Relaxation of quasi-convex integrals of arbitrary order

1994 ◽  
Vol 124 (5) ◽  
pp. 927-946 ◽  
Author(s):  
Micol Amar ◽  
Virginia De Cicco
Keyword(s):  
Integral Representation ◽  
Total Variation ◽  
Radon Measure ◽  
Arbitrary Order ◽  
Lower Semicontinuous ◽  
Integrable Functions ◽  
Lower Semicontinuous Envelope ◽  
Representation Result

An integral representation result is given for the lower semicontinuous envelope of the functional ʃΩf(∇ku)dxon the spaceBVk(Ω:ℝm) of the integrable functions, whose thef-th derivative in the sense of distributions is a Radon measure with bounded total variation.

Integral representation of functionals defined on curves of W1,p

1998 ◽  
Vol 128 (2) ◽  
pp. 193-217 ◽  
Author(s):  
Micol Amar ◽  
Giovanni Bellettini ◽  
Sergio Venturini
Keyword(s):  
Integral Representation ◽  
Representation Theorem ◽  
Lower Semicontinuous ◽  
Borel Function ◽  
Open Interval ◽  
Growth Conditions ◽  
The Family ◽  
Representation Result ◽  
Image Position

Let I ⊂ ℝ be a bounded open interval, (I) be the family of all open subintervals of I and let p > 1. The aim of this paper is to give an integral representation result for abstract functionals F: W1,p(I;ℝn) × (I) → [0, + ∞) which are lower semicontinuous and satisfy suitable properties. In particular, we prove an integral representation theorem for the Г-limit of a sequence {Fh}h, of functionals of the formwhere each fh is a Borel function satisfying proper growth conditions.

scholarly journals Connected perimeter of planar sets

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
François Dayrens ◽  
Simon Masnou ◽  
Matteo Novaga ◽  
Marco Pozzetta
Keyword(s):  
Minimization Problem ◽  
Existence Of Solutions ◽  
Lower Semicontinuous ◽  
Representation Formula ◽  
Steiner Trees ◽  
Lower Semicontinuous Envelope ◽  
Simply Connected

AbstractWe introduce a notion of connected perimeter for planar sets defined as the lower semicontinuous envelope of perimeters of approximating sets which are measure-theoretically connected. A companion notion of simply connected perimeter is also studied. We prove a representation formula which links the connected perimeter, the classical perimeter, and the length of suitable Steiner trees. We also discuss the application of this notion to the existence of solutions to a nonlocal minimization problem with connectedness constraint.

scholarly journals Relaxation of nonlinear elastic energies involving the deformed configuration and applications to nematic elastomers

2019 ◽  
Vol 25 ◽  
pp. 19 ◽  
Author(s):  
Carlos Mora-Corral ◽  
Marcos Oliva
Keyword(s):  
Sobolev Space ◽  
Elastic Energy ◽  
Lower Semicontinuous ◽  
Deformation Gradient ◽  
Variational Model ◽  
Nematic Elastomer ◽  
Nematic Elastomers ◽  
Lower Semicontinuous Envelope ◽  
Nonlinear Elastic

We start from a variational model for nematic elastomers that involves two energies: mechanical and nematic. The first one consists of a nonlinear elastic energy which is influenced by the orientation of the molecules of the nematic elastomer. The nematic energy is an Oseen–Frank energy in the deformed configuration. The constraint of the positivity of the determinant of the deformation gradient is imposed. The functionals are not assumed to have the usual polyconvexity or quasiconvexity assumptions to be lower semicontinuous. We instead compute its relaxation, that is, the lower semicontinuous envelope, which turns out to be the quasiconvexification of the mechanical term plus the tangential quasiconvexification of the nematic term. The main assumptions are that the quasiconvexification of the mechanical term is polyconvex and that the deformation is in the Sobolev space W1,p (with p > n − 1 and n the dimension of the space) and does not present cavitation.

