论文标题
各向异性慢速扩散的抛物线竖琴估计
Parabolic Harnack estimates for anisotropic slow diffusion
论文作者
论文摘要
我们证明了抛物线方程的阳性解决方案具有缓慢各向异性空间扩散的阳性溶液。确定其自然量表后，我们将问题减少到fokker-Planck方程并构建自相似的Barenblatt解决方案。我们利用翻译不变性，通过自我介绍方法在起源附近获得阳性，并推断出阳性的阳性膨胀。最终，这产生了量表不变的harnack不等式，这是通过扩散系数速度决定的各向异性几何形状。作为推论，我们推断出Hölder的连续性，椭圆形的不平等和liouville定理。
We prove a Harnack inequality for positive solutions of a parabolic equation with slow anisotropic spatial diffusion. After identifying its natural scalings, we reduce the problem to a Fokker-Planck equation and construct a self-similar Barenblatt solution. We exploit translation invariance to obtain positivity near the origin via a self-iteration method and deduce a sharp anisotropic expansion of positivity. This eventually yields a scale invariant Harnack inequality in an anisotropic geometry dictated by the speed of the diffusion coefficients. As a corollary, we infer Hölder continuity, an elliptic Harnack inequality and a Liouville theorem.
