标题： 椭圆哈密顿-雅可比方程解的分类 显示英文标题

标题： Classification of solutions of an elliptic Hamilton-Jacobi equation

摘要： 我们证明了在半空间中，对于$1<p\le 2$的零边界条件，扩散型哈密顿-雅可比（DHJ）方程$-\Delta u= |\nabla u|^p$的任何经典解必然是一个一维的解。 这改进了之前已知的结果，该结果需要额外的有上界假设。 结合现有的关于$p>2$的类似结果，我们的结果完成了半空间中方程（DHJ）狄利克雷问题的完整分类图景。 显示英文摘要

摘要： We show that any classical solution of the diffusive Hamilton-Jacobi (DHJ) equation $-\Delta u= |\nabla u|^p$ in a half-space with zero boundary conditions for $1<p\le 2$ is necessarily one-dimensional. This improves the previously known result, which required an extra assumption of boundedness from above. Combined with the existing analogous result for $p>2$, our result completes the full classification picture of the Dirichlet problem for equation (DHJ) in a half-space.