By Emerson A. M. Abreu, Paulo Cesar Carrião (auth.), Thierry Cazenave, David Costa, Orlando Lopes, Raúl Manásevich, Paul Rabinowitz, Bernhard Ruf, Carlos Tomei (eds.)

This quantity represents a huge survey of present examine within the fields of nonlinear research and nonlinear differential equations. it really is devoted to Djairo G. de Figueiredo at the party of his 70th birthday. the gathering of 34 study and survey articles displays the big variety of pursuits of Djairo de Figueiredo, together with:
- quite a few forms of nonlinear partial differential equations and platforms, particularly equations of elliptic, parabolic, hyperbolic and combined type;
- equations of Schrödinger, Maxwell, Navier-Stokes, Bernoulli-Euler, Seiberg-Witten, Caffarelli-Kohn-Nirenberg;
- life, strong point and multiplicity of solutions;
- severe Sobolev progress and attached phenomena;
- qualitative homes, regularity and form of solutions;
- inequalities, a-priori estimates and asymptotic behavior;
- a variety of functions to types comparable to asymptotic membranes, nonlinear plates and inhomogeneous fluids.
The contributions of such a lot of distinct mathematicians to this quantity rfile the significance and lasting impact of the mathematical study of Djairo de Figueiredo. The e-book therefore is a resource of latest rules and effects and may attract graduate scholars and mathematicians drawn to nonlinear problems.

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Extra resources for Contributions to Nonlinear Analysis: A Tribute to D.G. de Figueiredo on the Occasion of his 70th Birthday

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3. The semilinear Maxwell equations Here we report some results contained in [1] where a unitarian field theory, based on a semilinear perturbation of the Maxwell equations, has been introduced. We modify the usual Maxwell action in the empty space Sm (ϕ, A) = ∂A +∇ϕ ∂t 1 2 2 − |∇ × A| 2 dxdt in the following way: S= 1 2 2 ∂A 2 2 +∇ϕ − |∇ × A| + W |A| − ϕ2 ∂t dxdt (25) where W : R → R. 2 2 The argument of W is |A| − |ϕ| in order to make this expression invariant for the Poincar´e group and the equations consistent with special relativity.

24, Birkhauser Boston, 1996. Claudianor O. S. br Progress in Nonlinear Differential Equations and Their Applications, Vol. 66, 33–52 c 2005 Birkh¨ auser Verlag Basel/Switzerland A Unitarian Approach to Classical Electrodynamics: The Semilinear Maxwell Equations Vieri Benci and Donato Fortunato Dedicated to our friend Djairo 1. Introduction The study of the relation of matter and the electromagnetic field is a classical, intriguing problem both from physical and mathematical point of view. In the framework of a classical relativistic theory particles must be considered pointwise.

By the Hodge decomposition theorem, the vector field A : R3 → R3 in (61) can be splitted as A = u + v = u + ∇w (62) where u : R3 → R3 is a divergence free vector field (∇ · u = 0) and v : R3 → R3 is a potential vector field, v = ∇w (w scalar field). Since f is strictly convex, for every u with ∇ · u = 0, we can find a scalar field w0 which minimizes the functional w→ f (u + ∇w) . Set w0 = Φ (u) . Replacing (62) in (2) with w = Φ (u) , we get a new functional J(u) := E (u, Φ (u)) = 1 2 |∇u| − f (u + ∇Φ (u)) dx 2 which depends only on u.

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