论文标题
Hölder的规律性和liouville属性具有功率增长术语的非线性椭圆形不平等现象
Hölder regularity and Liouville properties for nonlinear elliptic inequalities with power-growth gradient terms
论文作者
论文摘要
本说明研究了局部积分梯度界限，以用于大量部分差分不平等的分布解，其散发形式和类似功率的一阶项的扩散。这些估计的应用是两个倍。首先，我们在此类的扩散PDE上显示了（尖锐的）全球Hölder分配半分布的规律性，其一阶术语具有超自然生长和右侧的一阶术语，并在适当的Morrey类中放置在有限和常规的开放式$ω$上。其次，我们为具有超线一阶术语的不平等的整个liouville属性提供了新的证明，而没有假设对相应均质偏差不等式的解决方案上有任何一侧绑定。我们还讨论了先前特性的一些扩展，以在亚riemannian几何形状中产生的问题，并讨论在非相入的完全差异歧管上构成的部分差异不平等，在大地测量球的适当面积生长条件下，在这两个方向上提供了新的结果。这些方法依赖于整体参数，并且不利用最大和比较原则。
This note studies local integral gradient bounds for distributional solutions of a large class of partial differential inequalities with diffusion in divergence form and power-like first-order terms. The applications of these estimates are two-fold. First, we show the (sharp) global Hölder regularity of distributional semi-solutions to this class of diffusive PDEs with first-order terms having supernatural growth and right-hand side in a suitable Morrey class posed on a bounded and regular open set $Ω$. Second, we provide a new proof of entire Liouville properties for inequalities with superlinear first-order terms without assuming any one-side bound on the solution for the corresponding homogeneous partial differential inequalities. We also discuss some extensions of the previous properties to problems arising in sub-Riemannian geometry and also to partial differential inequalities posed on noncompact complete Riemannian manifolds under appropriate area-growth conditions of the geodesic spheres, providing new results in both these directions. The methods rely on integral arguments and do not exploit maximum and comparison principles.
