On a new singular and degenerate extension of the p-Laplace operator

We study a novel degenerate and singular elliptic operator ${\tilde{\Delta }}_{(\tau ,\chi )}$ defined by ${\tilde{\Delta }}_{(\tau ,\chi )}u=\tau (x,Du)(\left|Du\right|{\Delta }_{1}u+\chi (x,Du){\Delta }_{\infty }u),$ where the singular weights $\tau (x,s)>0$ and $\chi (x,s)\ge 0$ are continuous functions on $\Omega \times R^n\setminus \left\{0\right\}$. The operator ${\tilde{\Delta }}_{(\tau ,\chi )}$ is an extension of ${\Delta }_{(p,q)}u={\left|Du\right|}^q{\Delta }_{1}u+(p-1){\left|Du\right|}^{p-2}{\Delta }_{\infty }u,p\ge 1,q\ge 0,$ introduced by the authors in Baravdishet al. (2020), which in turn is an extension of the $p$-Laplace operator ${\Delta }_p.$ We establish the well-posedness of the Neumann boundary value problem for the parabolic equation $u_t={\tilde{\Delta }}_{(\tau ,\chi )}u$ in the framework of viscosity solutions. For the solution $u$, the weight $\chi$ controls the evolution along the tangential and the normal directions, respectively, on the level surface of $u$. The weight $\tau$ controls the total speed of the evolution of $u$. We also prove the consistency and the convergence of the numerical scheme for the finite differences method of the parabolic equation above. Numerical simulations show that our novel nonlinear operator ${\tilde{\Delta }}_{(\tau ,\chi )}$ gives better results than both the Perona–Malik (Perona and Malik, 1990) and total variation (TV) methods (Chan and Shen, 2005) when applied to image enhancement.