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1.
José Maria Gomes 《Archiv der Mathematik》2007,88(3):269-278
Let Ω be a bounded convex domain in
. We consider constrained minimization problems related to the Euler-Lagrange equation
over classes of functions
(Ω) with convex super level sets. We then search for sufficient conditions ensuring that the minimizer obtained is a classical
solution to the above equation.
Supported by ESF activity “Global and geometrical aspects of nonlinear P.D.E.’s.”
Received: 4 April 2006 相似文献
2.
In this paper we prove a general and sharp Asymptotic Theorem for minimal surfaces in . As a consequence, we prove that there is no properly immersed minimal surface whose asymptotic boundary Γ∞ is a Jordan curve homologous to zero in such that Γ∞ is contained in a slab between two horizontal circles of with width equal to π. We construct vertical minimal graphs in over certain unbounded admissible domains taking certain prescribed finite boundary data and certain prescribed asymptotic
boundary data. Our admissible unbounded domains Ω in are non necessarily convex and non necessarily bounded by convex arcs; each component of its boundary is properly embedded
with zero, one or two points on its asymptotic boundary, satisfying a further geometric condition.
The first author wish to thank Laboratoire Géométrie et Dynamique de l’Institut de Mathématiques de Jussieu for the kind hospitality and support. The authors would like to thank CNPq, PRONEX of Brazil and Accord Brasil-France, for
partial financial support. 相似文献
3.
Suppose Ω is a smooth domain in Rm,N is a compact smooth Riemannian manifold, andZ is a fixed compact subset of Ω having finite (m − 3)-dimensional Minkowski content (e.g.,Z ism − 3 rectifiable). We consider various spaces of harmonic mapsu: Ω →N that have a singular setZ and controlled behavior nearZ. We study the structure of such spacesH and questions of existence, uniqueness, stability, and minimality under perturbation. In caseZ = 0,H is a Banach manifold locally diffeomorphic to a submanifold of the product of the boundary data space with a finite-dimensional
space of Jacobi fields with controlled singular behavior. In this smooth case, the projection ofu εH tou |ϖΩ is Fredholm of index 0.
R. H.’s research partially supported by the National Science Foundation. 相似文献
4.
We prove the parabolic counterpart of the almost monotonicity formula of Caffarelli, Jerison and Kening for pairs of functions
u
±(x, s) in the strip satisfying
We also establish a localized version of the formula as well as prove one of its variants. At the end of the paper we give
an application to a free boundary problem related to the caloric continuation of heat potentials.
The authors thank the STINT Foundation of Sweden for supporting their visits to Purdue University and the Royal Institute
of Technology, respectively, and they thank the universities for their hospitality during the visits. The second author also
acknowledges the support by NSF grant DMS-0401179. 相似文献
5.
This paper studies products of Hankel operators on the Hardy space. We show thatH
f
(1)*
H
f(2)
H
f
(3)*
=0 for all permutation if and only if eitherH
f1 orH
f2 orH
f3 is zero. Using Douglas' localization theorem and Izuchi's theorem on Sarason's three functions problem, we show that
is a sufficient condition forH
f
*
H
g
H
h
*
,H
g
*
H
f
H
h
*
, andH
f
*
H
h
H
g
*
to be compact.This work was partly supported by NSF grants. The second author was also partly supported by the Research Council of Vanderbilt University. 相似文献
6.
In this paper, we study a system of elliptic equations by applying the Limit Index Theory. Under some assumptions on nonlinear
part, we can obtain the existence of multiple solutions for the equations.
The research is supported by NNSF of China (10471024) and Fujian Provincial Natural Science Foundation of China (A0410015). 相似文献
7.
Jeffrey S. Geronimo Evans M. Harrell II Walter Van Assche 《Constructive Approximation》1988,4(1):403-417
We consider the abstract measures, known as thedensity- of- states measures, associated with the asymptotic distribution of eigenvalues of infinite banded Hermitian matrices. Two widely used definitions of these measures are shown to be equivalent, even in the unbounded case, and we prove that the density of states is invariant under certain, possibly unbounded, perturbations. Also considered are measures associated with the asymptotic distribution of eigenvalues of rescaled unbounded matrices. These measures are associated with the so-called contracted spectrum when the matrices are tridiagonal. Finally, we produce several examples clarifying the nature of the density of states.Communicated by Paul Nevai. 相似文献
8.
We consider a compact manifold X whose boundary is a locally trivial fiber bundle, and an associated pseudodifferential algebra
that models fibered cusps at infinity. Using tracelike functionals that generate the 0-dimensional Hochschild cohomology groups
we first express the index of a fully elliptic fibered cusp operator as the sum of a local contribution from the interior
of X and a term that comes from the boundary. This leads to an abstract answer to the index problem formulated in [11]. We
give a more precise answer for firstorder differential operators when the base of the boundary fiber bundle is S1. In particular, for Dirac operators associated to a metric of the form
near ∂X = {x = 0} with twisting bundle T we obtain
in terms of the integral of the Atiyah-Singer form in the interior of X, and the adiabatic limit of the η-invariant of the
restriction of the operator to the boundary. 相似文献
9.
