By Alberto Cialdea, Flavia Lanzara, Paolo Emilio Ricci

ISBN-10: 3764398973

ISBN-13: 9783764398972

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Chervova and D. 4) where the bold tensor index α runs through the values 0, 1, 2, 3, 4, whereas its non-bold counterpart α runs through the values 0, 1, 2, 3. 4) stands for a column of four zeros. 1 The coordinate x4 parametrises a circle of radius 2m . 3) means that the extended coframe ϑ experiences a full turn in the (ϑ1 , ϑ2 )-plane as we move along this circle, coming back to the starting point. We extend our metric as 1 Aα gαβ − m12 Aα Aβ m . 5) means that we view electromagnetism as a perturbation (shear) of the extended metric.

Barles, A weak Bernstein method for fully nonlinear elliptic equations, Diﬀerential Integral Equations 4 (1991), 241–262. [4] G. Barles, F. Da Lio, On the generalized Dirichlet problem for viscous Hamilton– Jacobi equations, J. Math. Pures Appl. (9) 83 (2004), no. 1, 53–75. [5] L. G. Crandall, M. Kocan, A. Swiech, On viscosity solutions of fully nonlinear equations with measurable ingredients, Comm. Pure Appl. Math. 49 (1996), no. 4, 365–397. [6] I. Capuzzo Dolcetta, F. Leoni, A. Porretta, Viscous Hamilton-Jacobi equations.

Lions in [12]. 1). Concerning the boundary behavior, if 1 < p ≤ 2 then u blows-up on ∂Ω, while if p > 2, then u is bounded and H¨ older continuous up to ∂Ω. This striking diﬀerence between the cases p ≤ 2 and p > 2 reﬂects a similar feature occurring in the study of the exit-time problem, see [4]. Namely if 1 < p ≤ 2, the Dirichlet problem can be solved in the classical sense for any boundary datum g, while if p > 2 this is no longer true and there can be loss of boundary condition. In that case, the best one can expect, in general, is that the Dirichlet condition u = g is satisﬁed only in a generalized sense, see [4].

### Analysis, partial differential equations and applications: The V.Maz'ya anniversary by Alberto Cialdea, Flavia Lanzara, Paolo Emilio Ricci

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