共查询到20条相似文献,搜索用时 31 毫秒
1.
In this paper, we consider a free boundary problem with volume constraint. We show that positive minimizer is locally Lipschitz
and the free boundary is analytic away from a singular set with Hausdorff dimension at most n − 8. 相似文献
2.
Séverine Rigot 《Calculus of Variations and Partial Differential Equations》2000,10(4):389-406
Quasi minimizers for the perimeter are measurable subsets G of such that
for all variations of G with and for a given increasing function such that . We prove here that, given , G a reduced quasi minimizer, and , there are , with , and , homeomorphic to a closed ball with radius t in , such that for some absolute constant . The constant above depends only on n, and . If moreover for some , we prove that we can find such a ball such that is a dimensional graph of class . This will be obtained proving that a quasi minimizer is equivalent to some set which satisfies the condition B. This condition
gives some kind of uniform control on the flatness of the boundary and then criterions proven by Ambrosio-Paolini and Tamanini
can be applied to get the required regularity properties.
Received: July 12, 1999 / Accepted: October 1, 1999 相似文献
3.
This paper is devoted to the autonomous Lagrange problem of the calculus of variations with a discontinuous Lagrangian. We
prove that every minimizer is Lipschitz continuous if the Lagrangian is coercive and locally bounded. The main difference
with respect to the previous works in the literature is that we do not assume that the Lagrangian is convex in the velocity.
We also show that, under some additional assumptions, the DuBois—Reymond necessary condition still holds in the discontinuous
case. Finally, we apply these results to deduce that the value function of the Bolza problem is locally Lipschitz and satisfies
(in a generalized sense) a Hamilton—Jacobi equation. 相似文献
4.
This paper is devoted to the autonomous Lagrange problem of the calculus of variations with a discontinuous Lagrangian. We
prove that every minimizer is Lipschitz continuous if the Lagrangian is coercive and locally bounded. The main difference
with respect to the previous works in the literature is that we do not assume that the Lagrangian is convex in the velocity.
We also show that, under some additional assumptions, the DuBois—Reymond necessary condition still holds in the discontinuous
case. Finally, we apply these results to deduce that the value function of the Bolza problem is locally Lipschitz and satisfies
(in a generalized sense) a Hamilton—Jacobi equation. 相似文献
5.
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
6.
Roberta Dal Passo Lorenzo Giacomelli Salvador Moll 《Calculus of Variations and Partial Differential Equations》2008,32(4):533-554
We consider rotationally symmetric 1-harmonic maps from D
2 to S
2 subject to Dirichlet boundary conditions. We prove that the corresponding energy—a degenerate non-convex functional with
linear growth—admits a unique minimizer, and that the minimizer is smooth in the bulk and continuously differentiable up to
the boundary. We also show that, in contrast with 2-harmonic maps, a range of boundary data exists such that the energy admits
more than one smooth critical point: more precisely, we prove that the corresponding Euler–Lagrange equation admits a unique
(up to scaling and symmetries) global solution, which turns out to be oscillating, and we characterize the minimizer and the
smooth critical points of the energy as the monotone, respectively non-monotone, branches of such solution.
R. Dal Passo passed away on 8th August 2007. Endowed with great strength, creativity and humanity, Roberta has been an outstanding
mathematician, an extraordinary teacher and a wonderful friend. Farewell, Roberta. 相似文献
7.
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
8.
Burglind Jöricke 《Journal of Geometric Analysis》1999,9(2):257-300
Let Ω be a bounded strictly pseudoconvex domain in ℂn, n ≥ 3, with boundary ∂Ω, of class C2. A compact subset K is called removable if any analytic function in a suitable small neighborhood of ∂Ω K extends to an analytic
function in Ω. We obtain sufficient conditions for removability in geometric terms under the condition that K is contained
in a generic C2 -submanifold M of co-dimension one in ∂Ω. The result uses information on the global geometry of the decomposition of a CR-manifold
into CR-orbits, which may be of some independent interest. The minimal obstructions for removability contained in M are compact
sets K of two kinds. Either K is the boundary of a complex variety of co-dimension one in Ω or it is an exceptional minimal
CR-invariant subset of M, which is a certain analog of exceptional minimal sets in co-dimension one foliations. It is shown
by an example that the latter possibility may occur as a nonremovable singularity set.
