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TY - CHAP
TI - Nonlinear Elliptic Equations with Measures Revisited
AU - Brezis, H.
AU - Marcus, M.
AU - Ponce, A. C.
A3 - Bourgain, Jean
A3 - Kenig, Carlos E.
A3 - Klainerman, S.
AB - Let Ω ⊂ ℝ^{N}be a bounded domain with smooth boundary. Let*g*: ℝ → ℝ be a continuous, nondecreasing function such that*g*(0) = 0. In this paper we are concerned with the problem\[\left\{ \begin{array}{rl} - \Delta u + g(u) = \mu & \text{in}\ \Omega,\\ u = 0 & \text{on}\ \partial \Omega, \\ \end{array} \right. \caption{(4.0.1)}\]where μ is a measure. The study of (4.0.1) when μ ∈*L*¹(Ω) was initiated by Brezis-Strauss [BS]; their main result asserts that for*every*μ ∈*L*¹ and*every g*as above, problem (4.0.1) admits a unique weak solution (see Theorem 4.B.2 in Appendix 4B below). The right concept of weak solution is the following:\[\left\{ \begin{array}{l} u \in L^1 (\Omega), g(u) \in L^1 (\Omega )\ \text{and}\\ - \int_{\Omega} u\Delta \zeta + \int_{\Omega } g(u)\zeta = \int_{\Omega} \zeta\, d\mu \quad \forall \zeta \in C^2 (\bar{\Omega}), \zeta = 0\ \text{on}\ \partial \Omega. \\ \end{array} \right. \caption{(4.0.2)}\]It will be convenient to write\[C_0(\bar{\Omega}) = \{ \zeta \in C(\bar{\Omega});\zeta = 0\ \text{on}\ \partial \Omega \} \]and\[C_0^2 (\bar{\Omega}) = \{ \zeta \in C^2 (\bar{\Omega});\zeta = 0\ \text{on}\ \partial \Omega \} \], and

EP - 110
PB - Princeton University Press
PY - 2007
SN - 9780691129556
SP - 55
T2 - Mathematical Aspects of Nonlinear Dispersive Equations (AM-163)
UR - http://www.jstor.org/stable/j.ctt7s1f9.7
Y2 - 2021/09/17/
ER -