标题： 半线性分数椭圆PDEs在开球上带有梯度非线性：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$\alpha \in (1,2)$指数下，于$d$维球体$d\geq 2$上存在粘性解的充分条件，这些PDE具有多项式梯度非线性项。我们的方法使用基于树的概率表示方法，通过$\alpha$稳定分支过程来表示解及其部分导数，使我们能够考虑目前确定性有限差分方法未涵盖的梯度非线性项。与现有文献中关于解的正则性的研究相比，对梯度非线性项没有施加多项式阶条件。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 of solutions and their partial derivatives using $\alpha$-stable branching processes, and allows us to take into account gradient nonlinearities not covered by deterministic finite difference methods so far. In comparison with the existing literature on the regularity of solutions, no polynomial order condition is imposed on gradient nonlinearities. 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.