走向现代数学学术报告 – 孙玉华教授（No. 999)

报告题目：Sharp Volume Tests for Quasilinear Elliptic Inequalities with Gradient Reactions

报 告 人：孙玉华 教授（南开大学）

邀 请 人：王大斌 教授

报告时间：2026年11月28日 9：00

报告地点：东海岸校区D209

报告摘要：We investigate positive weak solutions of

on complete noncompact Riemannian manifolds without curvature, volume doubling, or Poincare assumptions. For the superhomogeneous mixed reaction we derive a curvature-free integral volume test by combining two first-order estimates with a recurrence for the mass of a fixed density. The same approach organizes the pure-gradient and capacity criteria and yields sharp finite hierarchies of iterated-logarithmic growth bounds, with counterexamples on smooth model manifolds. In the subhomogeneous region, vanishing lower growth of forces the Liouville property. On the homogeneous line , we obtain the optimal strict lower-volume-entropy threshold for (and 1 for ), together with examples at equality. The existing six-region classification is not claimed as new; the advances concern refined volume hypotheses and their model sharpness. The tests remain valid for smooth weighted measures and for locally bounded positive Sobolev solutions. Sharpness is understood within the stated model families; no characterization from volume alone on an individual manifold is asserted.

报告人简介： 孙玉华，南开大学数学学院教授，博士生导师，国家级青年人才。本科毕业于吉林大学，在清华大学和德国比勒菲尔德大学获得博士学位。研究方向为黎曼流形及图上的分析，包括椭圆及抛物方程，在包括CPAM，Math. Annalen, JFA, CVPDE, JAM等国际著名期刊发表学术论文20余篇。 曾获得天津市数学会与统计学联合学术年会“青年学者奖”，入选天津市131创新型人才培养工程第三层次名单，主持多项国家自然科学基金。