First, we study the existence of solutions for a class of first order mean field games systems \begin{equation*} \left\{\begin{aligned} &H(x,u,Du)=F(x,m(t)),\quad &&x\in M,\ \forall\ t\in[0,T],\\ &\partial_t m-\text{div}\left(m\dfrac{\partial H}{\partial p}(x,u,Du)\right)=0,\quad &&(x,t)\in M\times(0,T],\\ &m(0)=m_0, \end{aligned}\right. \end{equation*} where the system comprises a stationary Hamilton-Jacobi equation in the contact case and an evolutionary continuity equation. Then, for any fixed $\lambda>0$, let $(u^\lambda,m^\lambda)$ be a solution of the system \begin{equation*} \left\{ \begin{aligned} &H(x,\lambda u^\lambda,Du^\lambda)=F(x,m^\lambda(t))+c(m^\lambda(t)),\quad &&x\in M,\ \forall t\in[0,T],\\ &\partial_t m^\lambda-\text{div}\left(m^\lambda\dfrac{\partial H}{\partial p}(x,\lambda u^\lambda,Du^\lambda)\right)=0,\quad &&(x,t)\in M\times(0,T],\\ &m(0)=m_0, \end{aligned}\right. \end{equation*} where $c(m^\lambda(t))$ is the Ma\~n\'e critical value of the Hamiltonian $H(x,0,p)-F(x,m^\lambda(t))$. We investigate the selection problem for the limit of $(u^\lambda,m^\lambda)$ as $\lambda$ tends to 0.

This paper deals with the obstacle problem for the fractional infinity Laplacian with nonhomogeneous term $f(u)$, where $f:\mathbb{R}^+ \mapsto \mathbb{R}^+$: $$\begin{cases} L[u]=f(u) &\qquad in \,\,\,\,\, \{u>0\}\\ u \geq 0 &\qquad in \,\,\,\,\, \Omega\\ u=g &\qquad on\,\,\,\,\, \partial \Omega\end{cases},$$ with [[EQUATION]]\\ Under the assumptions that $f$ is a continuous and monotone function and that the boundary datum $g$ is in $C^{0,\beta}(\partial\Omega)$ for some $0<\beta<\alpha$, we prove existence of a solution $u$ to this problem. Moreover, this solution $u$ is $\beta-$H\"olderian on $\overline{\Omega}$. Our proof is based on an approximation of $f$ by an appropriate sequence of functions $f_\varepsilon$ where we prove using Perron's method the existence of solutions $u_\varepsilon$, for every $\varepsilon>0$. Then, we show some uniform H\"older estimates on $u_\varepsilon$ that guarantee that $u_\varepsilon \rightarrow u$ where this limit function $u$ turns out to be a solution to our obstacle problem.

In this paper we study the optimal control of an initial-boundary value problem for the classical nonviscous Cahn–Hilliard system with zero Neumann boundary conditions. Phase field systems of this type govern the evolution of diffusive phase transition processes with conserved order parameter. For such systems, optimal control problems have been studied in the past. We deal with the situation when the cost functional of the optimal control problem contains a sparsity-enhancing nondifferentiable term like the L1-norm. We establish first-order necessary and second-order sufficient optimality conditions for locally optimal controls. Since in the case without viscosity the solutions to the state and adjoint systems are considerably less regular than in the viscous case, some technical difficulties have to be overcome, and additional conditions have to be imposed. In particular, we have to restrict ourselves to the case when the nonlinearity driving the phase separation is regular, while in the presence of a viscosity term also nonlinearities of logarithmic type could be admitted. In addition, the implicit function theorem, which was employed to establish the needed differentiability properties of the control-to-state operator in the viscous case, does not apply in our situation and has to be substituted by other arguments.

This article deals the optimal design of a region [[EQUATION]] of the boundary [[EQUATION]] of a given domain [[EQUATION]], supporting a particular type of boundary conditions of a physical boundary value problem. We develop suitable notions of shape and topological derivatives, accounting for the sensitivity of a quantity [[EQUATION]] depending on a region [[EQUATION]]: on the one hand, we evaluate the sensitivity of [[EQUATION]] with respect to ``small'' perturbations of the boundary of [[EQUATION]] within [[EQUATION]]. On the other hand, we appraise the sensitivity of [[EQUATION]] with respect to the addition of a ``small'' surface disk to the region [[EQUATION]]. The calculation of both derivatives raises original difficulties, due to the weakly singularities caused by the change in boundary conditions in a boundary value problem. These aspects are detailed in a simple setting based on the conductivity equation. We notably propose formal arguments to calculate our derivatives, and show how they can be generalized to more intricate contexts, such as those of the Helmholtz equation and the linear elasticity system. Our derivatives are incorporated into a recent numerical framework for tracking arbitrarily dramatic motions of a region [[EQUATION]] within a fixed ambient surface. Finally, various 3d numerical examples are presented.

