Mean value formulas are of great importance in the theory of partial differential equations: many very useful results are drawn, for instance, from the well-known equivalence between harmonic functions and mean value properties. In the nonlocal setting of fractional harmonic functions, such an equivalence still holds, and many applications are nowadays available. The nonlinear case, corresponding to the p-Laplace operator, has also been recently investigated, whereas the validity of a nonlocal, nonlinear, counterpart remains an open problem. In this paper, we propose a formula for the nonlocal, nonlinear mean value kernel, by means of which we obtain an asymptotic representation formula for harmonic functions in the viscosity sense, with respect to the fractional (variational) p-Laplacian (for p ≥ 2) and to other gradient-dependent nonlocal operators.

中文翻译：

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分数 p-拉普拉斯算子和梯度相关非局部算子的渐近展开

平均值公式在偏微分方程理论中非常重要：例如，从调和函数与平均值性质之间的众所周知的等价性中可以得出许多非常有用的结果。在分数调和函数的非局部设置中，这样的等价性仍然成立，并且现在有许多应用程序可用。非线性情况，对应于p-Laplace 算子，最近也被研究过，而非局部、非线性对应物的有效性仍然是一个悬而未决的问题。在本文中，我们提出了一个公式非局部、非线性均值核, 通过它，我们获得了粘度意义上的谐波函数的渐近表示公式，关于分数（变分）p-拉普拉斯算子（对于p ≥ 2) 和其他梯度相关的非局部算子。