Relaxation of functionals with linear growth: Interactions of emerging measures and free discontinuities

Abstract For an integral functional defined on functions ( u , v ) ∈ W 1 , 1 × L 1 {(u,v)\in W^{1,1}\times L^{1}} featuring a prototypical strong interaction term between u and v, we calculate its relaxation in the space of functions with bounded variations and Radon measures. Interplay between measures and discontinuities brings various additional difficulties, and concentration effects in recovery sequences play a major role for the relaxed functional even if the limit measures are absolutely continuous with respect to the Lebesgue one.

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On the algebraic treatment of the spin–orbit interaction

Canadian Journal of Physics ◽

10.1139/p95-032 ◽

1995 ◽

Vol 73 (3-4) ◽

pp. 241-244

Author(s):

Mayer Humi

Keyword(s):

Strong Interaction ◽

Symmetry Algebra ◽

Orbit Interaction ◽

Central Field ◽

Algebra Structure ◽

Spin Orbit ◽

Quadratic Operator ◽

Spin Orbit Interaction ◽

Algebraic Treatment ◽

Interaction Term

The spin–orbit interaction term in the Hamiltonian H of the heavier atoms or nuclei is a quadratic operator. It does not commute with the central field Hamiltonian H0 and in many cases represents a strong interaction term. In this paper we show how to accommodate this operator within an extended Lie algebra structure that contains the symmetry algebra of H0. The representations of this extended algebra can be used then to classify the states of the Hamiltonian H.

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Stochastic homogenization for an integral functional of a quasiconvex function with linear growth

Asymptotic Analysis ◽

10.3233/asy-1997-15203 ◽

1997 ◽

Vol 15 (2) ◽

pp. 183-202 ◽

Cited By ~ 5

Author(s):

Y. Abddaimi ◽

G. Michaille ◽

C. Licht

Keyword(s):

Linear Growth ◽

Integral Functional ◽

Quasiconvex Function ◽

Stochastic Homogenization

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An optimal design problem with non-standard growth and no concentration effects

Asymptotic Analysis ◽

10.3233/asy-211711 ◽

2021 ◽

pp. 1-28

Author(s):

Ana Cristina Barroso ◽

Elvira Zappale

Keyword(s):

Thin Films ◽

Optimal Design ◽

Integral Representation ◽

Design Problem ◽

Damage Evolution ◽

Singular Term ◽

Growth Conditions ◽

Optimal Design Problem ◽

Concentration Effects ◽

Absolutely Continuous

We obtain an integral representation for certain functionals arising in the context of optimal design and damage evolution problems under non-standard growth conditions and perimeter penalisation. Under our hypotheses, the integral representation includes a term which is absolutely continuous with respect to the Lebesgue measure and a perimeter term, but no additional singular term. We also study some dimension reduction problems providing results for the optimal design of thin films.

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Structured coagulation-fragmentation equation in the space of radon measures: Unifying discrete and continuous models

ESAIM Mathematical Modelling and Numerical Analysis ◽

10.1051/m2an/2021061 ◽

2021 ◽

Author(s):

Azmy S. Ackleh ◽

Rainey Lyons ◽

Nicolas Saintier

Keyword(s):

Regularity Result ◽

Stationary Solutions ◽

Fragmentation Model ◽

Model Parameters ◽

Physical Processes ◽

Absolutely Continuous ◽

Radon Measures ◽

Continuous Models ◽

Fragmentation Equation ◽

Well Posed

We present a structured coagulation-fragmentation model which describes the population dynamics of oceanic phytoplankton. This model is formulated on the space of Radon measures equipped with the bounded Lipschitz norm and unifies the study of the discrete and continuous coagulation-fragmentation models. We prove that the model is well-posed and show it can reduce down to the classic discrete and continuous coagulation-fragmentation models. To understand the interplay between the physical processes of coagulation and fragmentation and the biological processes of growth, reproduction, and death, we establish a regularity result for the solutions and use it to show that stationary solutions are absolutely continuous under some conditions on model parameters. We develop a semi-discrete approximation scheme which conserves mass and prove its convergence to the unique weak solution. We then use the scheme to perform numerical simulations for the model.

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Translates of L∞ functions and of bounded measures

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700025222 ◽

1964 ◽

Vol 4 (4) ◽

pp. 403-409 ◽

Cited By ~ 6

Author(s):

R. E. Edwards

Keyword(s):

Abelian Group ◽

Continuous Functions ◽

Group Element ◽

Locally Compact Abelian Group ◽

Space Of Continuous Functions ◽

Locally Compact ◽

Absolutely Continuous ◽

Radon Measures ◽

Uniformly Continuous Function ◽

Uniformly Continuous

D. A. Edwards has shown [1] that if X is a locally compact Abelian group and f ∈ L∞, then the translate fa of f varies continuously with α if and only if f is (equal l.a.e. to) a bounded, uniformly continuous function. He remarks that this is a sort of dual to part of a result due to Plessner and Raikov which asserts that an element μ of the space Mb of bounded Radon measures on X belongs to L1 (i.e., is absolutely continuous relative to Haar measure) if and only its translates vary continuously with the group element, the relevant topology on Mb being that defined by the natural norm of Mb as the dual of the space of continuous functions vanishing at infinity. The proof he uses (ascribed to Reiter) applies equally well in both cases, and also to the case in which X is non-Abelian. A brief examination shows that in the latter case it is ultimately immaterial whether left- or right-translates are considered; since the extra complexities of this case are principally terminological, we shall direct no further attention to it.

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