论文标题
线性椭圆和抛物线方程的解决方案梯度的指数界限
Exponential bounds for gradient of solutions to linear elliptic and parabolic equations
论文作者
论文摘要
在本文中，我们证明了对线性椭圆和抛物线方程的解决方案的全球梯度估计。对于足够平滑的有界凸域$ω\ subset \ mathbb {r}^n $，我们证明了w_0^{1，\ infty}中的解决方案$ ϕ \ in w_0^{1，\ infty}（ω）$ l^{\ infty}（ω; \ \ m athbb {r}）$，满足$ | \ nabla ϕ | _ {\ infty} \ leq c | f | f | _ {\ infty} $，带有正常常数$ c = \ exp（c = \ exp（c（c））我们还可以在抛物线环境中获得类似的估计。这些指数界限的证明取决于本·安德鲁斯（Ben Andrews）和朱莉·克鲁特巴克（Julie Clutterbuck）的一系列论文启发的全球梯度估计。这项工作是由兰迪斯猜想的双重版本激励的。
In this paper, we prove global gradient estimates for solutions to linear elliptic and parabolic equations. For a sufficiently smooth bounded convex domain $Ω\subset \mathbb{R}^N$, we show that a solution $ϕ\in W_0^{1,\infty}(Ω)$ to an appropriate elliptic equation $\mathcal{L} ϕ= F$, with $F \in L^{\infty}(Ω;\mathbb{R})$, satisfies $|\nabla ϕ|_{\infty} \leq C |F|_{\infty}$, with a positive constant $C = \exp(C(\mathcal{L})\text{diam}(Ω))$. We also obtain similiar estimates in the parabolic setting. The proof of these exponential bounds relies on global gradient estimates inspired by a series of papers by Ben Andrews and Julie Clutterbuck. This work is motivated by a dual version of the Landis conjecture.
