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1.
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
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We consider functionals of the calculus of variations of the form F(u)= ∝01 f(x, u, u′) dx defined for u ε W1,∞(0, 1), and we show that the relaxed functional
with respect to weak W1,1(0, 1) convergence can be written as
, where the additional term L(u), called the Lavrentiev term, is explicitly identified in terms of F. 相似文献
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6.
An optimal design problem with perimeter penalization 总被引:11,自引:0,他引:11
Luigi Ambrosio Giuseppe Buttazzo 《Calculus of Variations and Partial Differential Equations》1993,1(1):55-69
We study the optimal design problem of finding the minimal energy configuration for a mixture of two conducting materials when a perimeter penalization of the unknown domain is added. We show that in this situation an optimal domain exists and that, under suitable assumptions on the data, it is an open set.This work is part of the project EURHomogenization, contract SC1-CT91-0732 of the program SCIENCE of the Commission of the European Communities. 相似文献
7.
Given a bounded open subset of R
n, we prove the existence of a minimum point for a functional F defined on the family A() of all quasiopen subsets of , under the assumption that F is decreasing with respect to set inclusion and that F is lower semicontinuous on A() with respect to a suitable topology, related to the resolvents of the Laplace operator with Dirichlet boundary condition. Applications are given to the existence of sets of prescribed volume with minimal k
th eigenvalue (or with minimal capacity) with respect to a given elliptic operator. 相似文献
8.
Giuseppe Buttazzo Serena Guarino Lo Bianco Michele Marini 《Journal of Mathematical Analysis and Applications》2018,457(2):1153-1172
In this paper we generalize some classical estimates involving the torsional rigidity and the principal frequency of a convex domain to a class of functionals related to some anisotropic nonlinear operators. 相似文献
9.
Giuseppe Buttazzo Faustino Maestre Bozhidar Velichkov 《Journal of Optimization Theory and Applications》2018,177(3):743-769
The main purpose of this paper is to study the duality and penalty method for a constrained nonconvex vector optimization problem. Following along with the image space analysis, a Lagrange-type duality for a constrained nonconvex vector optimization problem is proposed by virtue of the class of vector-valued regular weak separation functions in the image space. Simultaneously, some equivalent characterizations to the zero duality gap property are established including the Lagrange multiplier, the Lagrange saddle point and the regular separation. Moreover, an exact penalization is also obtained by means of a local image regularity condition and a class of particular regular weak separation functions in the image space. 相似文献
10.
In this paper, we give some applications ofG-convergence and -convergence to the study of the asymptotic limits of optimal control problems. More precisely, given a sequence (P
h) of optimal control problems and a control problem (P), we determine some general conditions, involvingG-convergence and -convergence, under which the sequence of the optimal pairs of the problems (P
h) converges to the optimal pair of problem (P).The authors wish to thank Professor E. De Giorgi for many stimulating discussions. 相似文献