Xianling Fan Shao-Gao Deng 《NoDEA : Nonlinear Differential Equations and Applications》2009,16(2):255-271
We study the existence and multiplicity of positive solutions for the inhomogeneous Neumann boundary value problems involving
the p(x)-Laplacian of the form
where Ω is a bounded smooth domain in , and p(x) > 1 for with and φ ≢ 0 on ∂Ω. Using the sub-supersolution method and the variational method, under appropriate assumptions on f, we prove that, there exists λ* > 0 such that the problem has at least two positive solutions if λ = λ*, has at least one positive solution if λ = λ*, and has no positive solution if λ = λ*. To prove the result we establish a special strong comparison principle for the Neumann problems.
The research was supported by the National Natural Science Foundation of China 10371052,10671084). 相似文献
10.
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
11.
R. Monneau 《Journal of Geometric Analysis》2003,13(2):359-389
We study the obstacle problem in two dimensions. On the one hand we improve a result of L.A. Caffarelli and N.M. Rivière:
we state that every connected component of the interior of the coincidence set has at most N
0
singular points, where N
0
is only dependent on some geometric constants. Moreover, if the component is small enough, then this component has at most
two singular points. On the other hand, we prove in a simple case a conjecture of D.G. Schaeffer on the generic regularity
of the free boundary: for a family of obstacle problems in two dimensions continuously indexed by a parameter λ, the free
boundary of the solution uλ is analytic for almost every λ. Finally we present a new monotonicity formula for singular points.
Dedicated to Henri Berestycki and Alexis Bonnet. 相似文献
12.
Giovanna Cerami Mónica Clapp 《Calculus of Variations and Partial Differential Equations》2007,30(3):353-367
We prove the existence of a sign changing solution to the semilinear elliptic problem , in an exterior domain Ω having finite symmetries. 相似文献
13.
Lawrence C. Evans 《Calculus of Variations and Partial Differential Equations》2009,35(4):435-462
We introduce and make estimates for several new approximations that in appropriate asymptotic limits yield the key PDE for
weak KAM theory, namely a Hamilton–Jacobi type equation for a potential u and a coupled transport equation for a measure σ. We revisit as well a singular variational approximation introduced in Evans (Calc Vari Partial Differ Equ 17:159–177, 2003)
and demonstrate “approximate integrability” of certain phase space dynamics related to the Hamiltonian flow. Other examples
include a pair of strongly coupled PDE suggested by the Lions–Lasry theory (Lasry and Lions in Japan J Math 2:229–260, 2007)
of mean field games and a new and extremely singular elliptic equation suggested by sup-norm variational theory.
Supported in part by NSF Grant DMS-0500452. 相似文献
14.
Quo-Shin Chi 《Journal of Geometric Analysis》1991,1(4):373-382
It is proved that if the Besicovitch Covering Lemma is true on either a Hadamard manifold or a simply connected surface without
focal points that covers a compact quotient, then the manifold is the Euclidean space. As a corollary, the vanishing of the
topological entropy of a compact manifold of nonpositive curvature or of a compact surface without focal points is equivalent
to the validity of the Besicovitch Covering Lemma on the universal covering space of the manifold.
The author was partially supported by an NSF grant. 相似文献
15.
Existence of positive weak solutions with a prescribed singular set of semilinear elliptic equations
In this paper, we consider the problem of the existence of non-negative weak solution u of
having a given closed set S as its singular set. We prove that when
and S is a closed subset of Ω, then there are infinite many positive weak solutions with S as their singular set. Applying
this method to the conformal scalar curvature equation for n ≥ 9, we construct a weak solution
of
such that Sn is the singular set of u where L0 is the conformal Laplacian with respect to the standard metric of Sn. When n = 4 or 6, this kind of solution has been constructed by Pacard. 相似文献
16.
Dimitri Mugnai 《Calculus of Variations and Partial Differential Equations》2008,32(4):481-497
We show that a semilinear Dirichlet problem in bounded domains of in presence of subcritical exponential nonlinearities has four nontrivial solutions near resonance.
Research supported by the Italian National Project Metodi Variazionali ed Equazioni Differenziali Non Lineari. 相似文献
17.
N. M. Ivochkina 《Journal of Fixed Point Theory and Applications》2008,4(1):47-56
We adapt to degenerate m-Hessian evolution equations the notion of m-approximate solutions introduced by N. Trudinger for m-Hessian elliptic equations, and we present close to necessary and sufficient conditions guaranteeing the existence and uniqueness
of such solutions for the first initial boundary value problem.
Dedicated to Professor Felix Browder 相似文献
18.
In this paper we consider the equationu
tt
+u
xxxx
=0 for a vibrating rod supported at its end-points. We study the problem of determining the shape of the rod,u(·,0), knowingu(·,T). We see that the solution is unique ifT belongs to a certain dense subsetS of (0,+∞). IfT belongs toS, u(·,0) does not depend continuously onu(·,T). In this case, with the assumption of a bounded initial energy of the rod, we get stability estimates foru(·,0). 相似文献
19.
We consider here a class of nonlinear Dirichlet problems, in a bounded domain , of the form
investigating the problem of uniqueness of solutions. The functions (s) and
satisfy rather general assumptions of locally Lipschitz continuity (with possibly exponential growth) and the datum f is in L1(). Uniqueness of solutions is proved both for coercive a(x, s) and for the case of a(x, s) degenerating for s large. 相似文献
20.
Enrico Jabara 《Rendiconti del Circolo Matematico di Palermo》2001,50(3):393-404
LetG be a group and α an automorphism ofG; α is calledn-splitting if
for allg∈G. In this note we study the structure of finite groups admitting an-splitting automorphism of order 2.
相似文献