Stability for the infinity-Laplace equation with variable exponent. (Q454539)

The equation NEWLINE\[NEWLINE \Delta _{\infty (x)}u=\Delta _\infty u+| \nabla u| ^2\log (| \nabla u| ) \langle \nabla u, \nabla \log p(x)\rangle =0 NEWLINE\]NEWLINE arises as the limit \(k\to \infty \) of \(kp(x)\)-Laplacians NEWLINE\[NEWLINE \Delta _{kp(x)}u=\text{div}(| \nabla u| ^{kp(x)-2}\nabla u)=0. NEWLINE\]NEWLINE Here NEWLINE\[NEWLINE \Delta _\infty u=\langle D^2u\nabla u,\nabla u\rangle = \sum _{i,j} u_{x_i x_j}u_{x_i}u_{x_j} NEWLINE\]NEWLINE is the so-called infinity Laplacian. The authors consider the stability of viscosity solutions to the first equation with respect to perturbations in the function \(p(x)\). The results are qualitative: estimates for the difference of two solutions \(u_1\) and \(u_2\) with fixed boundary values in terms of the difference of the respective functions \(p_1(x)\) and \(p_2(x)\) are proved. The proofs are modifications of the arguments leading to a comparison principle for this type of equations.