• Jun Zheng

Articles written in Proceedings – Mathematical Sciences

• Remarks on Hausdorff Measure and Stability for the 𝑝-Obstacle Problem $(1 &lt; p &lt; 2)$

In this paper, we consider the obstacle problem for the inhomogeneous 𝑝-Laplace equation

$$\mathrm{div}(\nabla u|^{p-2}\nabla u)=f\cdot p\chi\{u&gt;0\},\quad 1 &lt; p &lt; 2,$$

where 𝑓 is a positive, Lipschitz function. We prove that the free boundary has finite $(N-1)$-Hausdorff measure and stability property, which completes previous works by Caffarelli (J. Fourier Anal. Appl. 4(4--5) (1998) 383--402) for $p=2$, and Lee and Shahgholian (J. Differ. Equ. 195 (2003) 14--24) for $2 &lt; p &lt; \infty$.

• Porosity of Free Boundaries in the Obstacle Problem for Quasilinear Elliptic Equations

In this paper, we establish growth rate of solutions near free boundaries in the identical zero obstacle problem for quasilinear elliptic equations. As a result, we obtain porosity of free boundaries, which is naturally an extension of the previous works by Karp et al. (J. Diff. Equ. 164 (2000) 110–117) for 𝑝-Laplacian equations, and by Zheng and Zhang (J. Shaanxi Normal Univ. 40(2) (2012) 11–13, 18) for 𝑝-Laplacian type equations.

• # Proceedings – Mathematical Sciences

Volume 131, 2021
All articles
Continuous Article Publishing mode

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019