Abstract. In this paper we study the dependence of the first eigenvalue of the infinity Laplace with respect to the domain. We prove that this first eigenvalue is continuous under some weak convergence conditions that are fulfilled when a sequence of domains converges in Hausdorff distance. Moreover, it is Lipschitz continuous but not differentiable when we considers deformations obtained via a vector field. Our results are illustrated with simple examples.

In this paper, we study solutions to the max–min mean value problem [Formula: see text] in the Sierpinski Gasket with a prescribed Dirichlet datum at the three vertices of the first triangle. In the previous mean value, formula [Formula: see text] is a vertex of one triangle at one stage in the construction of the Sierpinski Gasket and [Formula: see text] is the set of vertices that are adjacent to [Formula: see text] at that stage. For this problem, it was known that there are existence and uniqueness of a continuous solution, a comparison principle holds, and, moreover, solutions are Lipschitz continuous. Here we continue the analysis of this problem and prove that the solution is piecewise linear on the segments of the Sierpinski Gasket. Moreover, we also show for which values at the three vertices of the first triangle solutions to this mean value formula coincide with infinity harmonic functions.

Purpose -In this article the aim is to propose a new form to densify parallelepipeds of R N by sequences of a-dense curves with accumulated densities. Design/methodology/approach -This will be done by using a basic a-densification technique and adding the new concept of sequence of a-dense curves with accumulated density to improve the resolution of some global optimization problems. Findings -It is found that the new technique based on sequences of a-dense curves with accumulated densities allows to simplify considerably the process consisting on the exploration of the set of optimizer points of an objective function with feasible set a parallelepiped K of R N . Indeed, since the sequence of the images of the curves of a sequence of a-dense curves with accumulated density is expansive, in each new step of the algorithm it is only necessary to explore a residual subset. On the other hand, since the sequence of their densities is decreasing and tends to zero, the convergence of the algorithm is assured. Practical implications -The results of this new technique of densification by sequences of a-dense curves with accumulated densities will be applied to densify the feasible set of an objective function which minimizes the quadratic error produced by the adjustment of a model based on a beta probability density function which is largely used in studies on the transition-time of forest vegetation. Originality/value -A sequence of a-dense curves with accumulated density represents an original concept to be added to the set of techniques to optimize a multivariable function by the reduction to only one variable as a new application of a-dense curves theory to the global optimization.

In this paper we study the solutions to nonlinear mean-value formulas on fractal sets. We focus on the mean-value problem [Formula: see text] in the Sierpiński gasket with prescribed values [Formula: see text], [Formula: see text] and [Formula: see text] at the three vertices of the first triangle. For this problem we show existence and uniqueness of a continuous solution and analyze some properties like the validity of a comparison principle, Lipschitz continuity of solutions (regularity) and continuous dependence of the solution with respect to the prescribed values at the three vertices of the first triangle.