Further examples show that the germ of envelopes of holomorphy of neighborhoods of ∞Ω K for K ⊂ M may be multisheeted. A couple
of open problems are discussed. 相似文献
9.
Luca Granieri 《NoDEA : Nonlinear Differential Equations and Applications》2007,14(1-2):125-152
In recent years different authors ([4, 16, 17]) have noticed and investigated some analogy between Mather’s theory of minimal
measures in Lagrangian dynamic and the mass transportation (or Monge-Kantorovich) problem. We replace the closure and homological
constraints of Mather’s problem by boundary terms and we investigate the equivalence with the mass transportation problem.
An Hamiltonian duality formula for the mass transportation and the equivalence with Brenier’s formulation are also established. 相似文献
10.
I. Fonseca G. Leoni R. Paroni 《Calculus of Variations and Partial Differential Equations》2003,17(3):283-309
It is proved that if , with p > 1, if is bounded in , , and if in then provided is 2-quasiconvex and satisfies some appropriate growth and continuity condition. Characterizations of the 2-quasiconvex envelope
when admissible test functions belong to BHp are provided.
Received: 10 October 2001 / Accepted: 8 May 2002 / Published online: 17 December 2002 相似文献
11.
Stefan Hildebrandt Heiko von der Mosel 《Calculus of Variations and Partial Differential Equations》1999,9(3):249-267
Let be a two-dimensional parametric variational integral the Lagrangian F(x,z) of which is positive definite and elliptic, and suppose that is a closed rectifiable Jordan curve in . We then prove that there is a conformally parametrized minimizer of in the class of surfaces of the type of the disk B which are bounded by . An immediate consequence of this theorem is that the Dirichlet integral and the area functional have the same infima, a
result whose proof usually requires a Lichtenstein-type mapping theorem or else Morrey's lemma on -conformal mappings. In addition we show that the minimizer of is H?lder continuous in B, and even in if satisfies a chord-arc condition. In Section 1 it is described how our results are related to classical investigations, in
particular to the work of Morrey. Without difficulty our approach can be carried over to two-dimensional surfaces of codimension
greater than one.
Received July 20, 1998 / Accepted October 23, 1998 相似文献
12.
Given a compact, oriented Riemannian manifold M, without boundary, and a codimension-one homology class in H* (M, Z) (or, respectively, in H* (M, Zp) with p an odd prime), we consider the problem of finding a cycle of least area in the given class: this is known as the
homological Plateau’s problem.
We propose an elliptic regularization of this problem, by constructing suitable fiber bundles ξ (resp. ζ) on M, and one-parameter
families of functionals defined on the regular sections of ξ, (resp. ζ), depending on a small parameter ε.
As ε → 0, the minimizers of these functionals are shown to converge to some limiting section, whose discontinuity set is exactly
the minimal cycle desired. 相似文献
13.
We deal with variational problems on varying manifolds in ℝn. We represent each manifold by a positive measure μ, to which we associate a suitable notion of tangent space Tμ, of mean
curvature H(μ), and of Sobolev spaces with respect to μ on an open subset Ω ⊆ ℝn. We introduce the notions of weak and strong convergence for functions defined on varying manifolds, that is defined μh -a.e., being {μh} a weakly convergent sequence of measures. In this setting, we prove a strong-weak type compactness theorem for the pairs
(Pμ
h H(μh)), where Pμ
h are the projectors onto the tangent spaces Tμ
h. When μh belong to a suitable class of k-dimensional measures, having in particular a prescribed (k−1)-manifold as a boundary, we
enforce this result to study the convergence of energy functionals, possibly with a Dirichlet condition on ∂Ω. We also address
a perspective for optimization problems where the control variable is represented by a manifold with a prescribed boundary. 相似文献
14.
