Abstract
We consider integral functionals of the calculus of variations of the form \[ F ( u ) = ∫ 0 1 f ( x , u , u ′ , … , u ( n ) ) d x F(u) = \int \limits _0^1 {f(x,u,u’, \ldots ,{u^{(n)}})dx} \] defined for u ∈ W n , ∞ ( 0 , 1 ) u \in {W^{n,\infty }}(0,1) , and we show that the relaxed functional F F with respect to the weak W loc n , 1 ( 0 , 1 ) W_{{\text {loc}}}^{n,1}(0,1) convergence can be written as \[ F ¯ ( u ) = ∫ 0 1 f ( x , u , u ′ , … , u ( n ) ) d x + L ( u ) , \overline F (u) = \int \limits _0^1 {f(x,u,u’, \ldots ,{u^{(n)}})dx + L(u),} \] where the additional term L ( u ) L(u) , the Lavrentiev Gap, is explicitly identified in terms of F F .
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callSimilar Papers
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
