Variational Problems for a Class of Functionals on Convex Domains

Crasta, Graziano

doi:10.1006/jdeq.2000.4011

Cited by 10 publications

(20 citation statements)

References 14 publications

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“…Using the existence and uniqueness result obtained for (1), we prove that (6) admits a unique solution, provided that the ratio between the Lebesgue measure of Ω and the (N − 1)-dimensional Hausdorff measure of ∂Ω is strictly less than a constant related to M and g. This provides an extension of the results proved in [5,12] in the case g(u) = −u.…”

Section: Introductionmentioning

confidence: 48%

“…We underline that, if α(t) = H N −1 (∂Ω t ), then (H6) is satisfied (as a consequence of the Brunn-Minkowski theorem, see [5,Thm. 4.1]).…”

Section: Application To Variational Problems On Convex Domainsmentioning

confidence: 99%

“…Here The study of this non-convex scalar problem is mainly motivated by the following application (see [5,6,7,12]). Let Ω be a bounded convex open subset of R N , N ≥ 2, and denote by W Ω its inradius, that is the supremum of the radii of the balls contained in Ω.…”

Section: Introductionmentioning

confidence: 99%

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On the existence and uniqueness of minimizers for a class of integral functionals

Crasta¹,

Malusa²

2005

Nonlinear differ. equ. appl.

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We study the solvability of the minimization problem \[ \min_{\eta \in\Kd} \int_0^T \alpha(t)[f(|\eta'(t)|)+g(\eta(t))]\, dt\,, \] where $\Kd$ is a subset of $AC_{loc}[0,T[$ depending on the weight function $\alpha$. Neither the convexity nor the superlinearity of $f$ are required. The main application concerns the existence and uniqueness of minimizers to integral functionals on convex domains $\Omega \subset \R^{N}$, defined in the class of functions in $\Wuu (\Omega)$ depending only on the distance from the boundary of $\Omega$. As a corollary, when $\Omega$ is a ball we obtain the existence of radially symmetric solutions to nonconvex and noncoercive functionals

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Section: Introductionmentioning

confidence: 48%

“…We underline that, if α(t) = H N −1 (∂Ω t ), then (H6) is satisfied (as a consequence of the Brunn-Minkowski theorem, see [5,Thm. 4.1]).…”

Section: Application To Variational Problems On Convex Domainsmentioning

confidence: 99%

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On the existence and uniqueness of minimizers for a class of integral functionals

Crasta¹,

Malusa²

2005

Nonlinear differ. equ. appl.

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“…(Here AC loc [0, 1) denotes the set of measurable functions y : [0, 1] → R such that y is absolutely continuous on [0, r] for every 0 < r < 1.) Integrating by parts the term in y(t) (see [5,Lemma 5.6]), the minimum problem (4.2) can be rewritten as…”

Section: A Lower Bound For the Torsion By Means Of Web Functionsmentioning

confidence: 99%

“…We already dealt with this kind of functions in the previous papers [5]- [11], where we called them web functions. Here we consider the spaces By exploiting the strong properties of web functions, in Theorem 4.1 we write N (Ω) in a quite simple form in terms of the parallel sets.…”

Section: Introductionmentioning

confidence: 99%

Some estimates for the torsional rigidity of composite rods

Crasta

Fragalà

Gazzola

2007

Mathematische Nachrichten

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A well-known problem in elasticity consists in placing two linearly elastic materials (of different shear moduli) in a given plane domain Ω, so as to maximize the torsional rigidity of the resulting rod; moreover, the proportion of these materials is prescribed. Such a problem may not have a classical solution as the optimal design may contain homogenization regions, where the two materials are mixed in a microscopic scale. Then, the optimal torsional rigidity becomes difficult to compute. In this paper we give some different theoretical upper and lower bounds for the optimal torsional rigidity, and we compare them on some significant domains.

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On the role of energy convexity in the web function approximation

Crasta

Fragalà

Gazzola

2005

Nonlinear differ. equ. appl.

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For a given p > 1 and an open bounded convex set Ω ⊂ℝ2,we consider the minimization problem for the functional Jp (u) = ∫Ω(1/p|∇ μ|p - u)over W01,pp Ω.Since the energy of the unique minimizer u p 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 v p is available. Hence the problem of estimating the error one makes when approximating J p (u p ) by J p (v p ) 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 u p-v p converges to zero in W01,m(Ω) for all m<∞. These results reveal that the approximation of minima by means of web functions gains more and more precision as convexity in J p increases. © Birkhäuser Verlag, Basel 2005

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Variational Problems for a Class of Functionals on Convex Domains

Abstract: Let W be a bounded convex open subset of R N , N \ 2, and let J be the integral functional

Cited by 10 publications

References 14 publications

On the existence and uniqueness of minimizers for a class of integral functionals

On the existence and uniqueness of minimizers for a class of integral functionals

Some estimates for the torsional rigidity of composite rods

On the role of energy convexity in the web function approximation

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