Graziano Crasta Ilaria Fragalà Filippo Gazzola 《NoDEA : Nonlinear Differential Equations and Applications》2005,12(1):93-109
For a given p > 1 and an open bounded convex set
we consider the minimization problem for the functional
over
Since the energy of the unique minimizer up may not be computed explicitly, we restrict the minimization problem to the subspace of web functions, which depend only on the distance from the boundary δΩ. In this case, a representation formula for the unique minimizer vp is available. Hence the problem of estimating the error one makes when approximating Jp(up) by Jp(vp) arises. When Ω varies among convex bounded sets in the plane, we find an optimal estimate for such error, and we show that it is decreasing and infinitesimal with p. As p → ∞, we also prove that up − vp converges to zero in
for all m < ∞. These results reveal that the approximation of minima by means of web functions gains more and more precision as convexity in Jp increases. 相似文献
15.
Peter Kohlmann 《Geometriae Dedicata》1996,60(2):125-143
We consider noncompact, closed and convex sets with nonvoid interior in Euclidean space. It is shown that if such a set has one curvature measure sufficiently close to the boundary measure, then it is congruent to a product of a vector space and a compact convex body. Related stability and characterization theorems for orthogonal disc cylinders are proved. Our arguments are based on the Steiner-Schwarz symmetrization processes and generalized Minkowski integral formulas. 相似文献
16.
Keomkyo Seo 《Archiv der Mathematik》2008,90(2):173-180
Let C be a closed convex set in a complete simply connected Riemannian manifold M with sectional curvature bounded above by a nonpositive constant K. Assume that Σ is a compact minimal surface outside C such that Σ is orthogonal to ∂C along ∂Σ ∩ ∂C. If ∂Σ ∼ ∂C is radially connected from a point , then we prove a sharp relative isoperimetric inequality
where equality holds if and only if Σ is a geodesic half disk with constant Gaussian curvature K. We also prove the relative isoperimetric inequalities for minimal submanifolds outside a closed convex set in a higher-dimensional
Riemannian manifold.
Received: 3 February 2007 相似文献
17.
Given an open bounded connected subset Ω of ℝn, we consider the overdetermined boundary value problem obtained by adding both zero Dirichlet and constant Neumann boundary
data to the elliptic equation −div(A(|∇u|)∇u)=1 in Ω. We prove that, if this problem admits a solution in a suitable weak sense, then Ω is a ball. This is obtained under
fairly general assumptions on Ω and A. In particular, A may be degenerate and no growth condition is required. Our method of proof is quite simple. It relies on a maximum principle
for a suitable P-function, combined with some geometric arguments involving the mean curvature of ∂Ω. 相似文献
18.
Paolo Albano 《NoDEA : Nonlinear Differential Equations and Applications》2009,16(2):273-281
In an open bounded set Ω, we consider the distance function from ∂Ω associated to a Riemannian metric with C
1,1 coefficients. Assuming that Ω is convex near a boundary point x
0, we show that the distance function is differentiable at x
0 if and only if there exists the tangent space to ∂Ω at x
0. Furthermore, if the distance function is not differentiable at x
0 then there exists a Lipschitz continuous curve, with initial point at x
0, such that the distance function is not differentiable along such a curve.
相似文献
19.
Abstract In this paper we deal with the Dirichlet problem for the Laplace equation in a plane exterior domain Ω with a Lipschitz boundary. We prove that, if the boundary datum a is square summable, then the problem admits a solution which tends to a in the sense of nontangential convergence, is unique in a suitable function class and vanishes at infinity as r–k if and only if a satisfies k compatibility conditions, which we are able to explicit when Ω is the exterior of an ellipse.
Keywords: Dirichlet problem, Asymptotic behavior, Potential theory
Mathematics Subject Classification (2000): 31A05, 31A10 相似文献
20.
Sufficient conditions are obtained for wellposedness of convex minimum problems of the calculus of variations for multiple integrals under strong or weak perturbations of the boundary data. Problems with a unique minimizer as well as problems with several solutions are treated. Wellposedness under weak convergence of the boundary data in W1 p ω is proved if p >2 and a counterexample is exhibited if p =2. 相似文献