标题： 关于具有$L^1$数据的非局部 1-拉普拉斯方程的解 显示英文标题

标题： On the solutions of nonlocal 1-Laplacian equation with $L^1$-data

摘要： 我们研究由 $$ 2\text{P.V.}\int_{\mathbb{R}^N}\frac{u(x)-u(y)}{|u(x)-u(y)|} \frac{dy}{|x-y|^{N+s}}=f(x) \quad \textmd{for } x\in \Omega, $$ 给出的非局部 1-拉普拉斯方程在 $\mathbb R^N\backslash \Omega$ 上的狄利克雷边界条件 $u(x)=0$ 和非负 $L^1$-数据的解。 By investigating the asymptotic behaviour of renormalized solutions $u_p$ to the nonlocal $p$-Laplacian equations as $p$ goes to $1^+$, we introduce a suitable definition of solutions and prove that the limit function $u$ of $\{u_p\}$ is a solution of the nonlocal $1$-Laplacian equation above. 显示英文摘要

摘要： We study the solutions to a nonlocal 1-Laplacian equation given by $$ 2\text{P.V.}\int_{\mathbb{R}^N}\frac{u(x)-u(y)}{|u(x)-u(y)|} \frac{dy}{|x-y|^{N+s}}=f(x) \quad \textmd{for } x\in \Omega, $$ with Dirichlet boundary condition $u(x)=0$ in $\mathbb R^N\backslash \Omega$ and nonnegative $L^1$-data. By investigating the asymptotic behaviour of renormalized solutions $u_p$ to the nonlocal $p$-Laplacian equations as $p$ goes to $1^+$, we introduce a suitable definition of solutions and prove that the limit function $u$ of $\{u_p\}$ is a solution of the nonlocal $1$-Laplacian equation above.