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Jürgen Sprekels - One of the best experts on this subject based on the ideXlab platform.
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optimal Boundary control of a nonstandard viscous cahn hilliard system with Dynamic Boundary Condition
Nonlinear Analysis-theory Methods & Applications, 2018Co-Authors: Pierluigi Colli, Gianni Gilardi, Jürgen SprekelsAbstract:Abstract In this paper, we study an optimal Boundary control problem for a model for phase separation that was introduced by Podio-Guidugli (2006). The model consists of a strongly coupled system of nonlinear parabolic differential equations, in which products between the unknown functions and their time derivatives occur that are difficult to handle analytically. In contrast to the existing control literature about this PDE system, we consider here a Dynamic Boundary Condition involving the Laplace–Beltrami operator for the order parameter of the system, which models an additional nonconserving phase transition occurring on the surface of the domain. We show the Frechet differentiability of the associated control-to-state operator in appropriate Banach spaces and derive results on the existence of optimal controls and on first-order necessary optimality Conditions in terms of a variational inequality and the adjoint state system.
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Optimal Boundary control of a nonstandard viscous Cahn–Hilliard system with Dynamic Boundary Condition
Nonlinear Analysis, 2018Co-Authors: Pierluigi Colli, Gianni Gilardi, Jürgen SprekelsAbstract:Abstract In this paper, we study an optimal Boundary control problem for a model for phase separation that was introduced by Podio-Guidugli (2006). The model consists of a strongly coupled system of nonlinear parabolic differential equations, in which products between the unknown functions and their time derivatives occur that are difficult to handle analytically. In contrast to the existing control literature about this PDE system, we consider here a Dynamic Boundary Condition involving the Laplace–Beltrami operator for the order parameter of the system, which models an additional nonconserving phase transition occurring on the surface of the domain. We show the Frechet differentiability of the associated control-to-state operator in appropriate Banach spaces and derive results on the existence of optimal controls and on first-order necessary optimality Conditions in terms of a variational inequality and the adjoint state system.
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Optimal Boundary Control of a Nonstandard Cahn–Hilliard System with Dynamic Boundary Condition and Double Obstacle Inclusions
Springer INdAM Series, 2017Co-Authors: Pierluigi Colli, Jürgen SprekelsAbstract: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.
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Optimal Boundary control of a nonstandard Cahn-Hilliard system with Dynamic Boundary Condition and double obstacle inclusions
arXiv: Analysis of PDEs, 2017Co-Authors: Pierluigi Colli, Jürgen SprekelsAbstract: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] 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.
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Optimal Boundary control of a nonstandard viscous Cahn-Hilliard system with Dynamic Boundary Condition
arXiv: Analysis of PDEs, 2016Co-Authors: Pierluigi Colli, Gianni Gilardi, Jürgen SprekelsAbstract: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 Podio-Guidugli in Ric. Mat. 55 (2006), pp. 105-118. The model consists of a strongly coupled system of nonlinear parabolic differential equations, in which products between the unknown functions and their time derivatives occur that are difficult to handle analytically. In contrast to the existing control literature about this PDE system, we consider here a Dynamic Boundary Condition involving the Laplace-Beltrami operator for the order parameter of the system, which models an additional nonconserving phase transition occurring on the surface of the domain. We show the Frechet differentiability of the associated control-to-state operator in appropriate Banach spaces and derive results on the existence of optimal controls and on first-order necessary optimality Conditions in terms of a variational inequality and the adjoint state system.
Pierluigi Colli - One of the best experts on this subject based on the ideXlab platform.
