부산대학교 도서관

The authors prove smoothness of an imposed mean curvature flow. Moreprecisely, given a vector field $u$ and a compact smooth surface $M_0$ in$\Bbb{R}^n$, the imposed mean curvature flow$\{M_t\}_{t>0}\subset\Bbb{R}^n$ obeys $\lim_{t\to 0+}M_t=M_0$and $v=h+u^\perp$, where $v$, $h$ and $u^\perp$ respectivelymean the velocity, the mean curvature vector $h$ at any point of $M_t$and the projection to $M_t$ of a certain integrable vector$u$ of $\Bbb{R}^n$. Their main result is the following:\parMain theorem. Suppose an imposed mean curvature flow $\{M_t\}_{t>0}$belongs to $C^{1,\alpha}$ $(0<\alpha<1)$ and is expressed as a graph $x_n=f(x_1,x_2,\dots,x_{n-1},t)$. If $\partial_t f$ is of asigned Radon measure, then $\partial_t f,\nabla^2 f\in L^{p,q}$,i.e., $f$ is the strong solution to$$\frac{\partial_t f}{\sqrt{1+|\nablaf|^2}}=\roman{div}\left(\frac{\nabla f}{\sqrt{1+|\nablaf|^2}}\right) +u\cdot\frac{(-\nabla f,1)}{\sqrt{1+|\nabla f|^2}}.$$Here note that a smoothness in space variables is unlikely to beinherited by a smoothness in time one in the solution of such a PDE with strong nonlinearity.\par The authors show this claim, first by assuming $\partial_t f\in L^2$, andsecond by establishing Schauder estimates of $\nabla^2 f$ in $L^2$ underthe assumption of the theorem. Next they approximate a signed Radonmeasure by $L^2$-function and finally they pass to the limit.