Provider: JSTOR http://www.jstor.org Database: JSTOR Content: text/plain; charset="us-ascii" 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 ⊂ ℝNbe a bounded domain with smooth boundary. Letg: ℝ → ℝ be a continuous, nondecreasing function such thatg(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 ∈L¹(Ω) was initiated by Brezis-Strauss [BS]; their main result asserts that foreveryμ ∈L¹ andevery gas 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 -