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Vanishing diffusion in a Dynamic Boundary Condition for the Cahn-Hilliard equation
Nonlinear Differential Equations and Applications NoDEA, 2020Co-Authors: Pierluigi Colli, Takeshi FukaoAbstract:The initial Boundary value problem for a Cahn–Hilliard system subject to a Dynamic Boundary Condition of Allen–Cahn type is treated. The vanishing of the surface diffusion on the Dynamic Boundary Condition is the point of emphasis. By the asymptotic analysis as the diffusion coefficient tends to 0, one can expect that the solutions of the surface diffusion problem converge to the solution of the problem without the surface diffusion. This is actually the case, but the solution of the limiting problem naturally looses some regularity. Indeed, the system we investigate is rather complicate due to the presence of nonlinear terms including general maximal monotone graphs both in the bulk and on the Boundary. The two graphs are related each to the other by a growth Condition, with the Boundary graph that dominates the other one. In general, at the asymptotic limit a weaker form of the Boundary Condition is obtained, but in the case when the two graphs exhibit the same growth the Boundary Condition still holds almost everywhere.
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on a transmission problem for equation and Dynamic Boundary Condition of cahn hilliard type with nonsmooth potentials
arXiv: Analysis of PDEs, 2019Co-Authors: Pierluigi Colli, Takeshi FukaoAbstract:This paper is concerned with well-posedness of the Cahn-Hilliard equation subject to a class of new Dynamic Boundary Conditions. The system was recently derived in Liu-Wu (Arch. Ration. Mech. Anal. 233 (2019), 167-247) via an energetic variational approach and it naturally fulfills three physical constraints such as mass conservation, energy dissipation and force balance. The target problem examined in this paper can be viewed as a transmission problem that consists of Cahn-Hilliard type equations both in the bulk and on the Boundary. In our approach, we are able to deal with a general class of potentials with double-well structure, including the physically relevant logarithmic potential and the non-smooth double-obstacle potential. Existence, uniqueness and continuous dependence of global weak solutions are established. The proof is based on a novel time-discretization scheme for the approximation of the continuous problem. Besides, a regularity result is shown with the aim of obtaining a strong solution to the system.
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optimal Boundary control of a nonstandard viscous cahn hilliard system with Dynamic Boundary Condition
Nonlinear Analysis-theory Methods & Applications, 2018Co-Authors: Pierluigi Colli, Gianni Gilardi, Jürgen SprekelsAbstract:Abstract In this paper, we study an optimal Boundary control problem for a model for phase separation that was introduced by Podio-Guidugli (2006). The model consists of a strongly coupled system of nonlinear parabolic differential equations, in which products between the unknown functions and their time derivatives occur that are difficult to handle analytically. In contrast to the existing control literature about this PDE system, we consider here a Dynamic Boundary Condition involving the Laplace–Beltrami operator for the order parameter of the system, which models an additional nonconserving phase transition occurring on the surface of the domain. We show the Frechet differentiability of the associated control-to-state operator in appropriate Banach spaces and derive results on the existence of optimal controls and on first-order necessary optimality Conditions in terms of a variational inequality and the adjoint state system.
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Optimal Boundary control of a nonstandard viscous Cahn–Hilliard system with Dynamic Boundary Condition
Nonlinear Analysis, 2018Co-Authors: Pierluigi Colli, Gianni Gilardi, Jürgen SprekelsAbstract:Abstract In this paper, we study an optimal Boundary control problem for a model for phase separation that was introduced by Podio-Guidugli (2006). The model consists of a strongly coupled system of nonlinear parabolic differential equations, in which products between the unknown functions and their time derivatives occur that are difficult to handle analytically. In contrast to the existing control literature about this PDE system, we consider here a Dynamic Boundary Condition involving the Laplace–Beltrami operator for the order parameter of the system, which models an additional nonconserving phase transition occurring on the surface of the domain. We show the Frechet differentiability of the associated control-to-state operator in appropriate Banach spaces and derive results on the existence of optimal controls and on first-order necessary optimality Conditions in terms of a variational inequality and the adjoint state system.
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Optimal Boundary Control of a Nonstandard Cahn–Hilliard System with Dynamic Boundary Condition and Double Obstacle Inclusions
Springer INdAM Series, 2017Co-Authors: Pierluigi Colli, Jürgen SprekelsAbstract: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.
