An Epiperimetric Inequality for the Regularity of Some Free Boundary Problems: The 2-Dimensional Case

Using a direct approach, we prove a two-dimensional epiperimetric inequality for the one-phase problem in the scalar and vectorial cases and for the double-phase problem. From this we deduce, in dimension 2, the C1,α regularity of the free boundary in the scalar one-phase and double-phase problems, and of the reduced free boundary in the vectorial case, without any restriction on the sign of the component functions. Furthermore, we show that in the vectorial case each connected component of {|u|=0} might have cusps, but they must be a finite number. © 2018 Wiley Periodicals, Inc

On the logarithmic epiperimetric inequality for the obstacle problem

We give three different proofs of the log-epiperimetric inequality at singular points for the obstacle problem. In the first, direct proof, we write the competitor explicitly; the second proof is also constructive, but this time the competitor is given through the solution of an evolution problem on the sphere. We compare the competitors obtained in the different proofs and their relation to other similar results that appeared recently. Finally, in the appendix, we give a general theorem, which can be applied also in other contexts and in which the construction of the competitor is reduced to finding a flow satisfying two differential inequalities

Free boundary regularity for a multiphase shape optimization problem

In this paper we prove a C1,α regularity result in dimension two for almost-minimizers of the constrained one-phase Alt-Caffarelli and the two-phase Alt-Caffarelli-Friedman functionals for an energy with variable coefficients. As a consequence, we deduce the complete regularity of solutions of a multiphase shape optimization problem for the first eigenvalue of the Dirichlet Laplacian, up to the boundary of a fixed domain that acts as a geometric inclusion constraint. One of the main ingredients is a new application of the (one-phase) epiperimetric inequality up to the boundary of the constraint. While the framework that leads to this application is valid in every dimension, the epiperimetric inequality is known only in dimension two, thus the restriction on the dimension

Almost everywhere uniqueness of blow-up limits for the lower dimensional obstacle problem

We answer a question left open in [4] and [3], by proving that the blow-up of minimizers u of the lower dimensional obstacle problem is unique at generic point of the free boundary

Shaping and Dilating the Fitness Landscape for Parameter Estimation in Stochastic Biochemical Models

The parameter estimation (PE) of biochemical reactions is one of the most challenging tasks in systems biology given the pivotal role of these kinetic constants in driving the behavior of biochemical systems. PE is a non-convex, multi-modal, and non-separable optimization problem with an unknown fitness landscape; moreover, the quantities of the biochemical species appearing in the system can be low, making biological noise a non-negligible phenomenon and mandating the use of stochastic simulation. Finally, the values of the kinetic parameters typically follow a log-uniform distribution; thus, the optimal solutions are situated in the lowest orders of magnitude of the search space. In this work, we further elaborate on a novel approach to address the PE problem based on a combination of adaptive swarm intelligence and dilation functions (DFs). DFs require prior knowledge of the characteristics of the fitness landscape; therefore, we leverage an alternative solution to evolve optimal DFs. On top of this approach, we introduce surrogate Fourier modeling to simplify the PE, by producing a smoother version of the fitness landscape that excludes the high frequency components of the fitness function. Our results show that the PE exploiting evolved DFs has a performance comparable with that of the PE run with a custom DF. Moreover, surrogate Fourier modeling allows for improving the convergence speed. Finally, we discuss some open problems related to the scalability of our methodology

Biochemical parameter estimation vs. benchmark functions: A comparative study of optimization performance and representation design

© 2019 Elsevier B.V. Computational Intelligence methods, which include Evolutionary Computation and Swarm Intelligence, can efficiently and effectively identify optimal solutions to complex optimization problems by exploiting the cooperative and competitive interplay among their individuals. The exploration and exploitation capabilities of these meta-heuristics are typically assessed by considering well-known suites of benchmark functions, specifically designed for numerical global optimization purposes. However, their performances could drastically change in the case of real-world optimization problems. In this paper, we investigate this issue by considering the Parameter Estimation (PE) of biochemical systems, a common computational problem in the field of Systems Biology. In order to evaluate the effectiveness of various meta-heuristics in solving the PE problem, we compare their performance by considering a set of benchmark functions and a set of synthetic biochemical models characterized by a search space with an increasing number of dimensions. Our results show that some state-of-the-art optimization methods – able to largely outperform the other meta-heuristics on benchmark functions – are characterized by considerably poor performances when applied to the PE problem. We also show that a limiting factor of these optimization methods concerns the representation of the solutions: indeed, by means of a simple semantic transformation, it is possible to turn these algorithms into competitive alternatives. We corroborate this finding by performing the PE of a model of metabolic pathways in red blood cells. Overall, in this work we state that classic benchmark functions cannot be fully representative of all the features that make real-world optimization problems hard to solve. This is the case, in particular, of the PE of biochemical systems. We also show that optimization problems must be carefully analyzed to select an appropriate representation, in order to actually obtain the performance promised by benchmark results

Stellar population and the origin of intra-cluster stars around brightest cluster galaxies: the case of NGC 3311

Context. We investigate the stellar population and the origin of diffuse light around brightest cluster galaxies. Aims. We study the stellar population of the dynamically hot stellar halo of NGC 3311, the brightest galaxy in the Hydra I cluster, and that of photometric substructures in the diffuse light to constrain the origin of these components. Methods. We analyze absorption lines in medium-resolution, long-slit spectra in the wavelength range 4800-5800 angstrom obtained with FORS2 at the Very Large Telescope. We measure the equivalent width of Lick indices out to 20 kpc from the center of NGC 3311 and fit them with stellar population models that account for the [alpha/Fe] overabundance. Results. Stars in the dynamically hot halo of NGC 3311 are old (age > 13 Gyr), metal-poor ([Z/H] ~ -0.35), and alpha-enhanced ([alpha/Fe] ~ 0.48). Together with the high velocity dispersion, these measurements indicate that the stars in the halo were accreted from the outskirts of other early-type galaxies, with a possible contribution from dwarf galaxies. We identify a region in the halo of NGC 3311 associated with a photometric substructure where the stellar population is even more metal-poor ([Z/H] ~ -0.73). In this region, our measurements are consistent with a composite stellar population superposed along the line of sight, consisting of stars from the dynamically hot halo of NGC 3311 and stars stripped from dwarf galaxies. The latter component contributes < 28% to the local surface brightness. Conclusions. The build-up of diffuse light around NGC 3311 is on-going. Based on the observed stellar population properties, the dominant part of these stars may have come from the outskirts of bright early-type galaxies, while stars from stripped dwarf galaxies are presently being added.Comment: 8 pages, 4 figures. Accepted for publication in Astronomy & Astrophysic