论文标题
在一类带有多个解决方案的椭圆形边界问题上
On a class of elliptic free boundary problems with multiple solutions
论文作者
论文摘要
我们证明，某些类别的椭圆自由边界问题，其中包括流体动力学中的prandtl-batchelor问题，它具有两种不同的非平地解决方案，用于大型参数值。第一个解决方案是能量的全球最小化器。能量函数是非不同的，因此不能直接使用标准的变分参数来获得第二个非平凡解决方案。我们获得第二个解决方案，作为$ C^1 $功能的序列的界限，近似于能量。我们使用对相应能级的仔细估计来表明该极限既不是微不足道也不是最小化器。
We prove that a certain class of elliptic free boundary problems, which includes the Prandtl-Batchelor problem from fluid dynamics as a special case, has two distinct nontrivial solutions for large values of a parameter. The first solution is a global minimizer of the energy. The energy functional is nondifferentiable, so standard variational arguments cannot be used directly to obtain a second nontrivial solution. We obtain our second solution as the limit of mountain pass points of a sequence of $C^1$-functionals approximating the energy. We use careful estimates of the corresponding energy levels to show that this limit is neither trivial nor a minimizer.
