Analysis, partial differential equations and applications: by Alberto Cialdea, Flavia Lanzara, Paolo Emilio Ricci

By Alberto Cialdea, Flavia Lanzara, Paolo Emilio Ricci

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7) is simply a measure of deformations generated by rotations of spacetime points. We choose our Lagrangian density to be L(ϑ, ρ) := Tax where Tax 2 := 2 ρ 1 ax T Tax gακ gβλ gγμ . 3! 10) so our Lagrangian density L(ϑ, ρ) does not depend on x4 and can be viewed as a Lagrangian density in the original spacetime of dimension 1 + 3. This means, essentially, that we have performed a separation of variables in a nonlinear setting. Our action (variational functional) is L(ϑ, ρ) dx0 dx1 dx2 dx3 . Our field equations (Euler–Lagrange equations) are obtained by varying this action with respect to the coframe ϑ and density ρ.

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Anal. 74 (1980), no. 4, 335–353. L. -London (1982). L. Lions, Quelques remarques sur les probl`emes elliptiques quasilin´eaires du second ordre, (French) J. Analyse Math. 45 (1985), 234–254. L. E. Souganidis, Homogenization of degenerate second-order PDE in periodic and almost periodic environments and applications, Ann. Inst. H. Poincar´e Anal. Non Lin´eaire 22 (2005), no. 5, 667–677. [17] O. V. Radkevic, Second-order equations with non negative characteristic form, American Mathematical Society, Providence (1973).

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