Studies on the applications of the theory of viscosity solutions to some singular perturbation problems

Katsuyuki Ishii 研究了 Bence、Merriman 和 Osher 提出的平均曲率运动数值算法，并得到了以下结果： 1.我给出了收敛性证明，展示了如何从该算法导出平均曲率流动方程。 （这是与 Yoko Goto 和 Takayoshi Okawa 的共同工作。）2.我通过平均曲率获得了平滑运动情况下该算法的收敛速度。我还展示了在由曲率演化的圆的情况下的最优性。Kenji Maruo 研究了平面上的半线性简并椭圆偏微分方程。他得到了： 3.假设方程的系数是径向对称的，他证明了在无穷远的某些增长条件下，连续粘度解是径向对称的。Yasuo Kageyama 得到了收敛速度和修正伯恩斯坦多项式的一些性质

Katsuyuki Ishii studied a numerical algorithm for motion by mean curvature, which is proposed by Bence, Merriman and Osher and obtained the following results1.I gave a proof of convergence showing how the mean curvature flow equation is derived from this algorithm. (This is a joint work with Yoko Goto and Takayoshi Ogawa.)2.I obtained the rate of convergence of this algorithm in the case of smoth motion by mean curvature. I also showed the optimality in the case of a circle evolving by curvature.Kenji Maruo studied semilinear degenerate elliptic partial differential equations in the plane. He obtained the following3.Assuming that the coefficients of the equation are radially symmetric, he proved that, under some growth conditions at infinity, the continuous viscosity solutions are radially symmetricYasuo Kageyama obtained the rate of convergence and some properties of modified Bernstein polynomials