Takeshi Fukao - One of the best experts on this subject based on the ideXlab platform.
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Vanishing diffusion in a Dynamic Boundary Condition for the Cahn-Hilliard equation
Nonlinear Differential Equations and Applications NoDEA, 2020Co-Authors: Pierluigi Colli, Takeshi FukaoAbstract:The initial Boundary value problem for a Cahn–Hilliard system subject to a Dynamic Boundary Condition of Allen–Cahn type is treated. The vanishing of the surface diffusion on the Dynamic Boundary Condition is the point of emphasis. By the asymptotic analysis as the diffusion coefficient tends to 0, one can expect that the solutions of the surface diffusion problem converge to the solution of the problem without the surface diffusion. This is actually the case, but the solution of the limiting problem naturally looses some regularity. Indeed, the system we investigate is rather complicate due to the presence of nonlinear terms including general maximal monotone graphs both in the bulk and on the Boundary. The two graphs are related each to the other by a growth Condition, with the Boundary graph that dominates the other one. In general, at the asymptotic limit a weaker form of the Boundary Condition is obtained, but in the case when the two graphs exhibit the same growth the Boundary Condition still holds almost everywhere.
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Separation property and convergence to equilibrium for the equation and Dynamic Boundary Condition of Cahn-Hilliard type with singular potential
Asymptotic Analysis, 2020Co-Authors: Takeshi FukaoAbstract:We consider a class of Cahn–Hilliard equation that models phase separation process of binary mixtures involving nontrivial Boundary interactions in a bounded domain with non-permeable wall. The system is characterized by certain Dynamic type Boundary Conditions and the total mass, in the bulk and on the Boundary, is conserved for all time. For the case with physically relevant singular (e.g., logarithmic) potential, global regularity of weak solutions is established. In particular, when the spatial dimension is two, we show the instantaneous strict separation property such that for arbitrary positive time any weak solution stays away from the pure phases ± 1, while in the three dimensional case, an eventual separation property for large time is obtained. As a consequence, we prove that every global weak solution converges to a single equilibrium as t → ∞, by the usage of an extended Łojasiewicz–Simon inequality.
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A second-order accurate structure-preserving scheme for the Cahn-Hilliard equation with a Dynamic Boundary Condition.
arXiv: Numerical Analysis, 2020Co-Authors: Makoto Okumura, Takeshi Fukao, Daisuke Furihata, Shuji YoshikawaAbstract:We propose a structure-preserving finite difference scheme for the Cahn-Hilliard equation with a Dynamic Boundary Condition using the discrete variational derivative method (DVDM). In this approach, it is important and essential how to discretize the energy which characterizes the equation. By modifying the conventional manner and using an appropriate summation-by-parts formula, we can use a standard central difference operator as an approximation of an outward normal derivative on the discrete Boundary Condition of the scheme. We show that our proposed scheme is second-order accurate in space, although the previous structure-preserving scheme by Fukao-Yoshikawa-Wada (Commun. Pure Appl. Anal. 16 (2017), 1915-1938) is first-order accurate in space. Also, we show the stability, the existence, and the uniqueness of the solution for the proposed scheme. Computation examples demonstrate the effectiveness of the proposed scheme. Especially through computation examples, we confirm that numerical solutions can be stably obtained by our proposed scheme.
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separation property and convergence to equilibrium for the equation and Dynamic Boundary Condition of cahn hilliard type with singular potential
arXiv: Analysis of PDEs, 2019Co-Authors: Takeshi FukaoAbstract:We consider a class of Cahn-Hilliard equation that models phase separation process of binary mixtures involving nontrivial Boundary interactions in a bounded domain with non-permeable wall. The system is characterized by certain Dynamic type Boundary Conditions and the total mass, in the bulk and on the Boundary, is conserved for all time. For the case with physically relevant singular (e.g., logarithmic) potential, global regularity of weak solutions is established. In particular, when the spatial dimension is two, we show the instantaneous strict separation property such that for arbitrary positive time any weak solution stays away from the pure phases +1 and -1, while in the three dimensional case, an eventual separation property for large time is obtained. As a consequence, we prove that every global weak solution converges to a single equilibrium as the time goes to infinity, by the usage of an extended Lojasiewicz-Simon inequality.
