# Abelian categories with applications to rings and modules by M. Popescu By M. Popescu

Abelian different types with functions to earrings and Modules (London Mathematical Society Monographs)

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In Ω. 12 (Maximum Principle). Let u ∈ H01 (Ω). (1) Assume that Ω M(x)∇u · ∇v ≤ 0 for every positive v ∈ H01 (Ω). Then u ≤ 0. (2) Assume that Ω M(x)∇u · ∇v ≥ 0 for every positive v ∈ H01 (Ω). Then u ≥ 0. Proof. (1) Choose v = u+ , that is, the positive part of u as a test function. Then M(x)∇u+ · ∇u+ = Ω M(x)∇(u+ − u− ) · ∇u+ ≤ 0 . Ω Using the ellipticity of M , u+ = 0. (2) Choose v = u− , that is, the negative part of u as a test function. Then M(x)∇(u+ − u− ) · ∇u− = − 0≤ Ω M(x)∇u− · ∇u− . Ω The ellipticity of M gives u− = 0.

8. Using the previous theorem one sees immediately that there exists a solution u to u ∈ H01 (Ω) ∩ Lp−1 (Ω) : eu − 1 ∈ L1 (Ω) : −div(M(x)∇u) + |u|p−2 u = f u ∈ H01 (Ω), −div(M(x)∇u) + eu − 1 = f . 9. In Chapters 10 and 11, we shall again study approximating problems to get a priori estimates and then pass to the limit. 1) under the following hypotheses: 2N (1) f ∈ L N+2 (Ω); (2) g : R → R is increasing and continuous; there exists γ > 0 such that |g(s)| ≤ γ|s|a , a≤ N +2 . N −2 We recall that (1) M is elliptic, that is, there exists α > 0 such that M(x)ξ · ξ ≥ α|ξ|2 , ∀ ξ ∈ RN ; (2) M is bounded, meaning there exists β > 0 such that |M(x)| ≤ β , ∀ x ∈ Ω.

E. in E . From Beppo Levi’s theorem and from Step II , one has +∞ |f | = lim sn = lim n→∞ E meas{x ∈ E : |sn (x)| > t} dt n→∞ E 0 +∞ = lim dt n→∞ 0 χ{x∈E:|sn (x)|>t} . 8) 22 Preliminaries of real analysis Lebesgue’s theorem implies that χ{x∈E:|sn(x)|>t} = lim n→∞ E χAt ∩E . 9) E We set gn (t) = χ{x∈E:|sn(x)|>t} . 9) means that gn (t) → meas(At ∩ E) if n → ∞. To prove the result it suffices to prove that +∞ +∞ gn (t) dt = lim n→+∞ 0 meas(At ∩ E) dt . e. in (0, +∞); moreover |gn (t)| ≤ meas(E); it is sufficient to prove that meas(At ∩ E) belongs to L1 ((0, +∞)), and then apply Lebesgue’s theorem.

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