By Alberto Cialdea, Flavia Lanzara, Paolo Emilio Ricci

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7) becomes 1 Tax = (ϑ0 ∧ dϑ0 − ϑ1 ∧ dϑ1 − ϑ2 ∧ dϑ2 ). 4). 24) can be rewritten as 4m 1 Tax = T ax + ϑ ∧ ϑ2 ∧ ϑ4 . 25) 3 The coordinate x3 is redundant so Tax can be viewed as a 3-form in (1 + 3)dimensional Lorentzian space with local coordinates (x0 , x1 , x2 , x4 ). 26) 3! which is the Hodge dual of Tax . It is easy to see that we have Tax 2 = − ∗ Tax 2 . 27) we square a 1-form in (1 + 3)-dimensional Lorentzian space, so we took great care in getting the sign right. 28) ∗T 26 J. Burnett, O. Chervova and D.

We suggest an alternative mathematical model for the electron in which the dynamical variables are a coframe (ﬁeld of orthonormal bases) and a density. The electron mass and external electromagnetic ﬁeld are incorporated into our model by means of a Kaluza–Klein extension. Our Lagrangian density is proportional to axial torsion squared. The advantage of our approach is that it does not require the use of spinors, Pauli matrices or covariant diﬀerentiation. The only geometric concepts we use are those of a metric, diﬀerential form, wedge product and exterior derivative.

Assume further that A(·) is bounded and continuous, f (·) p p−1 is continuous and that d∂Ω f + is bounded. 34 I. Capuzzo Dolcetta Then, u is H¨ older continuous in Ω and p−2 |u(x) − u(y)| ≤ M |x − y| p−1 ∀ x, y ∈ Ω for some positive constant M depending only on ∂Ω , p , A p/p−1 d∂Ω (f + + λu− ) L∞ (Ω) L∞ (Ω) and . Before sketching the main steps of the proof, let us make some remarks on the above result. 2. As far as the H¨ older exponent is concerned, the value p−2 p−1 is the best one can expect in the assumptions of the above Theorem.