Global minimizing domains for the first eigenvalue of an elliptic operator with non-constant coefficients.
Global regularity for solutions of the minimal surface equation with continuous boundary values
Global solution of optimal shape design problems.
Gradient estimates and Harnack inequalities for solutions to the minimal surface equation
A gradient estimate for solutions to the minimal surface equation can be proved by Partial Differential Equations methods, as in [2]. In such a case, the oscillation of the solution controls its gradient. In the article presented here, the estimate is derived from the Harnack type inequality established in [1]. In our case, the gradient is controlled by the area of the graph of the solution or by the integral of it. These new results are similar to the one announced by Ennio De Giorgi in [3].
Gradient flow for the one-dimensional Mumford-Shah functional
Gradient flows in Wasserstein spaces and applications to crowd movement
Starting from a motivation in the modeling of crowd movement, the paper presents the topics of gradient flows, first in , then in metric spaces, and finally in the space of probability measures endowed with the Wasserstein distance (induced by the quadratic transport cost). Differently from the usual theory by Jordan-Kinderlehrer-Otto and Ambrosio-Gigli-Savaré, we propose an approach where the optimality conditions for the minimizers of the optimization problems that one solves at every time step...
Gradient flows in Wasserstein spaces and applications to crowd movement
Starting from a motivation in the modeling of crowd movement, the paper presents the topics of gradient flows, first in , then in metric spaces, and finally in the space of probability measures endowed with the Wasserstein distance (induced by the quadratic transport cost). Differently from the usual theory by Jordan-Kinderlehrer-Otto and Ambrosio-Gigli-Savaré, we propose an approach where the optimality conditions for the minimizers of the optimization problems that one solves at every time step...
Gradient flows of the entropy for jump processes
We introduce a new transport distance between probability measures on that is built from a Lévy jump kernel. It is defined via a non-local variant of the Benamou–Brenier formula. We study geometric and topological properties of this distance, in particular we prove existence of geodesics. For translation invariant jump kernels we identify the semigroup generated by the associated non-local operator as the gradient flow of the relative entropy w.r.t. the new distance and show that the entropy is...
Gradient theory of phase transitions with boundary contact energy
Ground states in complex bodies
A unified framework for analyzing the existence of ground states in wide classes of elastic complex bodies is presented here. The approach makes use of classical semicontinuity results, Sobolev mappings and cartesian currents. Weak diffeomorphisms are used to represent macroscopic deformations. Sobolev maps and cartesian currents describe the inner substructure of the material elements. Balance equations for irregular minimizers are derived. A contribution to the debate about the role of the balance...
Ground states in complex bodies
A unified framework for analyzing the existence of ground states in wide classes of elastic complex bodies is presented here. The approach makes use of classical semicontinuity results, Sobolev mappings and Cartesian currents. Weak diffeomorphisms are used to represent macroscopic deformations. Sobolev maps and Cartesian currents describe the inner substructure of the material elements. Balance equations for irregular minimizers are derived. A contribution to the debate about the role of the balance...
Gyroids of constant mean curvature.
H1/2 maps with values into the circle : minimal connections, lifting, and the Ginzburg–Landau equation
Hadamard incomplete sensitivity and shape optimization
Hausdorff dimension of uniformly non flat sets with topology.
Higher order Grassmann fibrations and the calculus of variations.
Higher-order phase transitions with line-tension effect
The behavior of energy minimizers at the boundary of the domain is of great importance in the Van de Waals-Cahn-Hilliard theory for fluid-fluid phase transitions, since it describes the effect of the container walls on the configuration of the liquid. This problem, also known as the liquid-drop problem, was studied by Modica in [Ann. Inst. Henri Poincaré, Anal. non linéaire 4 (1987) 487–512], and in a different form by Alberti et al. in [Arch. Rational Mech. Anal.u is a scalar density function and...
Higher-order phase transitions with line-tension effect
The behavior of energy minimizers at the boundary of the domain is of great importance in the Van de Waals-Cahn-Hilliard theory for fluid-fluid phase transitions, since it describes the effect of the container walls on the configuration of the liquid. This problem, also known as the liquid-drop problem, was studied by Modica in [Ann. Inst. Henri Poincaré, Anal. non linéaire4 (1987) 487–512], and in a different form by Alberti et al. in [Arch. Rational Mech. Anal.144 (1998) 1–46] for a first-order...
Hölder regularity of three-dimensional minimal cones in ℝⁿ
We show the local Hölder regularity of Almgren minimal cones of dimension 3 in ℝⁿ away from their centers. The proof is almost elementary but we use the generalized theorem of Reifenberg. In the proof, we give a classification of points away from the center of a minimal cone of dimension 3 in ℝⁿ, into types ℙ, 𝕐 and 𝕋. We then treat each case separately and give a local Hölder parameterization of the cone.
Hölder regularity of two-dimensional almost-minimal sets in
We give a different and probably more elementary proof of a good part of Jean Taylor’s regularity theorem for Almgren almost-minimal sets of dimension in . We use this opportunity to settle some details about almost-minimal sets, extend a part of Taylor’s result to almost-minimal sets of dimension in , and give the expected characterization of the closed sets of dimension in that are minimal, in the sense that for every closed set such that there is a bounded set so that out...