Reduced limits for nonlinear equations with measures
(2010) Journal of Functional Analysis — Vol. 258, n° 7, p. 2316-2372 (2010)
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- We consider equations (E) -Delta u + g(u) = mu in smooth bounded domains Omega subset of R-N, where g is a continuous nondecreasing function and mu is a finite measure in Omega. Given a bounded sequence of measures (mu(k)), assume that for each k >= 1 there exists a solution u(k) of (E) with datum mu(k) and zero boundary data. We show that if u(k) -> u(#) to in L-1(Omega), then u(#) is a solution of (E) relative to some finite measure mu(#). We call mu(#) the reduced limit of (mu(k)). This reduced limit has the remarkable property that it does not depend on the boundary data, but only on (mu(k)) and on g. For power nonlinearities g(t) = vertical bar t vertical bar(q-1)t, for all t is an element of R, we show that if (mu(k)) is nonnegative and bounded in W--2,W-q(Omega), then mu and mu(#) are absolutely continuous with respect to each other; we then produce an example where mu(#) not equal mu. (C) 2009 Elsevier Inc. All rights reserved.
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Marcus, M., & Ponce, A. (2010). Reduced limits for nonlinear equations with measures. Journal of Functional Analysis, 258(7), 2316-2372. https://doi.org/10.1016/j.jfa.2009.09.007 (Original work published 2010)