共查询到20条相似文献,搜索用时 31 毫秒
1.
Dorin Bucur 《Milan Journal of Mathematics》2007,75(1):379-398
This paper surveys the talk given by the author at the “Seminario Matematico e Fisico di Milano” in November 2006. It deals
with the existence question for shape optimization problems associated to the Dirichlet Laplacian. Existence of solutions
is seen from both geometrical and functional (γ-convergence) point of view and is discussed in relationship with the optimality conditions and numerical algorithms. Several
examples are given concerning isoperimetric inequalities for eigenvalues and shape control problems.
Received: August 2007 相似文献
2.
I. Hlaváček A. A. Novotny J. Sokołowski A. Żochowski 《Journal of Optimization Theory and Applications》2009,141(3):569-595
In this paper, a new approach to the derivation of the worst scenario and the maximum range scenario methods is proposed.
The derivation is based on the topological derivative concept for the boundary-value problems of elasticity in two and three
spatial dimensions. It is shown that the topological derivatives can be applied to the shape and topology optimization problems
within a certain range of input data including the Lamé coefficients and the boundary tractions. In other words, the topological
derivatives are stable functions and the concept of topological sensitivity is robust with respect to the imperfections caused by uncertain data. Two classes of integral shape functionals are considered, the
first for the displacement field and the second for the stresses. For such classes, the form of the topological derivatives
is given and, for the second class, some restrictions on the shape functionals are introduced in order to assure the existence
of topological derivatives. The results on topological derivatives are used for the mathematical analysis of the worst scenario
and the maximum range scenario methods. The presented results can be extended to more realistic methods for some uncertain
material parameters and with the optimality criteria including the shape and topological derivatives for a broad class of
shape functionals.
This research is partially supported by the Brazilian Agency CNPq under Grant 472182/2007-2, FAPERJ under Grant E-26/171.099/2006
(Rio de Janeiro) and Brazilian-French Research Program CAPES/COFECUB under Grant 604/08 between LNCC in Petrópolis and IECN
in Nancy, and by the Research Grant CNRS-CSAV between Institut Elie Cartan in Nancy and the Institut of Mathematics in Prague.
The support is gratefully acknowledged. 相似文献
3.
Isoperimetric inequalities,Wulff shape and related questions for strongly nonlinear elliptic operators 总被引:1,自引:0,他引:1
We investigate the first eigenvalue of a highly nonlinear class of elliptic operators which includes the p--Laplace operator
$\Delta_p u=\sum_i {{\partial}\over{\partial x_i}} (\vert\nabla u \vert^{p-2}{{\partial u}\over{\partial x_i}})$, the pseudo-p-Laplace operator
$\tilde\Delta_p u=\sum_i {{\partial}\over{\partial x_i}} (\vert {{\partial u}\over{\partial x_i}} \vert^{p-2} {{\partial u}\over{\partial x_i}})$ and others. We derive the positivity of the first eingefunction, simlicity of the first eigenvalue, Faber-Krahn and Payne-Rayner type inequalities.
In another chapter we address
the question of symmetry for positive solutions to more general equations.
Using a Pohozaev-type inequality and isoperimetric
inequalities as well as convex rearrangement methods
we generalize a symmetry result of Kesavan and Pacella.
Our optimal domains are level sets of a convex function
H
o.
They have the so-called Wulff shape associated with
H and only in special cases they are
Euclidean balls. 相似文献
4.
Using the principle of symmetric criticality for Motreanu–Panagiotopoulos type functionals we give some existence and multiplicity results for a class of variational–hemivariational inequalities on
L+M
.This work was partially supported by MEdC-ANCS, research project CEEX 2983/11.10.2005. 相似文献
5.
Alexander Mielke Tomáš Roubíček Ulisse Stefanelli 《Calculus of Variations and Partial Differential Equations》2008,31(3):387-416
This work uses the energetic formulation of rate-independent systems that is based on the stored-energy functionals and the dissipation distance . For sequences and we address the question under which conditions the limits q
∞ of solutions satisfy a suitable limit problem with limit functionals and , which are the corresponding Γ-limits. We derive a sufficient condition, called conditional upper semi-continuity of the stable sets, which is essential to guarantee that q
∞ solves the limit problem. In particular, this condition holds if certain joint recovery sequences exist. Moreover, we show that time-incremental minimization problems can be used to approximate the solutions. A first example
involves the numerical approximation of functionals using finite-element spaces. A second example shows that the stop and
the play operator converge if the yield sets converge in the sense of Mosco. The third example deals with a problem developing
microstructure in the limit k → ∞, which in the limit can be described by an effective macroscopic model.
