论文标题
分数$ p $ laplacian的渐近扩展和依赖梯度的非本地操作员
An asymptotic expansion for the fractional $p$-Laplacian and for gradient dependent nonlocal operators
论文作者
论文摘要
平均值公式在偏微分方程理论中至关重要：例如，从谐波函数和平均值属性之间的众所周知的等效性中得出了许多非常有用的结果。在分数谐波函数的非本地设置中，这种等价仍然存在，现在有许多应用程序可用。与$ p $ laplace运营商相对应的非线性案例最近也进行了调查，而非本地，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\ emph {非局部，非线性平均值核}的公式，我们通过该公式在粘度意义上获得了谐波函数的渐近表示公式，相对于分数（变性）$ p $ -laplacian（对于$ p $ laplacian）
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 now-days 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 \emph{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 $p\geq 2$) and to other gradient dependent nonlocal operators.
