论文标题
phragmén--lindelöf定理，用于具有非线性动力边界条件的弱椭圆方程
Phragmén--Lindelöf theorems for a weakly elliptic equation with a nonlinear dynamical boundary condition
论文作者
论文摘要
我们为完全非线性椭圆方程式建立了两个phragmén-lindelöf定理。我们考虑一个动态边界条件，其中包括空间变量和时间导数项。作为一个空间术语，我们考虑了一个非线性neumann型操作员，其在空间衍生项上的边界正常方向上具有严格的单调性。我们的第一个结果是在$ \ mathbb {r}^n $中的epigraph上的椭圆方程。因为我们假设一个良好的结构条件，其中包括一类宽类椭圆方程以及均匀的椭圆方程，我们可以从强大的最大原理中受益。第二个结果是对一个方向严格椭圆形的方程式。由于强大的最大原则不一定要适合此类方程式，因此我们采用通常用来证明最大原则的策略。考虑到平板上的这种方程式，我们可以通过严格满足粘度不平等的函数，然后获得矛盾来近似粘度下的物种。
We establish two Phragmén--Lindelöf theorems for a fully nonlinear elliptic equation. We consider a dynamic boundary condition that includes both spatial variable and time derivative terms. As a spatial term, we consider a non-linear Neumann-type operator with a strict monotonicity in the normal direction of the boundary on the spatial derivative term. Our first result is for an elliptic equation on an epigraph in $\mathbb{R}^n$. Because we assume a good structural condition, which includes wide classes of elliptic equations as well as uniformly elliptic equations, we can benefit from the strong maximum principle. The second result is for an equation that is strictly elliptic in one direction. Because the strong maximum principle need not necessarily hold for such equations, we adopt the strategy often used to prove the weak maximum principle. Considering such equations on a slab we can approximate the viscosity subsolutions by functions that strictly satisfy the viscosity inequality, and then obtain a contradiction.
