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1.
For a given convex subset Ω of Euclidean n-space, we consider the problem of minimizing the perimeter of subsets of Ω subject
to a volume constraint. The problem is to determine whether in general a minimizer is also convex. Although this problem is
unresolved, we show that if Ω satisfies a “great circle” condition, then any minimizer is convex. We say that Ω satisfies
a great circle condition if the largest closed ball B contained in Ω has a great circle that is contained in the boundary
of Ω. A great circle of B is defined as the intersection of the boundary of B with a hyperplane passing through the center
of B. 相似文献
2.
Yves Benoist 《Inventiones Mathematicae》2006,164(2):249-278
Divisible convex sets IV: Boundary structure in dimension 3
Let Ω be an indecomposable properly convex open subset of the real projective 3-space which is divisible i.e. for which there exists a torsion free discrete group Γ of projective transformations preserving Ω such that the quotient
M := Γ\Ω is compact. We study the structure of M and of ∂Ω, when Ω is not strictly convex:
The union of the properly embedded triangles in Ω projects in M onto an union of finitely many disjoint tori and Klein bottles which induces an atoroidal decomposition of M.
Every non extremal point of ∂Ω is on an edge of a unique properly embedded triangle in Ω and the set of vertices of these
triangles is dense in the boundary of Ω (see Figs. 1 to 4).
Moreover, we construct examples of such divisible convex open sets Ω.
相似文献
3.
V. G. Romanov 《Journal of Applied and Industrial Mathematics》2012,6(3):360-370
For an integrodifferential equation corresponding to a two-dimensional viscoelastic problem, we study the problem of defining the spatial part of the kernel involved in the integral term of the equation. The support of the sought function is assumed to belong to a compact domain Ω. As information for solving this inverse problem, the traces of the solution to the direct Cauchy problem and its normal derivative are given for some finite time interval on the boundary of Ω. An important feature in the statement of the problem is the fact that the solution of the direct problem corresponds to the zero initial data and a force impulse in time localized on a fixed straight line disjoint with Ω. The main result of the article consists in obtaining a Lipschitz estimate for the conditional stability of the solution to the inverse problem under consideration. 相似文献
4.
Let Ω ⊂ ℝd be a compact convex set of positive measure. A cubature formula will be called positive definite (or a pd-formula, for short)
if it approximates the integral ∫Ω f(x) dx of every convex function f from below. The pd-formulae yield a simple sharp error bound for twice continuously differentiable
functions. In the univariate case (d = 1), they are the quadrature formulae with a positive semidefinite Peano kernel of order
two. As one of the main results, we show that there is a correspondence between pd-formulae and partitions of unity on Ω.
This is a key for an investigation of pd-formulae without employing the complicated multivariate analogue of Peano kernels.
After introducing a preorder, we establish criteria for maximal pd-formulae. We also find a lower bound for the error constant
of an optimal pd-formula. Finally, we describe a phenomenon which resembles a property of Gaussian formulae. 相似文献
5.
Yves Benoist 《Geometriae Dedicata》2006,122(1):109-134
For any m ≥ 3, we construct properly convex open sets Ω in the real projective space
whose Hilbert metric is Gromov hyperbolic but is not quasiisometric to the hyperbolic space
. We show that such examples cannot exist for m = 2.
Some of our examples are divisible, i.e. there exists a discrete group Г of projective transformations preserving Ω with a
compact quotient Г\Ω. The open set Ω is strictly convex but the group Г is not isomorphic to any cocompact lattice in the
isometry group of
. 相似文献
6.
Lucio Damascelli Berardino Sciunzi 《Calculus of Variations and Partial Differential Equations》2006,25(2):139-159
We consider the Dirichlet problem for positive solutions of the equation −Δm (u) = f(u) in a bounded smooth domain Ω, with f positive and locally Lipschitz continuous. We prove a Harnack type inequality for the solutions of the linearized operator,
a Harnack type comparison inequality for the solutions, and exploit them to prove a Strong Comparison Principle for solutions
of the equation, as well as a Strong Maximum Principle for the solutions of the linearized operator. We then apply these results,
together with monotonicity results recently obtained by the authors, to get regularity results for the solutions. In particular
we prove that in convex and symmetric domains, the only point where the gradient of a solution u vanishes is the center of symmetry (i.e. Z≡{x∈ Ω ∨ D(u)(x) = 0 = {0} assuming that 0 is the center of symmetry). This is crucial in the study of m-Laplace equations, since Z is exactly the set of points where the m-Laplace operator is degenerate elliptic. As a corollary u ∈ C2(Ω∖{0}).
