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Evolution of nonparametric surfaces with speed depending on curvature. II: The mean curvature case. (English) Zbl 0808.53004
Let $$\Omega$$ be a bounded domain in $$\mathbb{R}^ n$$, $$n \geq 2$$ with $$C^ \infty$$ boundary $$\partial \Omega$$ and let $$Q$$ be the cylinder $$\Omega \times (0, \infty)$$. The authors investigate the solvability and asymptotic behavior of solutions of the following initial-boundary value problem $$u_ t = (1 + | Du|^ 2)^{1/2} H(u)$$ in $$Q$$, $$u(x,t) = 0$$ in $$\partial \Omega \times [0,\infty)$$, $$u(x,0) = u_ 0(x)$$ in $$\Omega$$, $$(u_ 0 \in C^ \infty _ 0(\Omega))$$, where $$H(u)=\text{div} (D(u) / \sqrt{1 + | Du|^ 2})$$, $$Du = \text{grad }u$$. This problem describes an evolution of the surface $$(x, u(x,t))$$, $$x \in \overline{\Omega}$$, propagating with normal speed equal to the mean curvature of the surface at the moment $$t$$. The case of the Gauss curvature is studied in the first author’s paper [Part I, Indiana Univ. Math. J. 40, 237-258 (1991; Zbl 0737.53002)].

##### MSC:
 53A05 Surfaces in Euclidean and related spaces 35G10 Initial value problems for linear higher-order PDEs 37C10 Dynamics induced by flows and semiflows
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##### References:
 [1] Bombieri, Arch. Rational Mech. Anal. 32 pp 255– (1969) [2] Chen, J. Diff. Geometry 33 pp 749– (1991) [3] Ecker, Ann. Math. 130 pp 450– (1989) [4] Ecker, Invent. Math. 105 pp 547– (1991) [5] Evans, J. Diff. Geometry 33 pp 635– (1991) [6] and , Motion of level sets by mean curvature III, preprint. [7] Gerhardt, J. Diff. Eq. 36 pp 139– (1980) [8] Huisken, J. Diff. Geometry 20 pp 237– (1984) [9] Huisken, J. Diff. Eq. 77 pp 369– (1989) [10] Jenkins, J. Reine Angew. Math. 299 pp 170– (1968) [11] Korevaar, Proc. Symposia Pure Math. 45 pp 81– (1986) · doi:10.1090/pspum/045.2/843597 [12] Ladyzhenskaya, Comm. Pure Appl. Math. 23 pp 677– (1970) [13] , and , Linear and Quasilinear Equations of Parabolic Type, AMS, Providence, 1968. [14] Lieberman, Ann. Scuola Norm. Sup. Pisa 13 pp 347– (1986) [15] Lichnewsky, J. Diff. Eq. 30 pp 340– (1978) [16] Oliker, Indiana Univ. J. 40 pp 237– (1991) [17] Self-similar solutions and asymptotic behavior of flows of nonparametric surfaces driven by Gauss or mean curvature, Proc. Symposia Pure Math., R. E. Greene and S. T. Yau, eds., to appear. · Zbl 0802.58054 [18] and , Evolution of nonparametric surfaces with speed depending on curvature, III: Some remarks on mean curvature and anisotropic flows, in: Degenerate Diffusions, , and , eds., IMA Volumes in Mathematics and its Applications, Vol. 47, Springer-Verlag, New York, to appear. · Zbl 0794.35087 [19] Osher, J. Computational Physics 79 pp 12– (1988) [20] Serrin, Philos. Trans. Royal Soc. London, Ser. A 264 pp 413– (1969) [21] Temam, Arch. Rational Mech. Anal. 44 pp 121– (1971)
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