论文标题
估计仅通过其径向衍生物来控制功能并应用于椭圆方程的稳定解决方案
Estimates controlling a function by only its radial derivative and applications to stable solutions of elliptic equations
论文作者
论文摘要
我们建立了两个新的估计值，这些估计仅通过其径向衍生物的$ l^1 $规范来控制一个函数（减去其平均值）。虽然室内估计适用于所有超谐功能，但边界版本更加精致。它要求该函数是具有非负，非稳定和凸非线性的半线性椭圆方程的稳定解。作为应用程序，我们的估计提供了量化证明，证明了[Cabre，Figalli，Ros-Oton和Serra，Acta Math在[Cabre，Figalli，Ros-Oton和Serra中建立的两个结果。 224（2020）]。我们记得，这项工作证明了在最佳尺寸$ n \ leq 9 $的最佳范围内，稳定解决方案的Hölder规律性。
We establish two new estimates which control a function (after subtracting its average) in $L^1$ by only the $L^1$ norm of its radial derivative. While the interior estimate holds for all superharmonic functions, the boundary version is much more delicate. It requires the function to be a stable solution of a semilinear elliptic equation with a nonnegative, nondecreasing, and convex nonlinearity. As an application, our estimates provide quantitative proofs of two results established by contradiction-compactness arguments in [Cabre, Figalli, Ros-Oton, and Serra, Acta Math. 224 (2020)]. We recall that this work proved the Hölder regularity of stable solutions to semilinear elliptic equations in the optimal range of dimensions $n \leq 9$.