Supported by MURST, Project “Metodi Variazionali ed Equazioni Differenziali Non Lineari.”
Mathematics Subject Classification (1991) 35B05, 35B65, 35J70 相似文献
7.
The finite element based approximation of a quasilinear elliptic equation of non monotone type with Neumann boundary conditions
is studied. Minimal regularity assumptions on the data are imposed. The consideration is restricted to polygonal domains of
dimension two and polyhedral domains of dimension three. Finite elements of degree k ≥ 1 are used to approximate the equation. Error estimates are established in the L
2(Ω) and H
1(Ω) norms for convex and non-convex domains. The issue of uniqueness of a solution to the approximate discrete equation is
also addressed. 相似文献
8.
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
9.
We construct and justify the asymptotics (as ε → +0) of a solution of the mixed boundary-value problem for the Poisson equation
in the domain obtained by joining two sets Ω+ and Ω- by a large number of thin (of width O (ε)) curvilinear strips (a hub and a rim with a large number of spokes). As a resulting
limit problem describing the principal terms of exterior expansions (in Ω± and in the set ω occupied by the strips) we take the problem of conjugating the partial differential equations and an ordinary
differential equation depending on a parameter. Bibliography: 16 titles; Illustrations: 1 figure.
Translated fromProblemy Matematicheskogo Analiza, No. 14, 1995, pp. 63–90. 相似文献
10.
LetG be a Lie group with Lie algebraL(G) and let Ω be a non-empty subset ofL(G). If Ω is interpreted as the set of controls, then the set of elements attainable from the identity for the system Ω is a
subsemigroup ofG.
A system Ω is called anon-overlapping control system if any element attainable for Ω is only attainable at one time. In this paper, we show that a compact convex generating nonoverlapping
control systems on a connected Lie group must be contained inX+E for someX∈L(G)\E, where E is a subspace of codimension one containing the commutator, and the homomorphism from the attainable semigroup intoR
+ extends continuously to the whole group in the case of solvable Lie groups.
This work is done under the support of TGRC-KOSEF. 相似文献
11.
V. G. Romanov 《Siberian Mathematical Journal》2011,52(4):682-695
We consider the integrodifferential system of equations of electrodynamics which corresponds to a dispersive nonmagnetic medium.
For this system we study the problem of determining the spatial part of the kernel of the integral term. This corresponds
to finding the part of dielectric permittivity depending nonlinearly on the frequency of the electromagnetic wave. We assume
that the support of dielectric permittivity lies in some compact domain Ω ⊂ ℝ3. In order to find it inside Ω we start with known data about the solution to the corresponding direct problem for the equations
of electrodynamics on the whole boundary of Ω for some finite time interval. On assuming that the time interval is sufficiently
large we estimate the conditional stability of the solution to this inverse problem. 相似文献
12.
Let Ω⊂ℝ
d
be a compact convex polytope of positive measure. We study cubature formulae on Ω which approximate the integral of every convex function f∈C(Ω) from above. They are called negative definite formulae or nd-formulae for short. In particular, we characterize nd-formulae by certain partitions of unity or, alternatively, by a class of positive
linear operators. For aiming at ‘good’ nd-formulae, we introduce three extremal properties named as minimal, best and optimal. We show that the Delaunay triangulation and one of its generalizations give access to efficient algorithms for computing
nd-formulae with one of these properties. 相似文献
13.
Magnus Bondesson 《Annali di Matematica Pura ed Applicata》1973,95(1):1-43
Summary We consider elliptic and parabolic difference operators and prove estimates in discrete Lp norms, 1<p<∞, which are analogues of known estimates for the corresponding differential operators. Let U be a solution in
a bounded domain Ω of an elliptic or parabolic differential equation and let Uh be a solution of the discrete equation. Using the estimates, we prove under mild regularity assumptions that if Uh converges to U in some discrete Lp normp>1, then the difference quotients of Uh converge uniformly (on compact subsets of Ω) to the corresponding derivatives of U.
Entrata in Redazione il 9 ottobre 1971. 相似文献
14.
Markos Katsoulakis Georgios T. Kossioris Fernando Reitich 《Journal of Geometric Analysis》1995,5(2):255-279
We study asharpinterface model for phase transitions which incorporates the interaction of the phase boundaries with the walls of a container Ω. In
this model, the interfaces move by their mean curvature and are normal to δΩ. We first establish local-in-time existence and
uniqueness of smooth solutions for the mean curvature equation with a normal contact angle condition. We then discuss global
solutions by interpreting the equation and the boundary condition in a weak (viscosity) sense. Finally, we investigate the
relation of the aforementioned model with atransitionlayer model. We prove that if Ω isconvex, the transition-layer solutions converge to the sharp-interface solutions as the thickness of the layer tends to zero. We
conclude with a discussion of the difficulties that arise in establishing this result in nonconvex domains.
