Abstract. We present a detailed analysis of one-dimensional Néel walls in thin uniaxial ferromagnetic films in the presence of an in-plane applied external field in the direction normal to the easy axis. Within the reduced one-dimensional thin film model, we formulate a non-local variational problem whose minimizers are given by one-dimensional Néel wall profiles. We prove existence, uniqueness (up to translations and reflections), regularity, strict monotonicity and the precise asymptotics of the decay of the minimizers in the considered variational problem.Mathematics Subject Classification: 78A30, 35Q60, 82D40

Objective: Anemia is a frequent comorbidity in hemodialysis patients that can be successfully treated by administering erythropoiesis-stimulating agents (ESAs). ESAs dosing is currently based on clinical protocols that often do not account for the high inter-and intra-individual variability in the patient's response. As a result, the hemoglobin level of some patients oscillates around the target range, which is associated with multiple risks and side-effects. This work proposes a methodology based on reinforcement learning (RL) to optimize ESA therapy. Methods: RL is a data-driven approach for solving sequential decision-making problems that are formulated as Markov decision processes (MDPs). Computing optimal drug administration strategies for chronic diseases is a sequential decision-making problem in which the goal is to find the best sequence of drug doses. MDPs are particularly suitable for modeling these problems due to their ability to capture the uncertainty associated with the outcome of the treatment and the stochastic nature of the underlying process. The RL algorithm employed in the proposed methodology is fitted Q iteration, which stands out for its ability to make an efficient use of data. Results:The experiments reported here are based on a computational model that describes the effect of ESAs on the hemoglobin level. The performance of the proposed method is evaluated and compared with the well-known Q-learning algorithm and with a standard protocol. Simulation results show that the performance of Q-learning is substantially lower than FQI and the protocol. When comparing FQI and the protocol, FQI achieves an increment of 27.6% in the proportion of patients that are within the targeted range of hemoglobin during the period of treatment. In addition, the quantity of drug needed is reduced by 5.13%, which indicates a more efficient use of ESAs. Conclusion: Although prospective validation is required, promising results demonstrate the potential of RL to become an alternative to current protocols.

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A variational model proposed in the physics literature to describe the onset of pattern formation in two-component bilayer membranes and amphiphilic monolayers leads to the analysis of a Ginzburg-Landau type energy, precisely, u → Z Ω » W (u) − q |∇u| 2 +˛∇ 2 u˛2 dx. When the stiffness coefficient −q is negative, one expects curvature instabilities of the membrane and, in turn, these instabilities generate a pattern of domains that differ both in composition and in local curvature. Scaling arguments motivate the study of the family of singular perturbed energies u → Fε(u, Ω) := Z Ω » 1 ε W (u) − qε|∇u| 2 + ε 3 |∇ 2 u| 2 dx. Here, the asymptotic behavior of {Fε} is studied using Γ-convergence techniques. In particular, compactness results and an integral representation of the limit energy are obtained.

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We consider the evolution of a polycrystalline material with three or more phases, in the presence of an even crystalline anisotropy. We analyze existence, uniqueness, regularity and stability of the flow. In particular, if the flow becomes unstable at a finite time, we prove that an additional segment (or even an arc) at the triple junction may develop in order to decrease the energy and make the flow stable at subsequent times. We discuss some examples of collapsing situations that lead to changes of topology, such as the collision of two triple junctions.

Abstract. In R m × R n−m , endowed with coordinates X = (x, y), we consider the PDE −div`α(x)|∇u(X)| p(x)−2 ∇u(X)´= f (x, u(X)).We prove a geometric inequality and a symmetry result.EV is supported by MIUR, project "Variational methods and Nonlinear Differential Equations". IntroductionThe purpose of this paper is to give some geometric results on the following problem:, with p(x) ≥ 2 for any x ∈ R m , and Ω is an open subset of R n .As well known, the operator in (1.1) comprises, as main example, the degenerate p(x)-Laplacian (and, in particular, the degenerate p-Laplacian). The motivation of this paper is the following. In [15], it was asked whether or not the level sets of bounded, monotone, global solutions offor X ∈ R n , are flat hyperplanes, at least when n ≤ 8.In spite of the marvelous progress performed in this direction (see, in particular, [43, 8, 31, 32, 7, 5, 46, 16]), part of the conjecture and many related problems are still unsolved (see [27]).In [47], the following generalization of (1.2) was taken into account:where, as above, the notation X = (x, y) ∈ R m × R n−m is used. 1We observe that when f (x, u) does not depend on x, then (1.3) reduces to a usual semilinar equation, of which (1.2) represents the chief example. When f (x, u) depends on x, the dependence on the space variable of f changes only with respect to a subset of the variables, namely the nonlinearity takes no dependence on y.In particular, for fixed u ∈ R, we have that f (x, u) is constant on the "vertical fibers" {x = c}, and for this the nonlinearity in (1.3) is called "fibered". Moreover, the model in (1.3) was considered in [47] as a sort of interpolation between the classical semilinear equation in (1.2) and the boundary reactions PDEs of [11,49], which are related to fractional power operators (see also [12]

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