A necessary and sufficient condition for the weak lower semicontinuity of one-dimensional non-local variational integrals

2006 ◽  
Vol 136 (4) ◽  
pp. 701-708 ◽  
Author(s):  
Jonathan Bevan ◽  
Pablo Pedregal
Keyword(s):  
Lower Semicontinuous ◽  
Short Note ◽  
Necessary And Sufficient Condition ◽  
One Dimensional ◽  
Lower Semicontinuous Envelope ◽  
Variational Integrals ◽  
Non Local ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Bounded Below

In this short note we prove that the functional I : W1,p(J;R) → R defined by is sequentially weakly lower semicontinuous in W1,p(J,R) if and only if the symmetric part W+ of W is separately convex. We assume that W is real valued, continuous and bounded below by a constant, and that J is an open subinterval of R. We also show that the lower semicontinuous envelope of I cannot in general be obtained by replacing W by its separately convex hull Wsc.

scholarly journals Chebyshev-Type Integral Inequalities for Continuous Fields of Operators Concerning Khatri–Rao Products and Synchronous Properties

Symmetry ◽  
2020 ◽  
Vol 12 (3) ◽  
pp. 422
Author(s):  
Arnon Ploymukda ◽  
Pattrawut Chansangiam
Keyword(s):  
Radon Measure ◽  
Geometric Mean ◽  
Integral Inequalities ◽  
Counting Measure ◽  
Integrable Functions ◽  
Discrete Inequalities ◽  
Finite Space ◽  
Continuous Fields ◽  
Adjoint Operators ◽  
Locally Compact Hausdorff Space

We consider bounded continuous fields of self-adjoint operators which are parametrized by a locally compact Hausdorff space Ω equipped with a finite Radon measure μ . Under certain assumptions on synchronous Khatri–Rao property of the fields of operators, we obtain Chebyshev-type inequalities concerning Khatri–Rao products. We also establish Chebyshev-type inequalities involving Khatri–Rao products and weighted Pythagorean means under certain assumptions of synchronous monotone property of the fields of operators. The Pythagorean means considered here are three classical symmetric means: the geometric mean, the arithmetic mean, and the harmonic mean. Moreover, we derive the Chebyshev–Grüss integral inequality via oscillations when μ is a probability Radon measure. These integral inequalities can be reduced to discrete inequalities by setting Ω to be a finite space equipped with the counting measure. Our results provide analog results for matrices and integrable functions. Furthermore, our results include the results for tensor products of operators, and Khatri–Rao/Kronecker/Hadamard products of matrices, which have been not investigated in the literature.

scholarly journals The Köthe Dual of an Abstract Banach Lattice

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
E. Jiménez Fernández ◽  
M. A. Juan ◽  
E. A. Sánchez-Pérez
Keyword(s):  
Integral Representation ◽  
Banach Lattice ◽  
Broad Class ◽  
Banach Lattices ◽  
Vector Measures ◽  
Integrable Functions ◽  
Suitable Definition ◽  
Integral Structures ◽  
Definition Of ◽  
Köthe Dual

We analyze a suitable definition of Köthe dual for spaces of integrable functions with respect to vector measures defined onδ-rings. This family represents a broad class of Banach lattices, and nowadays it seems to be the biggest class of spaces supported by integral structures, that is, the largest class in which an integral representation of some elements of the dual makes sense. In order to check the appropriateness of our definition, we analyze how far the coincidence of the Köthe dual with the topological dual is preserved.

Continuity of solutions of Schrödinger equations

1991 ◽  
Vol 110 (3) ◽  
pp. 581-597
Author(s):  
Mitsuru Nakai
Keyword(s):  
Euclidean Space ◽  
Total Variation ◽  
Open Subset ◽  
Radon Measure ◽  
Dimensional Euclidean Space ◽  
Schrödinger Equations ◽  
Kato Class ◽  
Continuity Of Solutions ◽  
Image Position ◽  
Newtonian Kernel

We denote by N(x, y) the Newtonian kernel on the d-dimensional Euclidean space (where d ≥ 2) so that N(x, y) = log|x–y|-1 for d = 2 and N(x, y) = |x−y|2−d for d ≥ 3. A signed Radon measure μ on an open subset Ω in d is said to be of Kato class iffor every y in Ω. where |μ| is the total variation measure of μ.

scholarly journals Radial representation of lower semicontinuous envelope

2014 ◽  
Vol 7 (1) ◽  
pp. 1-18 ◽  
Author(s):  
Omar Anza Hafsa ◽  
Jean-Philippe Mandallena
Keyword(s):  
Lower Semicontinuous ◽  
Lower Semicontinuous Envelope
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