On the equation \(\det \nabla \varphi =f\) prescribing \(\varphi =0\) on the boundary.
This interesting paper deals with the solvability of the prescribed Jacobian equation with zero Dirichlet boundary condition: \[ \begin{aligned} \det (\nabla \varphi ) &=f \text{ in }\Omega ,\\ \varphi &=0 \text{ on }\partial \Omega . \end{aligned} \tag{1} \] It is proved in particular that when \(\Omega \) is the ball, \(f\in C^1(\overline {\Omega })\), \(\int _{\Omega } f=0\) and \(| f | >0\) on \(\partial \Omega \), the problem can be solved with \(\varphi \in C^1(\Omega )\cap C^{0, 1/n}(\overline {\Omega })\). The main step in the proof involves a (constructive) radial integration argument as in the work [\textit{G. Cupini, B. Dacorogna} and \textit{O. Kneuss}, Calc. Var. Partial Differ. Equ. 36, No. 2, 251--283 (2009; Zbl 1180.35169)]. The paper also raises the interesting question whether (1) has a compactly supported solution when \(f\) is itself compactly supported. The author actually shows that, in this case, (1) can be solved by a compactly supported \(\varphi \) up to an arbitray \(C^1(\overline {\Omega })\) error. For a certain behavior of \(f\) near \(\partial \Omega \), O. Kneuss shows that his construction gives a solution that is \(C^1\) up to \(\partial \Omega \). Necessary conditions are also discussed. The article is well-written and very pleasant to read.
scientific article; zbMATH DE number 6148066
scientific article