论文标题
对于具有可变系数的运算符的半线性椭圆方程的稳定解决方案
Stable solutions to semilinear elliptic equations for operators with variable coefficients
论文作者
论文摘要
在本文中，我们扩展了[Cabré，Figalli，Ros-oton和Serra，Acta Math的稳定解决方案的内部规律性结果。 224（2020）]给具有可变系数的操作员。我们表明，半线性椭圆方程的稳定解决方案$ a_ {ij}（x）u_ {ij} + b_i（x）u_i + f（u）= 0 $在最佳范围内连续hölder连续$ n \ leq 9 $。我们的界限独立于c^1 $中的非线性$ f \，我们认为这是非负的。 我们工作的主要成就是使我们的估计中的常数取决于$ a_ {ij} $的$ c^1 $规范和$ c^0 $ norm of $ b_i $，而不是他们的$ c^2 $和$ c^1 $ norms，这是第一种计算方法所产生的。
In this paper we extend the interior regularity results for stable solutions in [Cabré, Figalli, Ros-Oton, and Serra, Acta Math. 224 (2020)] to operators with variable coefficients. We show that stable solutions to the semilinear elliptic equation $a_{ij}(x)u_{ij} + b_i(x) u_i + f(u) = 0$ are Hölder continuous in the optimal range of dimensions $n \leq 9$. Our bounds are independent of the nonlinearity $f \in C^1$, which we assume to be non-negative. The main achievement of our work is to make the constants in our estimates depend on the $C^1$ norm of $a_{ij}$ and the $C^0$ norm of $b_i$, instead of their $C^2$ and $C^1$ norms, respectively, which arise in a first approach to the computations.
