Poisson problems for semilinear Brinkman systems on Lipschitz domains in \(\mathbb{R}^n\)
The paper studies the Dirichlet and the Robin problem for the Brinkman system on a bounded domain \(\Omega \subset\mathbb R^n\) with connected Lipschitz boundary. First it is proved unique solvability of the Dirichlet and the Robin problem for a linear Brinkman system \(\Delta u -Pu-\nabla \pi =f\), \(\nabla \cdot u=g\) in Sobolev-Slobodetskij spaces \(L_s^2(\Omega ,\mathbb R^n) \times L_{s-1}^2(\Omega )\) with \(1/2<s<3/2\). Here \(P\in L^\infty (\Omega ,\mathbb R^n\times\mathbb R^n)\) is a matrix-valued function satisfying \(P(x)\geq 0\) in \(\Omega \). Then the authors prove the existence of a solution of the Dirichlet problem and the Robin problem for a semilinear Brinkman system \(\Delta u-P(x,u(x),\pi (x))u-\nabla \pi =f\) in \(\Omega \), \(\nabla \cdot u=g\) in \(\Omega \), where \(P\in L^\infty (\Omega \times\mathbb R^n\times\mathbb R,\mathbb R^n\times\mathbb R^n)\) satisfies the Carathéodory condition and \(P\geq 0\).
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