Research partially supported by LC06052 (MŠMT), MSM21620839 (MŠMT), A1077402 (GAČR), by the Deutsche Forschungsgemeinschaft
under MATHEON C18 and under SFB404 C7, by the European Union under HPRN-CT-2002-00284 Smart Systems, and by the Alexander von Humboldt-Stiftung. Both, TR and US gratefully acknowledge the kind hospitality of the WIAS, where
this research was initiated. 相似文献
6.
Weak vector equilibrium problems with bi-variable mappings from product space of two bounded complete locally convex Hausdorff topological vector spaces to another topological vector space are studied. The existence theorems of solutions are proved by the FKKM fixed point theorem. Viscosity principle of vector equilibrium problems is dealt with. The relations between solutions of the vector equilibrium problem and those of its perturbation problem are presented. 相似文献
7.
Lower semicontinuity for polyconvex functionals of the form ∫Ω
g(detDu)dx with respect to sequences of functions fromW
1,n
(Ω;ℝ
n
) which converge inL
1 (Ωℝ
n
) and are uniformly bounded inW
1,n−1 (Ω;ℝ
n
), is proved. This was first established in [5] using results from [1] on Cartesian Currents. We give a simple direct proof
which does not involve currents. We also show how the method extends to prove natural, essentially optimal, generalizations
of these results.
Supported by MURST, Gruppo Nazionale 40%
Partially supported by Australian Research Council 相似文献
8.
There is an obvious topological obstruction for a finite energy unimodular harmonic extension of a S
1-valued function defined on the boundary of a bounded regular domain of R
n
. When such extensions do not exist, we use the Ginzburg-Landau relaxation procedure. We prove that, up to a subsequence,
a sequence of Ginzburg-Landau minimizers, as the coupling parameter tends to infinity, converges to a unimodular harmonic
map away from a codimension-2 minimal current minimizing the area within the homology class induced from the S
1-valued boundary data. The union of this harmonic map and the minimal current is the natural generalization of the harmonic
extension.
Received December 3, 1998 / final version received May 10, 1999 相似文献
9.
The topological derivative is defined as the first term of the asymptotic expansion of a given shape functional with respect to a small parameter that measures the size of a singular domain perturbation. It has applications in many different fields such as shape and topology optimization, inverse problems, image processing and mechanical modeling including synthesis and/or optimal design of microstructures, fracture mechanics sensitivity analysis and damage evolution modeling. The topological derivative has been fully developed for a wide range of second order differential operators. In this paper we deal with the topological asymptotic expansion of a class of shape functionals associated with elliptic differential operators of order 2m , m?1. The general structure of the polarization tensor is derived and the concept of degenerate polarization tensor is introduced. We provide full mathematical justifications for the derived formulas, including precise estimates of remainders. 相似文献
10.
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
11.
In this paper we further develop the theory of the extended Timoshenko beam model, as first introduced in Part I [5] of this
work, with particular emphasis on applications of the model in formation theory [11], [12]. We begin with formal development
of the equilibrium equations of static formation theory in the context of the extended Timoshenko model, giving a rigorous
discussion of existence, uniqueness, and regularity of weak solutions with appropriate assumptions on the coefficients. We
continue to obtain the fundamental duality relationship in the context of weak solutions and indicate its usefulness in investigations
of approximate formability. Optimal formation problems and corresponding necessary conditions for optimality are discussed.
We conclude with a discussion of a particular problem of joint optimization of controls and actuator densities in the context
of a prismatic extended Timoshenko beam and we present the results of some computational studies.
Accepted 14 November 1996 相似文献
12.
