Existence of weak solutions to first-order stationary mean-field games with Dirichlet conditions
Existence of weak solutions to first-order stationary mean-field games with Dirichlet conditions
复制标题
狄利克雷条件下一阶平稳平均场博弈弱解的存在性
DOI：
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发表时间：
2018
影响因子：
1
通讯作者：
Teruo Tada
中科院分区：
文献类型：
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作者：
Rita Ferreira;D. Gomes;Teruo Tada
本文研究了具有Dirichlet边界条件的一阶平稳单调平均场对策。鉴于Hamilton-Jacobi方程可能不满足Dirichlet条件，本文建立了满足这些条件的mfg解的存在性。为了构造这些解，我们引入了一个单调正则化问题。利用Schaefer不动点定理和MFG的单调性，证明了正则化问题存在唯一的弱解。最后，我们取正则化问题解的极限，并利用Minty的方法证明了原正则化问题弱解的存在性。
In this paper, we study first-order stationary monotone mean-field games (MFGs) with Dirichlet boundary conditions. Whereas Dirichlet conditions may not be satisfied for Hamilton–Jacobi equations, here we establish the existence of solutions to MFGs that satisfy those conditions. To construct these solutions, we introduce a monotone regularized problem. Applying Schaefer’s fixed-point theorem and using the monotonicity of the MFG, we verify that there exists a unique weak solution to the regularized problem. Finally, we take the limit of the solutions of the regularized problem and, using Minty’s method, we show the existence of weak solutions to the original MFG.