We investigate finding a map $g$ within a function class $G$ that minimises an Optimal Transport (OT) cost between a target measure $\nu$ and the image by $g$ of a source measure $\mu$. This is relevant when an OT map from $\mu$ to $\nu$ does not exist or does not satisfy the desired constraints of $G$. We address existence and uniqueness for generic subclasses of $L$-Lipschitz functions, including gradients of (strongly) convex functions and typical Neural Networks. We explore a variant that approaches a transport plan, showing equivalence to a map problem in some cases. For the squared Euclidean cost, we propose alternating minimisation over a transport plan $\pi$ and map $g$, with the optimisation over $g$ being the $L^2$ projection on $G$ of the barycentric mapping $\overline{\pi}$. In dimension one, this global problem equates the $L^2$ projection of $\overline{\pi^*}$ onto $G$ for an OT plan $\pi^*$ between $\mu$ and $\nu$, but this does not extend to higher dimensions. We introduce a simple kernel method to find $g$ within a Reproducing Kernel Hilbert Space in the discrete case. We present numerical methods for $L$-Lipschitz gradients of $\ell$-strongly convex potentials, and study the convergence of SGD methods for Neural Networks.

A suite of input-to-state stability results are presented for a class of forced differential inclusions, so-called Lur'e inclusions. As a consequence, semi-global incremental input-to-state stability results for systems of forced Lur'e differential equations are derived. The results are in the spirit of the passivity theorem from control theory as both the linear and nonlinear components of the Lur'e inclusion (or equation) are assumed to satisfy passivity-type conditions. These results provide a basis for the analysis of forced Lur'e differential equations subject to (almost) periodic forcing terms and, roughly speaking, ensure the existence and attractivity of (almost) periodic state- and output-responses, comprising another focus of the present work. One ultimate aim of the study is to provide a robust and rigorous theoretical foundation for a well-defined and tractable ``frequency response'' of forced Lur'e systems.

Two flat sub-Lorentzian problems on the Martinet distribution are studied. For the first one, the attainable set has a nontrivial intersection with the Martinet plane, but for the second one it does not. Attainable sets, optimal trajectories, sub-Lorentzian distances and spheres are described.

In this paper, the asymptotic behavior of strongly coupled hyperbolic equations with one infinite memory term is investigated, one specific case of which is the model for describing the dynamical behavior of magnetic effected piezoelectric beams. A fractional operator is involved in the memory term depending on the parameter a ∈ [0, 1). By means of frequency domain analysis, it is proved that the system can be indirectly stabilized polynomially by only one infinite memory term located on one of these two strongly coupled PDEs, and the explicit decay rate given as t−1/2−2a is only dependent on the parameter a. When considering the exponentially decreasing kernel, a detailed spectral analysis for the system operator is further provided. Specifically, the asymptotic expressions of the spectrum for the system operator are derived. Based on the expressions of the spectrum, the optimality of the obtained decay rate is further verified for this system.

We generalize the shape optimization problem for the existence of stable equilibrium configurations of nematic and cholesteric liquid crystal drops surrounded by an isotropic solution to include a broader family of admissible domains with inner boundaries, allowing discontinuities in the director field across them. Within this setting, we prove the existence of optimal configurations under a volume constraint and show that the minimization problem is a natural generalization of that posed for regular domains.

We investigate the influence of surfactants on stabilizing the formation of interfaces in nonlocal anisotropic two-phase fluid at equilibrium. The analysis focuses on singularly perturbed non-local van der Waals–Cahn–Hillard-type energies, supplemented with a term that accounts for the interaction between the surfactant and the fluid. We derive by Γ-convergence the effective surface tension model as the thickness of the transition layer vanishes and show that it decreases when the surfactant segregates to the interface.