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on a transmission problem for equation and Dynamic Boundary Condition of cahn hilliard type with nonsmooth potentials
arXiv: Analysis of PDEs, 2019Co-Authors: Pierluigi Colli, Takeshi FukaoAbstract:This paper is concerned with well-posedness of the Cahn-Hilliard equation subject to a class of new Dynamic Boundary Conditions. The system was recently derived in Liu-Wu (Arch. Ration. Mech. Anal. 233 (2019), 167-247) via an energetic variational approach and it naturally fulfills three physical constraints such as mass conservation, energy dissipation and force balance. The target problem examined in this paper can be viewed as a transmission problem that consists of Cahn-Hilliard type equations both in the bulk and on the Boundary. In our approach, we are able to deal with a general class of potentials with double-well structure, including the physically relevant logarithmic potential and the non-smooth double-obstacle potential. Existence, uniqueness and continuous dependence of global weak solutions are established. The proof is based on a novel time-discretization scheme for the approximation of the continuous problem. Besides, a regularity result is shown with the aim of obtaining a strong solution to the system.
Hideki Sano - One of the best experts on this subject based on the ideXlab platform.
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Stability analysis of the telegrapher’s equations with Dynamic Boundary Condition
Systems & Control Letters, 2018Co-Authors: Hideki SanoAbstract:Abstract This paper is concerned with the stability analysis of a distributed parameter circuit with Dynamic Boundary Condition. The distributed parameter circuit is written by the telegrapher’s equations whose Boundary Condition is described by an ordinary differential equation. First of all, it is shown that, for any physical parameters of the circuit, the system operator generates an exponentially stable C 0 -semigroup on a Hilbert space. However, it is not clear whether the decay rate of the semigroup is the most precise one. In this paper, the spectral analysis is conducted for the system satisfying the distortionless Condition, and it is shown that the semigroup satisfies the spectrum determined growthCondition.
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stability analysis of the telegrapher s equations with Dynamic Boundary Condition
Systems & Control Letters, 2018Co-Authors: Hideki SanoAbstract:Abstract This paper is concerned with the stability analysis of a distributed parameter circuit with Dynamic Boundary Condition. The distributed parameter circuit is written by the telegrapher’s equations whose Boundary Condition is described by an ordinary differential equation. First of all, it is shown that, for any physical parameters of the circuit, the system operator generates an exponentially stable C 0 -semigroup on a Hilbert space. However, it is not clear whether the decay rate of the semigroup is the most precise one. In this paper, the spectral analysis is conducted for the system satisfying the distortionless Condition, and it is shown that the semigroup satisfies the spectrum determined growthCondition.
Gianni Gilardi - One of the best experts on this subject based on the ideXlab platform.
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optimal Boundary control of a nonstandard viscous cahn hilliard system with Dynamic Boundary Condition
Nonlinear Analysis-theory Methods & Applications, 2018Co-Authors: Pierluigi Colli, Gianni Gilardi, Jürgen SprekelsAbstract:Abstract In this paper, we study an optimal Boundary control problem for a model for phase separation that was introduced by Podio-Guidugli (2006). The model consists of a strongly coupled system of nonlinear parabolic differential equations, in which products between the unknown functions and their time derivatives occur that are difficult to handle analytically. In contrast to the existing control literature about this PDE system, we consider here a Dynamic Boundary Condition involving the Laplace–Beltrami operator for the order parameter of the system, which models an additional nonconserving phase transition occurring on the surface of the domain. We show the Frechet differentiability of the associated control-to-state operator in appropriate Banach spaces and derive results on the existence of optimal controls and on first-order necessary optimality Conditions in terms of a variational inequality and the adjoint state system.
