\(A\)-\(B\) quasiconvexity and implicit partial differential equations
This paper deals with the following boundary value problem in \(\Omega\) open subset of \(\mathbb{R} ^n\) for differential inclusion: \[ Bu(x)\in E \] with \(u=\varphi\) on \(\partial \Omega \) where \(\varphi \in C^{1} _{\text{pie}}(\overline \Omega , \mathbb{R}^N)\), \(E\) is a compact subset of \(\mathbb{R}^{mn}\) and B is a \(m n\) -valued first order differential operator. In the framework of \(A\)-\(B\)-quasiconvexity, the authors prove the existence of solution \(u\in C^{1} _{\text{pie}}(\overline \Omega , \mathbb{R}^N)\) with \(Bu \in L^\infty\) when \(\varphi\) satisfies the compatibility condition. Moreover the authors study in a very precise way the significant application of the results for the existence of exact equilibrium solutions in ferromagnetics.
scientific article; zbMATH DE number 1756807