The Bounded Slope Condition for Functionals Depending on <i>x</i>, <i>u</i>, and $\nablau$

Fiaschi, Alice; Treu, Giulia

doi:10.1137/110822980

Cited by 11 publications

(21 citation statements)

References 23 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…where C i are positive constants, while the exponent q is assumed to satisfy q = 2 * := 2n n−2 if n > 2 and q ∈ (1, +∞) if n ≤ 2; -g(0) = 0 (which is not restrictive up to a translation); -the solution u to J(Ω) is Lipschitz (we point out that, by strict convexity, u is unique, and we refer to [11,22,31,32] for some Lipschitz regularity results);…”

Section: Preliminariesmentioning

confidence: 99%

A Variational Method for Second Order Shape Derivatives

Bouchitté¹,

Fragalà²,

Lucardesi³

2016

SIAM J. Control Optim.

View full text Add to dashboard Cite

We consider shape functionals obtained as minima on Sobolev spaces of classical integrals having smooth and convex densities, under mixed Dirichlet-Neumann boundary conditions. We propose a new approach for the computation of the second order shape derivative of such functionals, yielding a general existence and representation theorem. In particular, we consider the p-torsional rigidity functional for p ≥ 2.Keywords: shape functionals, infimum problems, domain derivative, duality. MSC2010: 49Q10, 49K10, 49M29, 49J45.Our main results, which are intimately related to each other, are:-an existence result for J ′′ (Ω, V ) which is a quadratic form in V and can be represented as follows:here C(u, V ) · n is the normal trace of a suitable vector field (see (3.4)) depending on the solution u to J(Ω) and quadratically on the deformation V , whereas q(u, V ) is a nonlocal term which involves a further vector field B(u, V ) and a quadratic form Q(u, ·) depending on the second derivatives of f and g (see Theorem 3.4); we stress that, as detailed in Remark 3.6 below, formula (1.3) fits the general representation result for second order shape derivatives given in [36, Corollary 2.4];-a regularity result of type W 2,2 loc for the solution to J(Ω) (see Proposition 3.2); -a new necessary optimality condition, which involves the distributional divergence of the above mentioned vector field B(u, V ) (see Proposition 3.3).Both the regularity of the solution and the optimality condition on one hand seem to have an independent interest, and on the other hand are strictly related to the second order shape derivative. Namely, if the W 2,2 loc regularity of u stated in Proposition 3.2 extends up to the boundary, the field C(u, V ) turns out to admit a normal trace on the boundary as soon as the latter is piecewise C 1 ; moreover, in order to arrive at the expression (1.3) for J ′′ (Ω, V ), the optimality condition given in Proposition 3.3 is exploited as a crucial tool.The paper is organized as follows. After providing some notation and preliminary background in Section 2, we state in Section 3 our main results (Proposition 3.2, Proposition 3.3 and Theorem 3.4),

show abstract

Section: Preliminariesmentioning

confidence: 99%

A Variational Method for Second Order Shape Derivatives

Bouchitté¹,

Fragalà²,

Lucardesi³

2016

SIAM J. Control Optim.

View full text Add to dashboard Cite

show abstract

“…In recent years, many authors were investigating questions considering existence and regularity of minimizers to a more general class of problems of the form min f (Du) + g(x, u) dx : u ∈ W 1, p φ ( ) , (1.4) where -for a suitable domain ⊂ R n and p ≥ 1 -W 1, p φ ( ) denotes the space of Sobolev functions u ∈ W 1, p ( ) having trace equal to φ on ∂ . For a selection of results covering the case g = 0, we refer the interested reader to [6,[10][11][12]24,25], as well as to [7][8][9]14,23] for some results about the general case g = 0. In this context, the bounded slope condition was also considerably weakened in a few ways, one of them being a one-sided bounded slope condition (see [12]).…”

Section: Introductionmentioning

confidence: 99%

An evolutionary Haar-Rado type theorem

2021

View full text Add to dashboard Cite

In this paper, we study variational solutions to parabolic equations of the type $$\partial _t u - \mathrm {div}_x (D_\xi f(Du)) + D_ug(x,u) = 0$$ ∂ t u - div x ( D ξ f ( D u ) ) + D u g ( x , u ) = 0 , where u attains time-independent boundary values $$u_0$$ u 0 on the parabolic boundary and f, g fulfill convexity assumptions. We establish a Haar-Rado type theorem: If the boundary values $$u_0$$ u 0 admit a modulus of continuity $$\omega $$ ω and the estimate $$|u(x,t)-u_0(\gamma )| \le \omega (|x-\gamma |)$$ | u ( x , t ) - u 0 ( γ ) | ≤ ω ( | x - γ | ) holds, then u admits the same modulus of continuity in the spatial variable.

show abstract

“…In the present paper we are interested in the study of the Lipschitz regularity of minimizers of a class of functionals starting from the regularity of the boundary datum without assuming neither ellipticity nor the growth conditions on the lagrangian: the literature on this subject is extremely rich, we address the interested reader to [6,7,13,14,17,27,29,30,31] and references therein for an overview. Our analysis moves from a recent paper by Pinamonti et al [33] where the area functional for the t-graph of a function u ∈ W 1,1 (Ω) in the sub-Riemannian Heisenberg group H n = R n x × R n y × R t is investigated (see also further references in [33] on the Heisenberg's literature).…”

Section: Introductionmentioning

confidence: 99%