We discuss the local minimality of certain configurations for a nonlocal isoperimetric problem us... more We discuss the local minimality of certain configurations for a nonlocal isoperimetric problem used to model microphase separation in diblock copolymer melts. We show that critical configurations with positive second variation are local minimizers of the nonlocal area functional and, in fact, satisfy a quantitative isoperimetric inequality with respect to sets that are L^1-close. The link with local minimizers for the diffuse-interface Ohta-Kawasaki energy is also discussed. As a byproduct of the quantitative estimate, we get new results concerning periodic local minimizers of the area functional and a proof, via second variation, of the sharp quantitative isoperimetric inequality in the standard Euclidean case. As a further application, we address the global and local minimality of certain lamellar configurations.
Abstract.We discuss a variational problem defined on couples of functions that are constrained to... more Abstract.We discuss a variational problem defined on couples of functions that are constrained to take values into the 2-dimensional unit sphere. The energy functional contains, besides standard Dirichlet energies, a non-local interaction term that depends on the distance between the gradients of the two functions. Different gradients are preferred or penalized in this model, in dependence of the sign of the interaction term. In this paper we study the lower semicontinuity and the coercivity of the energy and we find an explicit representation formula for the relaxed energy. Moreover, we discuss the behavior of the energy in the case when we consider multifunctions with two leaves rather than couples of functions.
In order to apply the direct methods of the calculus of variations to this class of functionals, ... more In order to apply the direct methods of the calculus of variations to this class of functionals, a first problem to be solved is the identification of qualitative conditions on the supremand f which imply the lower semicontinuity with respect to a convergence weak enough to provide the compactness in a large number of situations, say the weak* L∞ convergence. This was already solved by Barron and Liu in [3] where they showed that a functional of the form (1) is weakly* L∞ sequentially lower semicontinuous if and only if the function
defined on the space SBV (Ω) of special functions of bounded variation are studied, where ∇u deno... more defined on the space SBV (Ω) of special functions of bounded variation are studied, where ∇u denotes the approximate gradient of u, and Su is the set of the discontinuity points of u. In a two-dimensional setting, Su represents the contours of the object in a picture and u is a smoothing of an imput image. Energies of the same form arise in fracture mechanics for brittle solids, where Su is interpreted as the crack surface and u as the displacement outside the fractured region ([4]). Problems involving functionals of this form are usually called free-discontinuity problems, after a terminology introduced by De Giorgi (see [11], [5], [7]). The Ambrosio and Tortorelli approach [6] provides a variational approximation of the Mumford and Shah functional (1) via elliptic functionals to obtain approximate smooth solutions and overcome the numerical problems due to surface detection. The unknown surface Su is substituted by an additional function variable v which approaches the characteris...
We discuss a variational problem defined on multifunctions with two leaves that are constrained t... more We discuss a variational problem defined on multifunctions with two leaves that are constrained to take values into the 2-dimensional unit sphere. The energy functional contains a non-local interaction term that depends on the distance between the gradients of the two leaves, and may represent a biaxial liquid crystal. Different gradients of the axes directions of the liquid crystal are preferred or penalized in this model, in dependence of the sign of the interaction term. In this paper we find an explicit representation formula for the relaxed energy. Introduction In the last decades there has been a growing interest in variational problems for vector valued mappings with geometric constraints, as e.g. for mappings defined between smooth manifolds isometrically embedded in Euclidean spaces. The most studied one is perhaps the minimization problem of the Dirichlet energy
Annali Della Scuola Normale Superiore Di Pisa-classe Di Scienze, 2001
We consider the integral ftinctional f f (x, Du) dx under non standard growth assumptions of (p, ... more We consider the integral ftinctional f f (x, Du) dx under non standard growth assumptions of (p, q)-type: namely, we assume that for some function p(x) > 1, a condition appearing in several models from mathematical physics. Under sharp assumptions on the continuous function p (x ) we prove partial regularity of minimizers in the vector-valued case u : R" -~ allowing for quasiconvex energy densities. This is, to our knowledge, the first regularity theorem for quasiconvex functionals under non standard growth conditions
1. IntroductionThe Lavrentiev phenomenon, or gap problem, has stirred renewed interest in recenty... more 1. IntroductionThe Lavrentiev phenomenon, or gap problem, has stirred renewed interest in recentyears as it challenges traditional theories in the Calculus of Variations. A prototypemodel, relevant to the study of cavitation in rubber-like materials (see [3,20,27,28],among others), assigns to each deformation
Proceedings of the Royal Society of Edinburgh: Section A Mathematics
We consider the total curvature of graphs of curves in high-codimension Euclidean space. We intro... more We consider the total curvature of graphs of curves in high-codimension Euclidean space. We introduce the corresponding relaxed energy functional and prove an explicit representation formula. In the case of continuous Cartesian curves, i.e. of graphs cu of continuous functions u on an interval, we show that the relaxed energy is finite if and only if the curve cu has bounded variation and finite total curvature. In this case, moreover, the total curvature does not depend on the Cantor part of the derivative of u. We treat the wider class of graphs of one-dimensional functions of bounded variation, and we prove that the relaxed energy is given by the sum of the length and total curvature of the new curve obtained by closing the holes in cu generated by jumps of u with vertical segments.
