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Weighted Miranda{Talenti inequality and applications to equa- tions with discontinuous coecients

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S. Leonardi

Weighted Miranda{Talenti inequality and applications to equa- tions with discontinuous coecients

Comment.Math.Univ.Carolinae 43,1 (2002) 43-59.

Abstract: Let Ω be an open bounded set inRn (n≥2), with C2 boundary, and Np,λ(Ω) (1< p <+∞, 0≤λ < n) be a weighted Morrey space.

In this note we prove a weighted version of the Miranda-Talenti inequality and we exploit it to show that, under a suitable condition of Cordes type, the Dirichlet problem: ( Pn

i,j=1aij(x)∂x∂2u

i∂xj =f(x)∈Np,λ(Ω) in Ω

u= 0 on ∂Ω

has a unique strong solution in the functional space

½

u∈W2,p∩Wo1,p(Ω) : ∂2u

∂xi∂xj ∈Np,λ(Ω), i, j= 1,2, . . . , n

¾ .

Keywords: Miranda-Talenti inequality, nonvariational elliptic equations, H¨older regularity

AMS Subject Classification: 35B45, 35B65, 35J25, 35J60, 35R05

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