Local solvability for nonlinear partial differential equations

✍ Scribed by F. Messina; L. Rodino

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 444 KB
• Volume: 47
• Category: Article
• ISSN: 0362-546X
• DOI: 10.1016/s0362-546x(01)00413-8

✦ Synopsis

In the introduction we give a short survey on known results concerning local solvability for nonlinear partial differential equations; the next sections will be then devoted to the proof of a new result in the same direction. Specifically we study the semilinear operator (F(u)=P(D) u+f\left(x, Q_{1}(D) u, \ldots, Q_{M}(D) u\right)) where (P, Q_{1}, \ldots, Q_{M}) are linear partial differential operators with constant coefficients and (f(x, v), x \in \mathbb{R}^{n}), (v \in \mathbb{C}^{M}), is a smooth function with respect to (x) and entire with respect to (v). Let (g) be in the Hörmander space (B_{p, k}) we want to solve locally near a point (x^{0} \in \mathbb{R}^{n}) the equation (F(u)=g).