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, 20 March 2003, Pages 46–70
Another approach to the heat flow for Plateau problem, , ,
Institute of Mathematics, Peking University, Beijing 100871, People's Republic of ChinaFor the minimal surfaces in Rn with Plateau boundary condition and establish the global existence and uniqueness of the flow as well as the continuous dependence of the initial datum.MSC35J65; 58A05KeywordsMinimal surface; Plateau problem1. IntroductionRecently, the authors studied the heat flow for minimal surfaces with Plateau boundary condition in the Euclidean spaces
and on general Riemmanian manifold . However, the proof used in
is somewhat complicate, it is the main purpose of this paper to present a fixed point argument in the study of the local existence of the heat flow for general Riemmanian manifold, based upon the estimates obtained in .Let (N,h) be a compact Riemmanian manifold, and let N?Rn be an isometric embedding. Assume that Γ&N is a C3 imbedded simple closed curve, and <img class="imgLazyJSB inlineImage" height="16" width="110" alt="Full-size image (<1 K)" title="Full-size image (<img height="16" border="0" style="vertical-align:bottom" width="110" alt="Full-size image (<1 K)" title="Full-size image ( is its parameterization, where D is the unit disk in R2.One uses the notations:<img class="imgLazyJSB inlineImage" height="26" width="107" alt="Full-size image (<1 K)" title="Full-size image (<img height="26" border="0" style="vertical-align:bottom" width="107" alt="Full-size image (<1 K)" title="Full-size image (, for a function ξ&T we do not distinguish the notations ξ(?) and ξ(ei?), if there is no confusion.<img class="imgLazyJSB inlineImage" height="23" width="330" alt="Full-size image (<1 K)" title="Full-size image (<img height="23" border="0" style="vertical-align:bottom" width="330" alt="Full-size image (<1 K)" title="Full-size image (,<img class="imgLazyJSB inlineImage" height="20" width="491" alt="Full-size image (<1 K)" title="Full-size image (<img height="20" border="0" style="vertical-align:bottom" width="491" alt="Full-size image (<1 K)" title="Full-size image (,<img class="imgLazyJSB inlineImage" height="22" width="535" alt="Full-size image (<1 K)" title="Full-size image (<img height="22" border="0" style="vertical-align:bottom" width="535" alt="Full-size image (<1 K)" title="Full-size image (.A(P)(&,&) is the second fundamental form of N at P, where N is considered as a hypersurface in Rn.The heat flow, we are going to study, is a solution X&W21,2(DT,N) (for T&0) of the following parabolic variational inequality:equation(FMS)<img class="imgLazyJSB inlineImage" height="83" width="386" alt="Full-size image (<1 K)" title="Full-size image (<img height="83" border="0" style="vertical-align:bottom" width="386" alt="Full-size image (<1 K)" title="Full-size image (where DT=[0,T]&D,ST=[0,T]&S, and the initial data X0 is given.As we have seen in
that the most complicate part in this aspect is to prove the local existence of the heat flow. Now, we consider the following inhomogeneous heat flow in Rn:equation(IHF)<img class="imgLazyJSB inlineImage" height="83" width="356" alt="Full-size image (<1 K)" title="Full-size image (<img height="83" border="0" style="vertical-align:bottom" width="356" alt="Full-size image (<1 K)" title="Full-size image (where f&L2(DT,Rn). After suitable modifications of the proof in , we can show that there exists a unique solution X&W21,2(DT,Rn) such that<img class="imgLazyJSB inlineImage" height="21" width="262" alt="Full-size image (<1 K)" title="Full-size image (<img height="21" border="0" style="vertical-align:bottom" width="262" alt="Full-size image (<1 K)" title="Full-size image (where C is a constant depending on Γ and the H1-norm of X0 only.With the aid of the a prior estimate, the local existence of
follows from the principle of contraction mappings.As a by-product, a heat flow for constant mean curvature surface with Plateau condition is also obtained.The paper is organized as follows. In
we investigate the inhomogeneous heat flow in Rn. The local existence result for the quasilinear flow
is obtained in . In , we give two examples, namely the constant mean curvature surfaces in Rn and the minimal surfaces in Sn. Under some small conditions, either the scale or the energy of the initial data, the flow exists globally. In the last section we deal with the heat flow on general Riemmanian manifolds.2. Inhomogeneous heat flowIn this section we establish the following theorem on the inhomogeneous heat flow.Theorem 2.1.
Given<img class="imgLazyJSB inlineImage" height="18" width="72" alt="Full-size image (<1 K)" title="Full-size image (<img height="18" border="0" style="vertical-align:bottom" width="72" alt="Full-size image (<1 K)" title="Full-size image (andf&L2([0,T]&D,Rn), there is a unique solutionX&W1,2([0,T]&D,Rn) for the inhomogeneous heat flowequation(IHF)<img class="imgLazyJSB inlineImage" height="82" width="356" alt="Full-size image (<1 K)" title="Full-size image (<img height="82" border="0" style="vertical-align:bottom" width="356" alt="Full-size image (<1 K)" title="Full-size image (Moreover, the trace<img class="imgLazyJSB inlineImage" height="22" width="195" alt="Full-size image (<1 K)" title="Full-size image (<img height="22" border="0" style="vertical-align:bottom" width="195" alt="Full-size image (<1 K)" title="Full-size image (, and the mapping (X0,t)?X(t,&) is continuous in theH1(D,Rn)-topology. Finally the a prior estimate holds:<img class="imgLazyJSB inlineImage" height="19" width="334" alt="Full-size image (<1 K)" title="Full-size image (<img height="19" border="0" style="vertical-align:bottom" width="334" alt="Full-size image (<1 K)" title="Full-size image (whereCdepends on the initial dataX0.
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No articles found.On the irredundant part of the first Piola-Kirchhoff stress tensor
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, April 1993, Pages 175-210
On the irredundant part of the first Piola-Kirchhoff stress tensor
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Lehrstuhl für Mathematik I, Universit?t Mannheim, Schloβ, D-6800 Mannheim, Germany
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Given is a deformable medium moving and deforming in a (nice) Riemannian manifold N maintaining the shape of a nice compact, bounded manifold M. The physical qualities of this medium are supposed to be characterized by a first Piola-Kirchhoff stress tensor α(j) which is required to depend on the actual configuration j, possibly in a non local way. We determine and investigate via the theory of elliptic operators the irredundant part of α(j). This part yields the same force densities on M as α(j) does. In case N is Euclidean, this irredundant part is the exact part of α(j) extracted via Hodge theory. The medium is equivalently described by α or by a vector field H on the configuration space. Moreover, the virtual work determined by these force densities is studied and discussed in detail. Finally a dynamics is set up within the framework of symplectic geometry in case δM = 0.
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