lavrentieff phenomenon
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Author(s):  
A. Corbo Esposito ◽  
T. Durante

The functional F(u) = ∫Bf(x, Du) is considered, where B is the unit ball in R2, u varies in the set of the locally Lipschitz functions on R2 and f belongs to a family containing, as model case, the following integrand: The computation of the relaxed functional F̄ is provided yielding an explicit representation formula.This formula nevertheless is not integral, because F̄ is not a measure and does not coincide with the obvious extension of F over all W1,p(B).This phenomenon is essentially due to the non-standard growth behaviour of f(x, z) in the variable z.


Author(s):  
Riccardo De Arcangelis

The paper provides an example of an integral functional in more than two dimensions, with a symmetric and positively defined quadratic integrand q, exhibiting the Lavrentieff phenomenon on a ball B and on a linear boundary datum u0, i.e. for whichThe example is also utilised to discuss nonidentity between some relaxation procedures for a quadratic integral functional and to provide a weighted Sobolev space in the Hilbert case in which smooth functions are not dense.


1992 ◽  
Vol 17 (9-10) ◽  
pp. 1503-1538 ◽  
Author(s):  
Esposito Antonio Corbo ◽  
Riccardo De Arcagelis

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