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Evolution Problems and Minimizing Movements

Ugo Gianazza, Massimo Gobbino, Giuseppe Savarè (1994)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

We recall the definition of Minimizing Movements, suggested by E. De Giorgi, and we consider some applications to evolution problems. With regards to ordinary differential equations, we prove in particular a generalization of maximal slope curves theory to arbitrary metric spaces. On the other hand we present a unifying framework in which some recent conjectures about partial differential equations can be treated and solved. At the end we consider some open problems.

Existence and iteration of positive solutions for a singular two-point boundary value problem with a p -Laplacian operator

De-xiang Ma, Weigao Ge, Zhan-Ji Gui (2007)

Czechoslovak Mathematical Journal

In the paper, we obtain the existence of symmetric or monotone positive solutions and establish a corresponding iterative scheme for the equation ( φ p ( u ' ) ) ' + q ( t ) f ( u ) = 0 , 0 < t < 1 , where φ p ( s ) : = | s | p - 2 s , p > 1 , subject to nonlinear boundary condition. The main tool is the monotone iterative technique. Here, the coefficient q ( t ) may be singular at t = 0 , 1 .

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