1,720,985 research outputs found
Uniform energy and density distribution: diblock copolymers' functional
No full textWe study a nonlocal variational problem arising in diblock copolymers models, whose energy is given by the Cahn–Hilliard functional plus a long-range interaction term. We prove that minimizers develop uniform energy and density distributions, thus justifying partially the highly regular microphase separation observed in diblock copolymers’ melts. We also give a new proof of the scaling law for the minimum energy. This work extends the techniques introduced in [1] where analogous results are proved for the sharp interface limit of the functional considered
Non-uniqueness of minimizers for strictly polyconvex functionals
Full text linkIn this note we solve a problem posed by Ball (in Philos Trans R Soc Lond Ser A 306(1496):557–611, 1982) about the uniqueness of smooth equilibrium solutions to boundary value problems for strictly polyconvex functionals,
where Ω is homeomorphic to a ball.
We give several examples of non-uniqueness. The main example is a boundary value problem with at least two different global minimizers, both analytic up to the boundary. All these examples are suggested by the theory of minimal surfaces
Mean-convex sets and minimal barriers
Full text linkA mean-convex set is locally a barrier for minimal surfaces but can fail to be a global barrier. In this note we suggest how to extend to general dimensions the results of a previous unpublished manuscript on the characterization of the global barriers for minimal surfaces
The local structure of the free boundary in the fractional obstacle problem
No full textBuilding upon the recent results in [M. Focardi and E. Spadaro, On the measure and the structure of the free boundary of the lower-dimensional obstacle problem, Arch. Ration. Mech. Anal. 230 2018, 1, 125-184] we provide a thorough description of the free boundary for the solutions to the fractional obstacle problem in Rn+1 with obstacle function φ (suitably smooth and decaying fast at infinity) up to sets of null Hn-1 measure. In particular, if φ is analytic, the problem reduces to the zero obstacle case dealt with in [M. Focardi and E. Spadaro, On the measure and the structure of the free boundary of the lower-dimensional obstacle problem, Arch. Ration. Mech. Anal. 230 2018, 1, 125-184] and therefore we retrieve the same results: Local finiteness of the (n-1)-dimensional Minkowski content of the free boundary (and thus of its Hausdorff measure), Hn-1-rectifiability of the free boundary, classification of the frequencies and of the blowups up to a set of Hausdorff dimension at most (n-2) in the free boundary. Instead, if φ ∈ Ck+1(Rn), k ≥ 2, similar results hold only for distinguished subsets of points in the free boundary where the order of contact of the solution with the obstacle function φ is less than k + 1
Regularity of solutions to nonlinear thin and boundary obstacle problems
Full text linkVariational inequalities with thin obstacles and Signorini-type boundary conditions are classical problems in the calculus of variations, arising in numerous applications. In the linear case many refined results are known, while in the nonlinear setting our understanding is still at a preliminary stage. In this paper we prove C1 regularity for the solutions to a general class of quasi-linear variational inequalities with thin obstacles and C1,α regularity for variational inequalities under Signorini-type conditions on the boundary of a domain
How a minimal surface leaves a thin obstacle
Full text linkWe prove the optimal regularity and a detailed analysis of the free boundary of the solutions to the thin obstacle problem for nonparametric minimal surfaces with flat obstacles
Long time behavior of discrete volume preserving mean curvature flows
Full text linkIn this paper we analyze the Euler implicit scheme for the volume preserving mean curvature flow. We prove the exponential convergence of the scheme to a finite union of disjoint balls with equal volume for any bounded initial set with finite perimeter
Monotonicity formulas for obstacle problems with Lipschitz coefficients
Full text linkWe prove quasi-monotonicity formulas for classical obstacle-type problems with energies being the sum of a quadratic form with Lipschitz coefficients, and a Hölder continuous
linear term.With the help of those formulas we are able to carry out the full analysis of the regularity of free-boundary points following the approaches by Caffarelli (J Fourier Anal
Appl 4(4–5), 383–402, 1998), Monneau (J Geom Anal 13(2), 359–389, 2003), and Weiss (Invent Math 138(1), 23–50, 1999)
Going Beyond Counting First Authors in Author Co-citation Analysis
Full text linkThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
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