共查询到20条相似文献,搜索用时 31 毫秒
1.
Emanuele Paolini 《manuscripta mathematica》1998,97(1):15-35
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
2.
Sven Winklmann 《manuscripta mathematica》2005,118(4):485-499
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
3.
Thierry De Pauw 《Journal of Geometric Analysis》2002,12(1):29-61
A concentrated (ξ, m) almost monotone measure inR
n
is a Radon measure Φ satisfying the two following conditions: (1) Θ
m
(Φ,x)≥1 for every x ∈spt (Φ) and (2) for everyx ∈R
n
the ratioexp [ξ(r)]r−mΦ(B(x,r)) is increasing as a function of r>0. Here ξ is an increasing function such thatlim
r→0-ξ(r)=0. We prove that there is a relatively open dense setReg (Φ) ∋spt (Φ) such that at each x∈Reg(Φ) the support of Φ has the following regularity property: given ε>0 and λ>0 there is an m dimensional spaceW ⊂R
n
and a λ-Lipschitz function f from x+W into x+W‖ so that (100-ε)% ofspt(Φ) ∩B (x, r) coincides with the graph of f, at some scale r>0 depending on x, ε, and λ. 相似文献
4.
Guido Cortesani 《Annali dell'Universita di Ferrara》1997,43(1):27-49
Let Ω be an open and bounded subset ofR
n
with locally Lipschitz boundary. We prove that the functionsv∈SBV(Ω,R
m
) whose jump setS
vis essentially closed and polyhedral and which are of classW
k, ∞ (S
v,R
m) for every integerk are strongly dense inGSBV
p(Ω,R
m
), in the sense that every functionu inGSBV
p(Ω,R
m
) is approximated inL
p(Ω,R
m
) by a sequence of functions {v
k{j∈N with the described regularity such that the approximate gradients ∇v
jconverge inL
p(Ω,R
nm
) to the approximate gradient ∇u and the (n−1)-dimensional measure of the jump setsS
v
j converges to the (n−1)-dimensional measure ofS
u. The structure ofS
v can be further improved in casep≤2.
Sunto Sia Ω un aperto limitato diR n con frontiera localmente Lipschitziana. In questo lavoro si dimostra che le funzioniv∈SBV(Ω,R m ) con insieme di saltoS v essenzialmente chiuso e poliedrale che sono di classeW k, ∞ (S v,R m ) per ogni interok sono fortemente dense inGSBV p(Ω,R m ), nel senso che ogni funzioneu∈GSBV p(Ω,R m ) è approssimata inL p(Ω,R m ) da una successione di funzioni {v j}j∈N con la regolaritá descritta tali che i gradienti approssimati ∇v jconvergono inL p(Ω,R nm ) al gradiente approssimato ∇u e la misura (n−1)-dimensionale degli insiemi di saltoS v jconverge alla misura (n−1)-dimensionale diS u. La struttura diS vpuó essere migliorata nel caso in cuip≤2.相似文献
5.
In this paper, we prove the validity of the Chern conjecture in affine geometry [18], namely that an affine maximal graph
of a smooth, locally uniformly convex function on two dimensional Euclidean space, R
2, must be a paraboloid. More generally, we shall consider the n-dimensional case, R
n
, showing that the corresponding result holds in higher dimensions provided that a uniform, “strict convexity” condition holds.
We also extend the notion of “affine maximal” to non-smooth convex graphs and produce a counterexample showing that the Bernstein
result does not hold in this generality for dimension n≥10.
Oblatum 16-IV-1999 & 4-XI-1999?Published online: 21 February 2000 相似文献
6.
We study the Navier-Stokes equations for compressible barotropic fluids in a bounded or unbounded domain Ω of R3. We first prove the local existence of solutions (ρ,u) in C([0,T*]; (ρ∞ +H3(Ω)) × under the assumption that the data satisfies a natural compatibility condition. Then deriving the smoothing effect of the
velocity u in t>0, we conclude that (ρ,u) is a classical solution in (0,T**)×Ω for some T** ∈ (0,T*]. For these results, the initial density needs not be bounded below away from zero and may vanish in an open subset (vacuum) of Ω. 相似文献
7.
Emilien Tarquini 《Monatshefte für Mathematik》2007,151(4):333-339
In this paper we consider the Gross-Pitaevskii equation iu
t
= Δu + u(1 − |u|2), where u is a complex-valued function defined on
, N ≥ 2, and in particular the travelling waves, i.e., the solutions of the form u(x, t) = ν(x
1 − ct, x
2, …, x
N
), where
is the speed. We prove for c fixed the existence of a lower bound on the energy of any non-constant travelling wave. This bound provides a non-existence
result for non-constant travelling waves of fixed speed having small energy. 相似文献
8.
Frank Duzaar Klaus Steffen 《Calculus of Variations and Partial Differential Equations》1993,1(4):355-406
We prove complete boundary regularity for energy minimizing integer multiplicity rectifiablen currents in
n+1 of prescribed mean curvatureH with boundaryB=
represented by an oriented smooth submanifold of dimensionn – 1 in sun+1. We also give applications to the Plateau problem for surfaces with prescribed mean curvature.This article was processed by the author using the LaTEX style filepljour1 from Springer-Verlag. 相似文献
9.
We consider a C
1 smooth surface with prescribed p (or H)-mean curvature in the 3-dimensional Heisenberg group. Assuming only the prescribed p-mean curvature we show that any characteristic curve is C
2 smooth and its (line) curvature equals − H in the nonsingular domain. By introducing characteristic coordinates and invoking the jump formulas along characteristic
curves, we can prove that the Legendrian (or horizontal) normal gains one more derivative. Therefore the seed curves are C
2 smooth. We also obtain the uniqueness of characteristic and seed curves passing through a common point under some mild conditions,
respectively. These results can be applied to more general situations. 相似文献
10.
