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A singular equation with positive and free boundary solutions.

Juan DávilaMarcelo Montenegro — 2003

RACSAM

Para 0 < β < 1 consideramos la ecuación -Δu = χ (-u + λf(x, u)) en Ω­ con condición de borde tipo Dirichlet. Esta ecuación posee una solución maximal u ≥ 0 para todo λ > 0. Si λ es menor que una cierta constante λ*, u se anula en el interior del dominio creando una frontera libre, y para λ > λ* esta solución es positiva en Ω­ y estable. Establecemos la regularidad de u incluso en presencia de una frontera libre. Para λ ≥ λ* la solución del problema parabólico singular...

A Hardy type inequality for W 0 m , 1 ( Ω ) functions

Hernán CastroJuan DávilaHui Wang — 2013

Journal of the European Mathematical Society

We consider functions u W 0 m , 1 ( Ω ) , where Ω N is a smooth bounded domain, and m 2 is an integer. For all j 0 , 1 k m - 1 , such that 1 j + k m , we prove that i u ( x ) d ( x ) m - j - k W 0 k , 1 ( Ω ) with k ( i u ( x ) d ( x ) m - j - k ) L 1 ( Ω ) C u W m , 1 ( Ω ) , where d is a smooth positive function which coincides with dist ( x , Ω ) near Ω , and l denotes any partial differential operator of order l .

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