Communicated by David Kinderlehrer 相似文献
15.
We define a convex extension of a lower semi-continuous function to be a convex function that is identical to the given function
over a pre-specified subset of its domain. Convex extensions are not necessarily constructible or unique. We identify conditions
under which a convex extension can be constructed. When multiple convex extensions exist, we characterize the tightest convex
extension in a well-defined sense. Using the notion of a generating set, we establish conditions under which the tightest
convex extension is the convex envelope. Then, we employ convex extensions to develop a constructive technique for deriving
convex envelopes of nonlinear functions. Finally, using the theory of convex extensions we characterize the precise gaps exhibited
by various underestimators of $x/y$ over a rectangle and prove that the extensions theory provides convex relaxations that
are much tighter than the relaxation provided by the classical outer-linearization of bilinear terms.
Received: December 2000 / Accepted: May 2002 Published online: September 5, 2002
RID="*"
ID="*" The research was funded in part by a Computational Science and Engineering Fellowship to M.T., and NSF CAREER award
(DMI 95-02722) and NSF/Lucent Technologies Industrial Ecology Fellowship (NSF award BES 98-73586) to N.V.S.
Key words. convex hulls and envelopes – multilinear functions – disjunctive programming – global optimization 相似文献
16.
We consider a class of unconstrained nonsmooth convex optimization problems, in which the objective function is the sum of
a convex smooth function on an open subset of matrices and a separable convex function on a set of matrices. This problem
includes the covariance selection problem that can be expressed as an ℓ
1-penalized maximum likelihood estimation problem. In this paper, we propose a block coordinate gradient descent method (abbreviated
as BCGD) for solving this class of nonsmooth separable problems with the coordinate block chosen by a Gauss-Seidel rule. The
method is simple, highly parallelizable, and suited for large-scale problems. We establish global convergence and, under a
local Lipschizian error bound assumption, linear rate of convergence for this method. For the covariance selection problem,
the method can terminate in O(n3/e){O(n^3/\epsilon)} iterations with an e{\epsilon}-optimal solution. We compare the performance of the BCGD method with the first-order methods proposed by Lu (SIAM J Optim
19:1807–1827, 2009; SIAM J Matrix Anal Appl 31:2000–2016, 2010) for solving the covariance selection problem on randomly generated instances. Our numerical experience suggests that the
BCGD method can be efficient for large-scale covariance selection problems with constraints. 相似文献
17.
Let Ω be a bounded convex domain and let ω be a finite Blaschke product of order N = 1, 2, .... It is known that the elliptic
differential equation
admits a one-to-one solution normalized by ƒ(0) = 0, ƒz(0) > 0 and maps the open unit disc
onto a convex (n + 2)-gon whose vertices belong to ∂Ω. In this article it is shown that this solution is unique. 相似文献
18.
Summary In this paper we study the Dirichlet problem for the minimal surface equation in a open set Ω without any assumption about
the regularity of ϖΩ. We prove an existence theorem using only the pseudoconvexity of Ω.
Riassunto In questo lavoro studiamo il problema di Dirichlet per l'equazione delle superfici minime in un aperto Ω diR n sulla cui frontiera non si fa nessuna ipotesi di regolarità. Si ottiene un teorema di esistenza usando la sola pseudoconvessità di Ω.相似文献
19.
A geometric permutation induced by a transversal line of a finite family of disjoint convex sets in ℝd is the order in which the transversal meets the members of the family. It is known that the maximal number of geometric permutations
in families of n disjoint translates of a convex set in ℝ3 is 3. We prove that for d ≥ 3 the maximal number of geometric permutations for such families in ℝd is Ω(n). 相似文献
20.
Hui-ling LI & Ming-xin WANG Department of Mathematics Southeast University Nanjing China 《中国科学A辑(英文版)》2007,50(4):590-608
This paper deals with the properties of positive solutions to a quasilinear parabolic equation with the nonlinear absorption and the boundary flux. The necessary and sufficient conditions on the global existence of solutions are described in terms of different parameters appearing in this problem. Moreover, by a result of Chasseign and Vazquez and the comparison principle, we deduce that the blow-up occurs only on the boundary (?)Ω. In addition, for a bounded Lipschitz domainΩ, we establish the blow-up rate estimates for the positive solution to this problem with a= 0. 相似文献