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Non-parametric mean curvature evolution with boundary conditions. (English) Zbl 0686.34013
An evolution equation, $$\dot u+Au=0$$ in $$\Omega$$ $$\times [0,t]$$ has previously been investigated where the surfaces, $$m_ t=graph$$ u($$\cdot,t)$$ moves in the $$x^{n+1}$$-direction with speed dictated by their mean curvature, $$H=-Au$$. This paper replaces this speed to be in the direction of their unit normal equal to the mean curvature. With this focus in mind the paper begins with the boundary value problem, (1) $$\dot u+(1+| Du|^ 2)^{1/2}Au=0$$ in $$\Omega$$ $$\times [0,\infty]$$, (2) $$a^ i(Du)\cdot \gamma_ i=0$$ on $$\partial \Omega \times [0,\infty]$$, (3), $$u(\cdot,0)=u_ 0$$. The operator A in equation (1) is the minimal surface operator given by $A=-D_ i(a^ i(p)),\quad a^ i=p^ i(1+| p|^ 2)^{-1\setminus 2}$ and $$\Omega$$ is a bounded domain in $${\mathbb{R}}^ n$$ with $$\partial \Omega$$ being of class, $$C^{2,\alpha}$$. The outer unit normal to $$\partial \Omega$$ is denoted by $$\gamma =(\gamma_ 1,\gamma_ 2,...,\gamma_ n)$$. The paper then continues to prove in a well written organized style two principal theorems. The first one requires that the boundary condition given in equation (3) satisfies $$u_ 0\in C^{2,\alpha}({\bar \Omega})$$ and $$a^ i(Du_ 0)\cdot \gamma_ i=0$$ on $$\partial \Omega$$. Under this hypothesis it is then shown that the problem (1), (2), (3) has a smooth solution u and $$u_ t=u(\cdot,t)$$ converges to a constant function as $$t\to \infty$$. The proof is elegant and requires several technical details. The other theorem replaces the boundary condition given in equation (2) with $$u_ 0=\phi$$ on $$\partial \Omega$$ where $$\phi \in C^{2,\alpha}({\bar \Omega})$$. In this situation the problem then has again a smooth solution u and $$u_ t=u(\cdot,t)$$ converges to the solution of the minimal surface equation with boundary data $$\phi$$ as $$t\to \infty$$.
Reviewer: J.Schmeelk

##### MSC:
 34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations 35G10 Initial value problems for linear higher-order PDEs 35K25 Higher-order parabolic equations
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##### References:
 [1] Brakke, K.A., The motion of a surface by its Mean curvature, () · Zbl 0521.90039 [2] Ecker, K., Estimates for evolutionary surfaces of prescribed Mean curvature, Math. Z., 180, 179-192, (1982) · Zbl 0469.35053 [3] Gerhardt, C., Evolutionary surfaces of prescribed Mean curvature, J. differential equations, 36, 139-172, (1980) · Zbl 0485.35053 [4] Gerhardt, C., Global regularity of the solutions to the capillarity problem, Ann. sci. norm. sup. Pisa ser. IV, 4, 343-362, (1977) [5] Huisken, G., Flow by Mean curvature of convex surfaces into spheres, J. differential geom., 20, 237-266, (1984) · Zbl 0556.53001 [6] Huisken, G., Contracting convex hypersurfaces in riemannian manifolds by their Mean curvature, Invent. math., 84, 463-480, (1986) · Zbl 0589.53058 [7] Lichnewski, A.; Temam, R., Surfaces minimales d’évolution: le concept de pseudosolution, C.R. acad. sci. Paris, 284, 853-856, (1977) · Zbl 0354.35053 [8] Lieberman, G.M., The first initial boundary value problem for quasilinear second order parabolic equations, Ann. sci. norm. sup. Pisa ser. IV, 8, 347-387, (1986) · Zbl 0655.35047 [9] Michael, J.H.; Simon, L.M., Sobolev and Mean value inequalities on generalized submanifolds of $$R$$^n, Comm. pure appl. math., 26, 361-379, (1973) · Zbl 0256.53006 [10] Serrin, J., The problem of Dirichlet for quasilinear elliptic differential equations with many independent variables, Philos. trans. roy. soc. London ser. A, 264, 413-496, (1969) · Zbl 0181.38003 [11] Stampacchia, G., Équations elliptiques du second ordre à coefficients discontinus, (1966), Les Presses de l’Université Montréal · Zbl 0151.15501 [12] Ural’ceva, N.N., The solvability of the capillarity problem, Vestnik leningrad univ. mat. mekh. astronom., 4, 54-64, (1973)
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