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1.
The partitioning problem for a smooth convex bodyB 3 consists in to study, among surfaces which divideB in two pieces of prescribed volume, those which are critical points of the area functional.We study stable solutions of the above problem: we obtain several topological and geometrical restrictions for this kind of surfaces. In the case thatB is a Euclidean ball we obtain stronger results.Antonio Ros is partially supported by DGICYT grant PB91-0731 and Enaldo Vergasta is partially supported by CNPq grant 202326/91-8.  相似文献   

2.
Let H:R3R be a C1 mapping such that H(p)→H>0 as ∣p∣→. We show that when H satisfies some global conditions then there exists an H-bubble, namely a sphere S in R3 such that the mean curvature of S at any regular point pS equals H(p).  相似文献   

3.
We discuss existence and multiplicity of positive solutions of the prescribed mean curvature problem
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4.
Given an integralm-currentT 0 in ℝ m+k and a tensorH of typ (m, 1) on ℝ m+k with values orthogonal to each of its arguments we prove the existence of an integralm-currentT with boundary ∂T=∂T 0 having prescribed mean curvature vectorH, i. e. is a solution of   相似文献   

5.
We give an algorithm for finding finite element approximations to surfaces of prescribed variable mean curvature, which span a given boundary curve. We work in the parametric setting and prove optimal estimates in the H1 norm. The estimates are verified computationally.  相似文献   

6.
Summary We consider—in the setting of geometric measure theory—hypersurfacesT (of codimension one) with prescribed boundaryB in Euclideann+1 space which maximize volume (i.e.T together with a fixed hypersurfaceT 0 encloses oriented volume) subject to a mass constraint. We prove existence and optimal regularity of solutionsT of such variational problems and we show that, on the regular part of its support,T is a classical hypersurface of constant mean curvature. We also prove that the solutionsT become more and more spherical as the valuem of the mass constraint approaches ∞. This work was done at the Centre for Mathematics and its Applications at the Australian National University, Canberra while the author was a visiting member This article was processed by the author using the LATEX style filecljour1 from Springer-Verlag.  相似文献   

7.
An elementary existence proof based on variational and finite dimensional approximation methods is proposed for nontrivial solutions of the generalized prescribed mean curvature boundary value problem
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8.
We study a variational approach, called Generalized Minimizing Movemenents (GMM) and proposed by E. De Giorgi, to evolution of hypersurfaces by mean curvature in the case of a Dirichlet boundary datum. We prove an existence theorem of a GMM when on the initial solid are made suitable geometric hypotheses.
Sunto Si studia un approccio variazionale, detto Movimenti Minimizzanti Generalizzati (GMM) e proposto da Ennio De Giorgi, per l’evoluzione di una ipersuperficie secondo la curvatura media con un dato al bordo di tipo Dirichlet. Viene provato un teorema di esistenza quando sul solido iniziale siano fatte opportune ipotesi di tipo geometrico.
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9.
It is proved the existence and uniqueness of graphs with prescribed mean curvature in Riemannian submersions fibered by flow lines of a vertical Killing vector field.  相似文献   

10.
We consider immersed hypersurfaces :Mn→ℝn+1 with prescribed anisotropic mean curvature . Such hypersurfaces can be characterized as critical points of parametric functionals of the type with an elliptic Lagrangian F depending on normal directions and a smooth vectorfield Q satisfying . We establish curvature estimates for stable hypersurfaces of dimension n≤5, provided F is C3-close to the area integrand.  相似文献   

11.
12.
We study and solve the Dirichlet problem for graphs of prescribed mean curvature in Rn+1 over general domains Ω without requiring a mean convexity assumption. By using pieces of nodoids as barriers we first give sufficient conditions for the solvability in case of zero boundary values. Applying a result by Schulz and Williams we can then also solve the Dirichlet problem for boundary values satisfying a Lipschitz condition.  相似文献   

13.
We consider graphs with prescribed mean curvature and flat normal bundle. Using techniques of Schoen et al. (Acta Math 134:275–288, 1975) and Ecker and Huisken (Ann Inst H Poincaré Anal Non Linèaire 6:251–260, 1989), we derive the interior curvature estimate
up to dimension n ≤ 5, where C is a constant depending on natural geometric data of Σ only. This generalizes previous results of Smoczyk et al. (Calc Var Partial Differ Equs 2006) and Wang (Preprint, 2004) for minimal graphs with flat normal bundle.  相似文献   

14.
Let be a minimal set with mean curvature in L n that is a minimum of the functional , where is open and . We prove that if then can be parametrized over the (n−1)-dimensional disk with a C α mapping with C α inverse. Received: 11 July 1997 / Revised version: 24 February 1998  相似文献   

15.
The author was partially funded by NSF grants DMS85-53231(PYI), DMS-92-07704, and by the IHES.  相似文献   

16.
We consider the evolution of fronts by mean curvature in the presence of obstacles. We construct a weak solution to the flow by means of a variational method, corresponding to an implicit time-discretization scheme. Assuming the regularity of the obstacles, in the two-dimensional case we show existence and uniqueness of a regular solution before the onset of singularities. Finally, we discuss an application of this result to the positive mean curvature flow.  相似文献   

17.
We consider the problem of determining the existence of absolute apriori gradient bounds of nonparametric hypersurfaces of constant mean curvature in ann-dimensional sphereB R, 1>R>R 0 (n) , (R 0 (n) being a constant depending only onn), without imposing boundary conditions or bounds of any sort.
Sunto Consideriamo il problema di determinare stime a priori di gradienti di ipersuperfici non parametriche di curvatura media costante in una sferan-dimensionaleB R, 1>R>R 0 (n), (R 0 (n) essendo una costante che dipende solo dan), senza imporre condizioni al contorno o limiti di altro tipo.
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18.
We consider large solutions of annular type to the volume constrained Douglas problem. They are conformally immersed H-surfaces. By rescaling we set the volume functional at one while the boundary curves shrink to the origin. We show that the solutions become spherical in a precise manner. Spherical bubbling may fail if the conformality condition is dropped. We also discuss the rotationally symmetric annular solutions to the H-surface equation and consider some illustrative examples. Received: 2 May 2000 / Accepted: 23 January 2001 / Published online: 4 May 2001  相似文献   

19.
We study the existence of radial ground state solutions for the problem
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20.
We consider a certain variational problem on Caccioppoli partitions with countably many components, which models immiscible fluids as well as variational image segmentation, and generalizes the well-known problem with prescribed mean curvature. We prove existence and regularity results, and finally show some explicit examples of minimizers. Received: 7 June 2001 / Revisied version: 8 October 2001  相似文献   

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