标题： 半线性分数椭圆偏微分方程在开球上的梯度非线性：解的存在性和概率表示 显示英文标题

标题： Semilinear fractional elliptic PDEs with gradient nonlinearities on open balls: existence of solutions and probabilistic representation

摘要： 我们提供了分数半线性椭圆PDE在$\alpha \in (1,2)$索引下，于$d$维球体$d\geq 2$上具有多项式梯度非线性的粘性解存在的充分条件。 我们的方法基于$\alpha$稳定分支过程的树状概率表示，使我们能够考虑迄今为止确定性有限差分方法未涵盖的梯度非线性项。 数值示例展示了该方法在维度$d=10$中的准确性，解决了在高维设置中使用确定性有限差分方法时遇到的挑战。 显示英文摘要

摘要： We provide sufficient conditions for the existence of viscosity solutions of fractional semilinear elliptic PDEs of index $\alpha \in (1,2)$ with polynomial gradient nonlinearities on $d$-dimensional balls, $d\geq 2$. Our approach uses a tree-based probabilistic representation based on $\alpha$-stable branching processes, and allows us to take into account gradient nonlinearities not covered by deterministic finite difference methods so far. Numerical illustrations demonstrate the accuracy of the method in dimension $d=10$, solving a challenge encountered with the use of deterministic finite difference methods in high-dimensional settings.