A. B. Levy 《Set-Valued Analysis》1993,1(4):379-392
Epi-derivatives have many applications in optimization as approached through nonsmooth analysis. In particular, second-order epi-derivatives can be used to obtain optimality conditions and carry out sensitivity analysis. Therefore the existence of second-order epi-derivatives for various classes of functions is a topic of considerable interest. A broad class of composite functions on
n
called fully amenable functions (which include general penalty functions composed withC
2 mappings, possibly under a constraint qualification) are now known to be twice epi-differentiable. Integral functionals appear widely in problems in infinite-dimensional optimization, yet to date, only integral functionals defined by convex integrands have been shown to be twice epi-differentiable, provided that the integrands are twice epi-differentiable. Here it is shown that integral functionals are twice epi-differentiable even without convexity, provided only that their defining integrands are twice epi-differentiable and satisfy a uniform lower boundedness condition. In particular, integral functionals defined by fully amenable integrands are twice epi-differentiable under mild conditions on the behavior of the integrands.This work was supported in part by the National Science Foundation under grant DMS-9200303. 相似文献
13.
A. Visintin 《Calculus of Variations and Partial Differential Equations》2007,29(2):239-265
Nguetseng’s notion of two-scale convergence is reviewed, and some related properties of integral functionals are derived. The coupling of two-scale convergence with
convexity and monotonicity is then investigated, and a two-scale version is provided for compactness by strict convexity. The div-curl lemma of Murat and Tartar is also extended to two-scale convergence, and applications are outlined. 相似文献
14.
João M. Bezerra do Ó Olímpio H. Miyagaki Sérgio H.M. Soares 《Journal of Differential Equations》2010,248(4):722-744
In this paper we establish the existence of standing wave solutions for quasilinear Schrödinger equations involving critical growth. By using a change of variables, the quasilinear equations are reduced to semilinear one, whose associated functionals are well defined in the usual Sobolev space and satisfy the geometric conditions of the mountain pass theorem. Using this fact, we obtain a Cerami sequence converging weakly to a solution v. In the proof that v is nontrivial, the main tool is the concentration-compactness principle due to P.L. Lions together with some classical arguments used by H. Brezis and L. Nirenberg (1983) in [9]. 相似文献
15.
M.A. Sychev 《Calculus of Variations and Partial Differential Equations》2001,13(2):213-229
Given a compact set we consider the differential inclusion
We show how to use the main idea of the method of convex integration [ N], [G], [K] (to control convergence of the gradients of a sequence of approximate solutions by appropriate selection of the sequence)
to obtain an optimal existence result. We compare this result with the ones available by the Baire category approach applied
to the set of admissible functions with topology.
A byproduct of our result is attainment in the minimization problems
with integrands L having quasiaffine quasiconvexification that was, in fact, the reason of our interest to differential inclusions. This result
can be considered as a first step towards characterization of those minimization problems which are solvable for all boundary
data. This problem was solved in [S1] in the scalar case m=1.
Received November 5, 1998 / Accepted July 17, 2000 / Published online December 8, 2000 相似文献
16.
Explicit exact solutions for a new generalized Hamiltonian amplitude equation with nonlinear terms of any order 总被引:1,自引:0,他引:1
Making use of a proper transformation and a generalized ansatz, we consider a new generalized Hamiltonian amplitude equation with nonlinear terms of any order, iux + utt + (|u|p + |u|2p)u + uxt = 0. As a result, many explicit exact solutions, which include kink-shaped soliton solutions, bell-shaped soliton solutions, periodic wave solutions, the combined formal solitary wave solutions and rational solutions, are obtained.Received: April 4, 2002 相似文献
17.
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
19.
Andrés I. Ávila Jianfu Yang 《NoDEA : Nonlinear Differential Equations and Applications》2006,12(4):459-479
We proved a multiplicity result for a nonlinear elliptic system in RN. The functional related to the system is strongly indefinite. We investigated the relation between the number of solutions
and the topology of the set of the global maxima of the coefficients. 相似文献
20.
Martin Schechter 《Journal of Differential Equations》2009,246(8):3398-3415
For many equations arising in practice, the solutions are critical points of functionals. In previous papers we have shown that there are pairs of subsets, called sandwich pairs, that can produce critical points even though they do not separate the functional. All that is required is that the functional be bounded from above on one of the sets and bounded from below on the other, with no relationship needed between the bounds. This provides a distinct advantage in applications. The present paper discusses the situation in which one cannot find sandwich pairs for which the functional is bounded below on one set and bounded above on the other. We develop a method which can deal with such situations and apply it to problems in partial differential equations. 相似文献