论文标题
粗糙系数的椭圆特征值问题的最佳域
Optimal domains for elliptic eigenvalue problems with rough coefficients
论文作者
论文摘要
我们证明，在给定度量的所有开放式集合中，我们证明了一个开放式集合最小化具有有界，可测量系数的椭圆算子的第一个dirichlet特征值。我们的证明是基于一种自由边界方法：我们将最佳集合上的本征功能表征为受惩罚功能的最小化器，并由于Hölder估算本质功能而得出了最佳集合的开放性。我们还证明，最佳特征功能最多可以从自由边界线性地增长，即在自由边界点处是Lipschitz的连续。
We prove the existence of an open set minimizing the first Dirichlet eigenvalue of an elliptic operator with bounded, measurable coefficients, over all open sets of a given measure. Our proof is based on a free boundary approach: we characterize the eigenfunction on the optimal set as the minimizer of a penalized functional, and derive openness of the optimal set as a consequence of a Hölder estimate for the eigenfunction. We also prove that the optimal eigenfunction grows at most linearly from the free boundary, i.e. it is Lipschitz continuous at free boundary points.
