We study an optimal control problem for a quasi-linear elliptic equation with anisotropic p-Laplace operator in its principal part and $ L^1 $-control in coefficient of the low-order term. We assume that the matrix of anisotropy belongs to BMO-space. Since we cannot expect to have a solution of the state equation in the classical Sobolev space, we introduce a suitable functional class in which we look for solutions and prove existence of optimal pairs using an approximation procedure and compactness arguments in variable spaces.

中文翻译：

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在最佳状态 \ begin {document} $ L ^ 1 $ \ end {document}拟线性Dirichlet边值问题的系数控制 \ begin {document} $ BMO $ \ end {document}各向异性 \ begin {document} $ p $ \ end {document}-拉普拉斯

我们研究了一个拟线性椭圆方程的最优控制问题，该方程的主要部分是各向异性p-Laplace算符，低阶项的系数为$ L ^ 1 $ -control。我们假设各向异性矩阵属于BMO空间。由于我们不能期望在经典的Sobolev空间中得到状态方程的解，因此我们引入合适的泛函类，在其中寻找解并使用逼近过程和可变空间中的紧实度参数证明最优对的存在。