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Optimal Boundary control of a nonstandard viscous Cahn–Hilliard system with Dynamic Boundary Condition
Nonlinear Analysis, 2018Co-Authors: Pierluigi Colli, Gianni Gilardi, Jürgen SprekelsAbstract:Abstract In this paper, we study an optimal Boundary control problem for a model for phase separation that was introduced by Podio-Guidugli (2006). The model consists of a strongly coupled system of nonlinear parabolic differential equations, in which products between the unknown functions and their time derivatives occur that are difficult to handle analytically. In contrast to the existing control literature about this PDE system, we consider here a Dynamic Boundary Condition involving the Laplace–Beltrami operator for the order parameter of the system, which models an additional nonconserving phase transition occurring on the surface of the domain. We show the Frechet differentiability of the associated control-to-state operator in appropriate Banach spaces and derive results on the existence of optimal controls and on first-order necessary optimality Conditions in terms of a variational inequality and the adjoint state system.
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Optimal Boundary control of a nonstandard viscous Cahn-Hilliard system with Dynamic Boundary Condition
arXiv: Analysis of PDEs, 2016Co-Authors: Pierluigi Colli, Gianni Gilardi, Jürgen SprekelsAbstract: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 Podio-Guidugli in Ric. Mat. 55 (2006), pp. 105-118. The model consists of a strongly coupled system of nonlinear parabolic differential equations, in which products between the unknown functions and their time derivatives occur that are difficult to handle analytically. In contrast to the existing control literature about this PDE system, we consider here a Dynamic Boundary Condition involving the Laplace-Beltrami operator for the order parameter of the system, which models an additional nonconserving phase transition occurring on the surface of the domain. We show the Frechet differentiability of the associated control-to-state operator in appropriate Banach spaces and derive results on the existence of optimal controls and on first-order necessary optimality Conditions in terms of a variational inequality and the adjoint state system.
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Global existence for a nonstandard viscous Cahn-Hilliard system with Dynamic Boundary Condition
arXiv: Analysis of PDEs, 2016Co-Authors: Pierluigi Colli, Gianni Gilardi, Jürgen SprekelsAbstract:In this paper, we study a model for phase segregation taking place in a spatial domain that was introduced by Podio-Guidugli in Ric. Mat. 55 (2006), pp. 105-118. The model consists of a strongly coupled system of nonlinear parabolic differential equations, in which products between the unknown functions and their time derivatives occur that are diffcult to handle analytically. In contrast to the existing literature about this PDE system, we consider here a Dynamic Boundary Condition involving the Laplace-Beltrami operator for the order parameter. This Boundary Condition models an additional nonconserving phase transition occurring on the surface of the domain. Different well-posedness results are shown, depending on the smoothness properties of the involved bulk and surface free energies.
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A Boundary control problem for a possibly singular phase field system with Dynamic Boundary Conditions
Journal of Mathematical Analysis and Applications, 2016Co-Authors: Pierluigi Colli, Gianni Gilardi, Gabriela MarinoschiAbstract:Abstract This paper deals with an optimal control problem related to a phase field system of Caginalp type with a Dynamic Boundary Condition for the temperature. The control placed in the Dynamic Boundary Condition acts on a part of the Boundary. The analysis carried out in this paper proves the existence of an optimal control for a general class of potentials, possibly singular. The study includes potentials for which the derivatives may not exist, these being replaced by well-defined subdifferentials. Under some stronger assumptions on the structure parameters and on the potentials (namely for the regular and the logarithmic case having single-valued derivatives), the first order necessary optimality Conditions are derived and expressed in terms of the Boundary trace of the first adjoint variable.
