Nonlinear elliptic equations with measures revisited
By Haim Brezis, M. Marcus, Augusto C. Ponce. 2007
123 citations recorded by the source corpus.
We study the existence of solutions of the nonlinear problem $$ \left\{ \begin{alignedat}{2} -\Delta u + g(u) & = \mu & & \quad \text{in } \Omega,\\ u & = 0 & & \quad \text{on } \partial \Omega, \end{alignedat} \right. $$ where $\mu$ is a Radon measure and $g : \mathbb{R} \to \mathbb{R}$ is a nondecreasing continuous function with $g(0) = 0$. This equation need not have a solution for every measure $\mu$, and we say that $\mu$ is a good measure if the Dirichlet problem above admits a solution. We show that for every $\mu$ there exists a largest good measure $\mu^* \leq \mu$. This reduced measure has a number of remarkable properties.
Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/53f4ee8790d1f768b85c01388729cf58cc150ba5
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In Applied Mathematics
In Computational Theory and Mathematics
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Citations in the corpus
- Diffuse measures and nonlinear parabolic equations (2011, unknown): “by the stationary case (see [9]), if one looks for a “general solvability” result (i.”
- On singular elliptic equations with measure sources (2015, background): “sition to prove Theorem 2.14. ON SINGULAR ELLIPTIC EQUATIONS WITH MEASURE SOURCES 11 Proof of Theorem 2.14. As we have fu−γ ∈ L1(Ω), then the uniqueness is an easy consequence of Proposition 4.B.3 in [12]. In fact, if u1 and u2 are two solutions of (2.21) in the sense of Definition 2.12 with data µ1 ”