We discuss the local minimality of certain configurations for a nonlocal isoperimetric problem us... more We discuss the local minimality of certain configurations for a nonlocal isoperimetric problem used to model microphase separation in diblock copolymer melts. We show that critical configurations with positive second variation are local minimizers of the nonlocal area functional and, in fact, satisfy a quantitative isoperimetric inequality with respect to sets that are L^1-close. The link with local minimizers for the diffuse-interface Ohta-Kawasaki energy is also discussed. As a byproduct of the quantitative estimate, we get new results concerning periodic local minimizers of the area functional and a proof, via second variation, of the sharp quantitative isoperimetric inequality in the standard Euclidean case. As a further application, we address the global and local minimality of certain lamellar configurations.
Abstract.We discuss a variational problem defined on couples of functions that are constrained to... more Abstract.We discuss a variational problem defined on couples of functions that are constrained to take values into the 2-dimensional unit sphere. The energy functional contains, besides standard Dirichlet energies, a non-local interaction term that depends on the distance between the gradients of the two functions. Different gradients are preferred or penalized in this model, in dependence of the sign of the interaction term. In this paper we study the lower semicontinuity and the coercivity of the energy and we find an explicit representation formula for the relaxed energy. Moreover, we discuss the behavior of the energy in the case when we consider multifunctions with two leaves rather than couples of functions.
In order to apply the direct methods of the calculus of variations to this class of functionals, ... more In order to apply the direct methods of the calculus of variations to this class of functionals, a first problem to be solved is the identification of qualitative conditions on the supremand f which imply the lower semicontinuity with respect to a convergence weak enough to provide the compactness in a large number of situations, say the weak* L∞ convergence. This was already solved by Barron and Liu in [3] where they showed that a functional of the form (1) is weakly* L∞ sequentially lower semicontinuous if and only if the function
defined on the space SBV (Ω) of special functions of bounded variation are studied, where ∇u deno... more defined on the space SBV (Ω) of special functions of bounded variation are studied, where ∇u denotes the approximate gradient of u, and Su is the set of the discontinuity points of u. In a two-dimensional setting, Su represents the contours of the object in a picture and u is a smoothing of an imput image. Energies of the same form arise in fracture mechanics for brittle solids, where Su is interpreted as the crack surface and u as the displacement outside the fractured region ([4]). Problems involving functionals of this form are usually called free-discontinuity problems, after a terminology introduced by De Giorgi (see [11], [5], [7]). The Ambrosio and Tortorelli approach [6] provides a variational approximation of the Mumford and Shah functional (1) via elliptic functionals to obtain approximate smooth solutions and overcome the numerical problems due to surface detection. The unknown surface Su is substituted by an additional function variable v which approaches the characteris...
We discuss a variational problem defined on multifunctions with two leaves that are constrained t... more We discuss a variational problem defined on multifunctions with two leaves that are constrained to take values into the 2-dimensional unit sphere. The energy functional contains a non-local interaction term that depends on the distance between the gradients of the two leaves, and may represent a biaxial liquid crystal. Different gradients of the axes directions of the liquid crystal are preferred or penalized in this model, in dependence of the sign of the interaction term. In this paper we find an explicit representation formula for the relaxed energy. Introduction In the last decades there has been a growing interest in variational problems for vector valued mappings with geometric constraints, as e.g. for mappings defined between smooth manifolds isometrically embedded in Euclidean spaces. The most studied one is perhaps the minimization problem of the Dirichlet energy
Annali Della Scuola Normale Superiore Di Pisa-classe Di Scienze, 2001
We consider the integral ftinctional f f (x, Du) dx under non standard growth assumptions of (p, ... more We consider the integral ftinctional f f (x, Du) dx under non standard growth assumptions of (p, q)-type: namely, we assume that for some function p(x) > 1, a condition appearing in several models from mathematical physics. Under sharp assumptions on the continuous function p (x ) we prove partial regularity of minimizers in the vector-valued case u : R" -~ allowing for quasiconvex energy densities. This is, to our knowledge, the first regularity theorem for quasiconvex functionals under non standard growth conditions
1. IntroductionThe Lavrentiev phenomenon, or gap problem, has stirred renewed interest in recenty... more 1. IntroductionThe Lavrentiev phenomenon, or gap problem, has stirred renewed interest in recentyears as it challenges traditional theories in the Calculus of Variations. A prototypemodel, relevant to the study of cavitation in rubber-like materials (see [3,20,27,28],among others), assigns to each deformation
Proceedings of the Royal Society of Edinburgh: Section A Mathematics
We consider the total curvature of graphs of curves in high-codimension Euclidean space. We intro... more We consider the total curvature of graphs of curves in high-codimension Euclidean space. We introduce the corresponding relaxed energy functional and prove an explicit representation formula. In the case of continuous Cartesian curves, i.e. of graphs cu of continuous functions u on an interval, we show that the relaxed energy is finite if and only if the curve cu has bounded variation and finite total curvature. In this case, moreover, the total curvature does not depend on the Cantor part of the derivative of u. We treat the wider class of graphs of one-dimensional functions of bounded variation, and we prove that the relaxed energy is given by the sum of the length and total curvature of the new curve obtained by closing the holes in cu generated by jumps of u with vertical segments.
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Papers by Emilio Acerbi