In this paper,for the solution of the torsion problem about the equation Δu=-2 with homogeneous Dirichlet boundary conditions in a bounded convex domain in Rn,we find a superharmonic function which implies the strict...In this paper,for the solution of the torsion problem about the equation Δu=-2 with homogeneous Dirichlet boundary conditions in a bounded convex domain in Rn,we find a superharmonic function which implies the strict concavity of u ^(1/2) and give some convexity estimates.It is a generalization of Makar-Limanov’s result(Makar-Limanov(1971))and Ma-Shi-Ye’s result(Ma et al.(2012)).展开更多
For the Monge-Amp`ere equation det D2u = 1, the authors find new auxiliary curvature functions which attain their respective maxima on the boundary. Moreover, the upper bounded estimates for the Gauss curvature and th...For the Monge-Amp`ere equation det D2u = 1, the authors find new auxiliary curvature functions which attain their respective maxima on the boundary. Moreover, the upper bounded estimates for the Gauss curvature and the mean curvature of the level sets for the solution to this equation are obtained.展开更多
基金supported by National Key Research and Development Project (Grant No. SQ2020YFA070080)National Natural Science Foundation of China (Grant Nos. 11871255 and 11721101)supported by National Natural Science Foundation of China (Grant Nos. 11971137 and 11771396)
文摘In this paper,for the solution of the torsion problem about the equation Δu=-2 with homogeneous Dirichlet boundary conditions in a bounded convex domain in Rn,we find a superharmonic function which implies the strict concavity of u ^(1/2) and give some convexity estimates.It is a generalization of Makar-Limanov’s result(Makar-Limanov(1971))and Ma-Shi-Ye’s result(Ma et al.(2012)).
基金supported by the Chinese Universities Scientific Fund(No.WK0010000028)supported by the National Science Fund for Distinguished Young Scholars of China and Wu Wen-Tsun Key Laboratory of Mathematics+1 种基金partially supported by the National Natural Science Foundation of China(Nos.11101110,11326144)the Foundation of Harbin Normal University(No.KGB201224)
文摘For the Monge-Amp`ere equation det D2u = 1, the authors find new auxiliary curvature functions which attain their respective maxima on the boundary. Moreover, the upper bounded estimates for the Gauss curvature and the mean curvature of the level sets for the solution to this equation are obtained.
