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 70^{th} 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.

**Read Online or Download Contributions to Nonlinear Analysis: A Tribute to D.G. de Figueiredo on the Occasion of his 70th Birthday PDF**

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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**

**Example text**

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.