On sliding methods for mixed local and nonlocal equations and Gibbons' conjecture

We investigate elliptic and parabolic equations involving mixed local and nonlocal operators of the form $(-\mathrm{\Delta }{)}^s-\mathrm{\Delta }$, as well as their parabolic counterparts with both the Marchaud fractional time derivative and the classical first-order derivative. A major difficulty in this setting stems from the coexistence of operators with different nonlocal structures and incompatible scaling properties, which obstruct the direct use of classical sliding methods. To address this issue, we develop a refined sliding method suited to mixed local-nonlocal operators. As key technical ingredients, we establish new generalized weighted average inequalities, narrow region principles, and maximum principles in bounded and unbounded domains. These tools enable us to derive monotonicity and one-dimensional symmetry results for mixed elliptic equations in bounded domains, half-spaces, and the whole space, and to extend the analysis to parabolic equations with mixed time derivatives. As an application, we resolve the Gibbons’ conjecture for a class of mixed fractional equations.