:1308.5617) to the (simpler) Allen-Cahn case, is the following: we use the results that were recently established by Colli, Gilardi, Sprekels in the preprint arXiv:1407.3916 [math.AP] 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 (non-differentiable) double obstacle potentials.

math.AP↗
arXiv2014

The Allen-Cahn equation with dynamic boundary conditions and mass constraints

The Allen-Cahn equation, coupled with dynamic boundary conditions, has recently received a good deal of attention. The new issue of this paper is the setting of a rather general mass constraint which may involve either the solution inside the domain or its trace on the boundary. The system of nonlinear partial differential equations can be formulated as variational inequality. The presence of the constraint in the evolution process leads to additional terms in the equation and the boundary condition containing a suitable Lagrange multiplier. A well-posedness result is proved for the related initial value problem.

math.AP↗
arXiv2014

On a Cahn-Hilliard type phase field system related to tumor growth

The paper deals with a phase field system of Cahn-Hilliard type. For positive viscosity coefficients, the authors prove an existence and uniqueness result and study the long time behavior of the solution by assuming the nonlinearities to be rather general. In a more restricted setting, the limit as the viscosity coefficients tend to zero is investigated as well.

math.AP↗
arXiv2014

Sharp interface control in a Penrose-Fife model

In this paper we study a singular control problem for a system of PDEs describing a phase-field model of Penrose-Fife type. The main novelty of this contribution consists in the idea of forcing a sharp interface separation between the states of the system by using heat sources distributed in the domain and at the boundary. We approximate the singular cost functional with a regular one, which is based on the Legendre-Fenchel relations. Then, we obtain a regularized control problem for which we compute the first order optimality conditions using an adapted penalization technique. The proof of some convergence results and the passage to the limit in these optimality conditions lead to the characterization of the desired optimal controller.

math.AP↗
arXiv2013

Analysis of a time discretization scheme for a nonstandard viscous Cahn-Hilliard system

In this paper we propose a time discretization of a system of two parabolic equations describing diffusion-driven atom rearrangement in crystalline matter. The equations express the balances of microforces and microenergy; the two phase fields are the order parameter and the chemical potential. The initial and boundary-value problem for the evolutionary system is known to be well posed. Convergence of the discrete scheme to the solution of the continuous problem is proved by a careful development of uniform estimates, by weak compactness and a suitable treatment of nonlinearities. Moreover, for the difference of discrete and continuous solutions we prove an error estimate of order one with respect to the time step.

math.AP↗
arXiv2013

A deep quench approach to the optimal control of an Allen-Cahn equation with dynamic boundary condition and double obstacle potentials

In this paper, we investigate optimal control problems for Allen-Cahn variational inequalities with a dynamic boundary condition involving double obstacle potentials and the Laplace-Beltrami operator. The approach covers both the cases of distributed controls and of boundary controls. The cost functional is of standard tracking type, and box constraints for the controls are prescribed. We prove existence of optimal controls and derive first-order necessary conditions of optimality. The general strategy is the following: we use the results that were recently established by two of the authors 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 (non-differentiable) double obstacle potentials.

math.AP↗
arXiv2013

Existence of solutions for a mathematical model related to solid-solid phase transitions in shape memory alloys

We consider a strongly nonlinear PDE system describing solid-solid phase transitions in shape memory alloys. The system accounts for the evolution of an order parameter (related to different symmetries of the crystal lattice in the phase configurations), of the stress (and the displacement), and of the absolute temperature. The resulting equations present several technical difficulties to be tackled: in particular, we emphasize the presence of nonlinear coupling terms, higher order dissipative contributions, possibly multivalued operators. As for the evolution of temperature, a highly nonlinear parabolic equation has to be solved for a right hand side that is controlled only in L^1. We prove the existence of a solution for a regularized version, by use of a time discretization technique. Then, we perform suitable a priori estimates which allow us pass to the limit and find a weak global-in-time solution to the system.

math.AP↗
arXiv2013

A continuous dependence result for a nonstandard system of phase field equations

The present note deals with a nonstandard systems of differential equations describing a two-species phase segregation. This system naturally arises in the asymptotic analysis carried out recently by the same authors, as the diffusion coefficient in the equation governing the evolution of the order parameter tends to zero. In particular, an existence result has been proved for the limit system in a very general framework. On the contrary, uniqueness was shown by assuming a constant mobility coefficient. Here, we generalize this result and prove a continuous dependence property in the case that the mobility coefficient suitably depends on the chemical potential.

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