The paper studies the evolution by mean curvature of hypersurfaces with fixed boundary by exploiting the variational approach called ``generalized minimizing movements'' proposed by E. De Giorgi. For every bounded, open subset $A$ of ${\Bbb R}^n$ satisfying suitable hypotheses, the authors consider a family of integral functionals ${\scr F}_\lambda$, $\lambda>0$, defined on pairs of measurable subsets of $A$ having finite perimeter in $A$. The functionals ${\scr F}_\lambda$ are close relatives of those already considered by F. J. Almgren, Jr., J. E. Taylor and L. Wang [SIAM J. Control Optim. 31 (1993), no. 2, 387--438; MR1205983 (94h:58067)] and now include an extra term which penalizes those sets whose trace on the boundary $\partial A$ of $A$ differs from a given subset $\Gamma$ of $\partial A$. Starting from a suitable initial condition $E_0$, the authors recursively define for every $\lambda>0$ a sequence of sets $E_\lambda (k)$, $k\in{\Bbb N}$, such that $E_\lambda (0)=E_0$ and $E_\lambda (k+1)$ minimizes ${\scr F}_\lambda (·\,,E_\lambda (k))$ among all subsets of $A$ having finite perimeter in $A$. Then they prove that, for some sequence $\lambda_j\to\infty$, there are sets of finite perimeter $E(t)$, $t>0$, such that $E_{\lambda_j}([\lambda_jt])\to E(t)$ as $j\to\infty$ in the sense of strong $L^1(A)$ convergence of the corresponding characteristic functions. Here, $[a]$ denotes the integer part of $a\geq 0$. According to De Giorgi's original definition, the family of sets $E(t)$, $t\geq 0$, is a generalized minimizing movement associated with the functionals ${\scr F}_\lambda$ and the main point here is that each set $E(t)$ satisfies the boundary condition, i.e., the trace of $E(t)$ on $\partial A$ is $\Gamma$. The authors emphasize that the relationship between the functionals ${\scr F}_\lambda$ here considered and those appearing in the paper by Almgren, Taylor and Wang mentioned above suggests that the generalized minimizing movements associated with ${\scr F}_\lambda$ can be considered as candidates for being classical motion by mean curvature.
Generalized Minimizing Movements for the Mean Curvature Flow with Dirichlet Boundary Condition.
MASSARI, Umberto;
1999
Abstract
The paper studies the evolution by mean curvature of hypersurfaces with fixed boundary by exploiting the variational approach called ``generalized minimizing movements'' proposed by E. De Giorgi. For every bounded, open subset $A$ of ${\Bbb R}^n$ satisfying suitable hypotheses, the authors consider a family of integral functionals ${\scr F}_\lambda$, $\lambda>0$, defined on pairs of measurable subsets of $A$ having finite perimeter in $A$. The functionals ${\scr F}_\lambda$ are close relatives of those already considered by F. J. Almgren, Jr., J. E. Taylor and L. Wang [SIAM J. Control Optim. 31 (1993), no. 2, 387--438; MR1205983 (94h:58067)] and now include an extra term which penalizes those sets whose trace on the boundary $\partial A$ of $A$ differs from a given subset $\Gamma$ of $\partial A$. Starting from a suitable initial condition $E_0$, the authors recursively define for every $\lambda>0$ a sequence of sets $E_\lambda (k)$, $k\in{\Bbb N}$, such that $E_\lambda (0)=E_0$ and $E_\lambda (k+1)$ minimizes ${\scr F}_\lambda (·\,,E_\lambda (k))$ among all subsets of $A$ having finite perimeter in $A$. Then they prove that, for some sequence $\lambda_j\to\infty$, there are sets of finite perimeter $E(t)$, $t>0$, such that $E_{\lambda_j}([\lambda_jt])\to E(t)$ as $j\to\infty$ in the sense of strong $L^1(A)$ convergence of the corresponding characteristic functions. Here, $[a]$ denotes the integer part of $a\geq 0$. According to De Giorgi's original definition, the family of sets $E(t)$, $t\geq 0$, is a generalized minimizing movement associated with the functionals ${\scr F}_\lambda$ and the main point here is that each set $E(t)$ satisfies the boundary condition, i.e., the trace of $E(t)$ on $\partial A$ is $\Gamma$. The authors emphasize that the relationship between the functionals ${\scr F}_\lambda$ here considered and those appearing in the paper by Almgren, Taylor and Wang mentioned above suggests that the generalized minimizing movements associated with ${\scr F}_\lambda$ can be considered as candidates for being classical motion by mean curvature.I documenti in SFERA sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
