论文标题
特殊拉格朗日电位方程中比较原理的反例
Counterexamples to the comparison principle in the special Lagrangian potential equation
论文作者
论文摘要
对于每个$ k = 0，\ dots，n $，我们构建一个连续阶段$ f_k $，带有$ f_k（0）=（n-2k）\fracπ{2} $，以及粘度子和supersolutions $ v_k $，$ v_k $，$ u_k $，e elliptic pde pde pde pde $ \ sum_ = 1} i = 1}^n \ rctan（ f_k（x）$使得$ v_k-u_k $在原点上具有孤立的最大值。 对于任意连续阶段$ f \colonΩ\ to（-nπ/2，nπ/2）$，比较原理是否会在二阶方程中保持比较原理是一个空的问题。我们的例子表明不是。
For each $k = 0,\dots,n$ we construct a continuous phase $f_k$, with $f_k(0) = (n-2k)\fracπ{2}$, and viscosity sub- and supersolutions $v_k$, $u_k$, of the elliptic PDE $\sum_{i=1}^n \arctan(λ_i(D^2 w)) = f_k(x)$ such that $v_k-u_k$ has an isolated maximum at the origin. It has been an open question whether the comparison principle would hold in this second order equation for arbitrary continuous phases $f\colonΩ\to (-nπ/2,nπ/2)$. Our examples show it does not.
