题目：Yau's Conjecture, Landis' Conjecture and Nodel Set for Elliptic and Parabolic Equations

报告人：刘祖汉 教授 （扬州大学）

地点：腾讯会议室

时间：2024年12月10日 星期二 14:30-15:30

摘要：We investigate the structure of the nodal set of solutions to equations of the form

$$-\mathrm{div}(A(x)\nabla u)=\lambda_+(u^+)^{q-1}-\lambda_-(u^-)^{q-1}, ~\hbox{~in~} B_1,$$

where $\lambda_+,\lambda_->0$, $q\in[1,2)$, and $u^+:=\max\{u,0\}$, $u^-:=\max\{-u,0\}$ are respectively the positive and the negative part of $u$. This collection of nonlinearities includes the unstable two-phase membrane problem $q=1$ as well as sublinear equations for $1<q<2$. Under suitable assumptions on $A(x)$, we prove the local behavior of solutions close to the nodal set, the complete classification of the admissible vanishing orders. Moreover, we study the regularity of the nodal set and we prove the estimates on the Hausdorff dimension of the nodal set, as well as a partial stratification theorem for the singular set. One of the main points in the proof are suitable generalizations of a family of Almgren-type and a 2-parameter family of Weiss-type monotonicity formulas for solutions of such systems. Our work generalizes previous results, where the case $A(x)=Id$ (i.e. the operator is the Laplacian) was treated.

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