Reduced limits for nonlinear equations with measures |
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Authors: | Moshe Marcus Augusto C Ponce |
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Institution: | a Technion, Department of Mathematics, Haifa 32000, Israel b Université catholique de Louvain, Département de mathématique, Chemin du Cyclotron 2, 1348 Louvain-la-Neuve, Belgium |
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Abstract: | We consider equations (E) −Δu+g(u)=μ in smooth bounded domains Ω⊂RN, where g is a continuous nondecreasing function and μ is a finite measure in Ω. Given a bounded sequence of measures (μk), assume that for each k?1 there exists a solution uk of (E) with datum μk and zero boundary data. We show that if uk→u# in L1(Ω), then u# is a solution of (E) relative to some finite measure μ#. We call μ# the reduced limit of (μk). This reduced limit has the remarkable property that it does not depend on the boundary data, but only on (μk) and on g. For power nonlinearities g(t)=|t|q−1t, ∀t∈R, we show that if (μk) is nonnegative and bounded in W−2,q(Ω), then μ and μ# are absolutely continuous with respect to each other; we then produce an example where μ#≠μ. |
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Keywords: | Semilinear elliptic equations Outer measure Equidiffuse sequence of measures Diffuse limit Biting lemma Inverse maximum principle Kato's inequality |
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