论文标题
通过椭圆规律性的当地一阶平均野战游戏的古典和弱解决方案
Classical and weak solutions to local first-order mean field games through elliptic regularity
论文作者
论文摘要
我们研究了局部，一阶前回向野外游戏系统的规律性和适应性，假设具有多项式增长的成本函数和二次增长的哈密顿量。我们认为系统和终端数据严格在密度中是单调的，并根据是否存在运行成本函数的下限研究了两种不同的制度。这项工作依赖于P.-L.引起的转换。狮子会产生带有倾斜边界条件的椭圆形偏微分方程，当耦合从下方不绑定时，这是严格的椭圆形。在这种情况下，我们证明解决方案是平滑的。当问题是退化的椭圆形时，我们获得了与Cardaliaguet和P.J. Graber所获得的弱溶液的存在和唯一性，对于与密度无关的末端条件的情况。薄弱的解决方案显示出是严格椭圆问题的经典解决方案的粘性限制。
We study the regularity and well-posedness of the local, first-order forward-backward mean field games system, assuming a polynomially growing cost function and a Hamiltonian of quadratic growth. We consider systems and terminal data that are strictly monotone in the density and study two different regimes depending on whether there exists a lower bound for the running cost function. The work relies on a transformation due to P.-L. Lions, which gives rise to an elliptic partial differential equation with oblique boundary conditions, that is strictly elliptic when the coupling is unbounded from below. In this case, we prove that the solution is smooth. When the problem is degenerate elliptic, we obtain existence and uniqueness of weak solutions analogous to those obtained by P. Cardaliaguet and P.J. Graber for the case of a terminal condition that is independent of the density. The weak solutions are shown to arise as viscous limits of classical solutions to strictly elliptic problems.
