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A direct proof of the Caffarelli-Kohn-Nirenberg theorem

Jörg Wolf (2008)

Banach Center Publications

In the present paper we give a new proof of the Caffarelli-Kohn-Nirenberg theorem based on a direct approach. Given a pair (u,p) of suitable weak solutions to the Navier-Stokes equations in ℝ³ × ]0,∞[ the velocity field u satisfies the following property of partial regularity: The velocity u is Lipschitz continuous in a neighbourhood of a point (x₀,t₀) ∈ Ω × ]0,∞ [ if l i m s u p R 0 1 / R Q R ( x , t ) | c u r l u × u / | u | | ² d x d t ε * for a sufficiently small ε * > 0 .

A Dirichlet problem with asymptotically linear and changing sign nonlinearity.

Marcello Lucia, Paola Magrone, Huan-Song Zhou (2003)

Revista Matemática Complutense

This paper deals with the problem of finding positive solutions to the equation -∆[u] = g(x,u) on a bounded domain 'Omega' with Dirichlet boundary conditions. The function g can change sign and has asymptotically linear behaviour. The solutions are found using the Mountain Pass Theorem.

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