On viscosity solutions of certain Hamilton-Jacobi equations: Regularity results and generalized Sard's theorems
On viscosity solutions of certain Hamilton-Jacobi equations: Regularity results and generalized Sard's theorems
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DOI：
10.1080/03605300701382522
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发表时间：
2008-01-01
影响因子：
1.9
通讯作者：
Rifford, Ludovic
中科院分区：
文献类型：
--
作者：
Rifford, Ludovic
在哈密顿量的一般假设下，我们证明了流形M上对应的哈密顿-雅可比方程的任何黏性解在其奇异集闭包之外是局部半洞和C-loc（1,1）的。此外，我们证明了在附加的假设和低维情况下，该Hamilton-Jacobi方程的任何粘度解都满足广义Sard定理。因此，这种函数的几乎每一个水平集都是M中的局部Lipschitz超曲面。
Under usual assumptions on the Hamiltonian, we prove that any viscosity solution of the corresponding Hamilton-Jacobi equation on the manifold M is locally semiconcave and C-loc(1,1) outside the closure of its singular set (which is nowhere dense in M). Moreover, we prove that, under additional assumptions and in low dimension, any viscosity solution of that Hamilton-Jacobi equation satisfies a generalized Sard theorem. In consequence, almost every level set of such a function is a locally Lipschitz hypersurface in M.