On the Lipschitz continuity of certain quasiregular mappings between smooth Jordan domains

We first investigate the Lipschitz continuity of $(K,K^{\mathrm{\prime }})$-quasiregular $C^2$ mappings between two Jordan domains with smooth boundaries, satisfying certain partial differential inequalities concerning Laplacian. Then two applications of the obtained result are given: As a direct consequence, we get the Lipschitz continuity of $\rho$-harmonic $(K,K^{\mathrm{\prime }})$-quasiregular mappings, and as the other application, we study the Lipschitz continuity of $(K,K^{\mathrm{\prime }})$-quasiconformal self-mappings of the unit disk, which are the solutions of the Poisson equation $\mathrm{\Delta }w=g$. These results generalize and extend several recently obtained results by Kalaj, Mateljevi'{c} and Pavlovi'{c}.