Viscosity solutions to second order partial differential equations on Riemannian manifolds
Viscosity solutions to second order partial differential equations on Riemannian manifolds
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DOI：
10.1016/j.jde.2008.03.030
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发表时间：
2006-12
影响因子：
2.4
通讯作者：
D. Azagra;J. Ferrera;B. Sanz
中科院分区：
文献类型：
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作者：
D. Azagra;J. Ferrera;B. Sanz
我们证明了定义在有限维黎曼流形M上的一类完全非线性二阶偏微分方程F(x，u，Du，d2u)=0的粘性解的比较、唯一性和存在性结果。在假设函数F是退化椭圆的，即二阶导数不增，且关于变量x一致连续的假设下，得到了最好的结果，而如果另外要求F以一致连续的方式依赖于d2u，则建立了比较结果。
We prove comparison, uniqueness and existence results for viscosity solutions to a wide class of fully nonlinear second order partial differential equations F(x,u,du,d2u)=0 defined on a finite-dimensional Riemannian manifold M. Finest results (with hypothesis that require the function F to be degenerate elliptic, that is nonincreasing in the second order derivative variable, and uniformly continuous with respect to the variable x) are obtained under the assumption that M has nonnegative sectional curvature, while, if one additionally requires F to depend on d2u in a uniformly continuous manner, then comparison results are established with no restrictive assumptions on curvature.