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A relaxation result for autonomous integral functionals with discontinuous non-coercive integrand

Carlo MaricondaGiulia Treu — 2004

ESAIM: Control, Optimisation and Calculus of Variations

Let L : N × N be a borelian function and consider the following problems inf F ( y ) = a b L ( y ( t ) , y ' ( t ) ) d t : y A C ( [ a , b ] , N ) , y ( a ) = A , y ( b ) = B ( P ) inf F * * ( y ) = a b Ł ( y ( t ) , y ' ( t ) ) d t : y A C ( [ a , b ] , N ) , y ( a ) = A , y ( b ) = B · ( P * * ) We give a sufficient condition, weaker then superlinearity, under which inf F = inf F * * if L is just continuous in x . We then extend a result of Cellina on the Lipschitz regularity of the minima of ( P ) when L is not superlinear.

A relaxation result for autonomous integral functionals with discontinuous non-coercive integrand

Carlo MaricondaGiulia Treu — 2010

ESAIM: Control, Optimisation and Calculus of Variations

Let L : N × N be a Borelian function and consider the following problems inf F ( y ) = a b L ( y ( t ) , y ' ( t ) ) d t : y A C ( [ a , b ] , N ) , y ( a ) = A , y ( b ) = B ( P ) inf F * * ( y ) = a b L * * ( y ( t ) , y ' ( t ) ) d t : y A C ( [ a , b ] , N ) , y ( a ) = A , y ( b ) = B · ( P * * ) We give a sufficient condition, weaker then superlinearity, under which inf F = inf F * * if is just continuous in . We then extend a result of Cellina on the Lipschitz regularity of the minima of when is not superlinear.

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