By Alberto Cialdea, Flavia Lanzara, Paolo Emilio Ricci
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Additional info for Analysis, partial differential equations and applications: The V.Maz'ya anniversary
Swiech, On viscosity solutions of fully nonlinear equations with measurable ingredients, Comm. Pure Appl. Math. 49 (1996), no. 4, 365–397.  I. Capuzzo Dolcetta, F. Leoni, A. Porretta, Viscous Hamilton-Jacobi equations. In: Proceedings of the 6th International Congress on Industrial and Applied Mathematics, Z¨ urich (2007).  I. Capuzzo Dolcetta, F. Leoni, A. Porretta, H¨ older estimates for degenerate elliptic equations with coercive Hamiltonians, to appear in Trans. Amer. Math. Soc.  M.
Choosing a common language In order to compare the two models described in Sections 3 and 4 we need to choose a common mathematical language. We choose the language of bispinors. 5) where η a˙ = a˙ b˙ ηb˙ , ab = a˙ b˙ = ab = a˙ b˙ = 0 1 . 6) Note that throughout this paper we assume that the density ρ does not vanish. 5) are invariant under the change of bispinor ﬁeld ξ a → ξ a eiϕ , ηb˙ → ηb˙ e−iϕ where ϕ : M → R is an arbitrary scalar function. 5) do not feel the argument of the complex scalar s.
2). The coordinate x3 is redundant so T ax can be viewed as a 3-form in (1 + 2)-dimensional Lorentzian space with local coordinates (x0 , x1 , x2 ). Hence, we can deﬁne the scalar ∗T ax := 1 3! 21) Dirac Equation 25 which is the Hodge dual of T ax . 21) can be rewritten as 1 ∗T ax = T ax · (ϑ0 ∧ ϑ1 ∧ ϑ2 ) = (ϑ0 ∧ dϑ0 − ϑ1 ∧ dϑ1 − ϑ2 ∧ dϑ2 ) · (ϑ0 ∧ ϑ1 ∧ ϑ2 ) 3 1 0 1 2 1 = (ϑ ∧ ϑ ) · dϑ + (ϑ ∧ ϑ2 ) · dϑ0 + (ϑ2 ∧ ϑ0 ) · dϑ1 . 19) we arrive at the compact formula L± Dir (ξ) = 3 ∗ T ax ∓ m ρ . 7) we get − − 16 32m L+ Dir (ξ)LDir (ξ) = (∗T ax )2 − m2 ρ .
Analysis, partial differential equations and applications: The V.Maz'ya anniversary by Alberto Cialdea, Flavia Lanzara, Paolo Emilio Ricci