Harmonic maps with potential 总被引:8,自引:0,他引:8
Ali Fardoun Andrea Ratto 《Calculus of Variations and Partial Differential Equations》1997,5(2):183-197
Let (M,g) and (N,h) be two Riemannian manifolds, and G:N →ℝ a given function. If f:M → N is a smooth map, we set E
G
(f)=12 ∫M [∣df∣2− 2G(f)]dv
g. We establish some variational properties and some existence results for the functional E
G
(f): in particular, we analyse the case of maps into a sphere.
Received April 29, 1996 / Accepted May 28, 1996 相似文献
11.
Fei-Tsen Liang 《Annali dell'Universita di Ferrara》2002,48(1):189-217
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.相似文献
12.
Denoting Δ? the Laplacian operator on the (2N+1)-dimensional Heisenberg group ?
N
, we prove some nonexistence results for solutions of inequalities of the three types
in ?
N
and ?
N
×ℝ}+, with a∈L
∞, when 1<p≤p
0, where p
0 depends on N and the type of equation.
Received: 17 June 1999 相似文献
13.
We consider symmetric simple exclusion processes with L=&ρmacr;N
d
particles in a periodic d-dimensional lattice of width N. We perform the diffusive hydrodynamic scaling of space and time. The initial condition is arbitrary and is typically far
away form equilibrium. It specifies in the scaling limit a density profile on the d-dimensional torus. We are interested in the large deviations of the empirical process, N
−
d
[∑
L
1δ
xi
(·)] as random variables taking values in the space of measures on D[0.1]. We prove a large deviation principle, with a rate function that is more or less universal, involving explicity besides
the initial profile, only such canonical objects as bulk and self diffusion coefficients.
Received: 7 September 1997 / Revised version: 15 May 1998 相似文献
14.
Jaigyoung Choe Mohammad Ghomi Manuel Ritoré 《Calculus of Variations and Partial Differential Equations》2007,29(4):421-429
We prove that the area of a hypersurface Σ which traps a given volume outside a convex domain C in Euclidean space R
n
is bigger than or equal to the area of a hemisphere which traps the same volume on one side of a hyperplane. Further, when
C has smooth boundary ∂C, we show that equality holds if and only if Σ is a hemisphere which meets ∂C orthogonally. 相似文献
15.
Adimurthi Maria J. Esteban 《NoDEA : Nonlinear Differential Equations and Applications》2005,12(2):243-263
In this paper we prove new Hardy-like inequalities with optimal potential singularities for functions in W1,p(Ω), where Ω is either bounded or the whole space
and also some extensions to arbitrary Riemannian manifolds. We also study the spectrum of perturbed Schr?dinger operators
for perturbations which are just below the optimality threshold for the corresponding Hardy inequality. 相似文献
16.
Federica Dragoni 《NoDEA : Nonlinear Differential Equations and Applications》2007,14(3-4):429-441
We investigate the behavior, as ε → 0+, of ε log w
ε (t, x) where w
ε are solutions of a suitable family of subelliptic heat equations. Using the Large Deviation Principle, we show that the limiting
behavior is described by the metric inf-convolution w.r.t. the associated Carnot-Carathéodory distance.
相似文献
17.
It is well known that the Helmholtz decomposition of Lq-spaces fails to exist for certain unbounded smooth planar domains unless q = 2, see [2], [9]. As recently shown [6], the Helmholtz projection does exist for general unbounded domains of uniform C2-type in
if we replace the space Lq, 1 < q < ∞, by L2 ∩ Lq for q > 2 and by Lq + L2 for 1 < q < 2. In this paper, we generalize this new approach from the three-dimensional case to the n-dimensional case, n ≥ 2. By these means it is possible to define the Stokes operator in arbitrary unbounded domains of uniform
C2-type.
Received: 15 February 2006 相似文献
18.
A class of spiral minimal surfaces in E3 is constructed using a symmetry reduction. The reduction leads to a cubic-nonlinear ODE whose phase portrait is described using an auxiliary Riccati's equation and the Warzewski topological principle for its solutions. The new surfaces are invariant with respect to the composition of rotation and dilation. The solutions are obtained in parametric form through the Legendre and the Weierstrass representations, and also their asymptotic behaviour is described. 相似文献
19.
In [3] we studied p-mean curvature and the associated p-minimal surfaces in the Heisenberg group from the viewpoint of PDE and differential geometry. In this paper, we look into the problem through the variational formulation. We study a generalized p-area and associated ( p-) minimizers in general dimensions. We prove the existence and investigate the uniqueness of minimizers. Since this is reduced to solving a degenerate elliptic equation, we need to consider the effect of the singular set and this requires a careful study. We define the notion of weak solution and prove that in a certain Sobolev space, a weak solution is a minimizer and vice versa. We also give many interesting examples in dimension 2. An intriguing point is that, in dimension 2, a C
2-smooth solution from the PDE viewpoint may not be a minimizer. However, this statement is true for higher dimensions due to the relative smallness of the size of the singular set. 相似文献
20.
We consider the zero-velocity stationary problem of the Navier–Stokes equations of compressible isentropic flow describing
the distribution of the density ϱ of a fluid in a spatial domain Ω⊂ℝ
N
driven by a time-independent potential external force b=∇F. A sharp condition in terms of F is given for the problem to possess a unique nonnegative solution ϱ having a prescribed mass m > 0.
Received: 20 October 1997 相似文献