Corrigendum to “Nonconvex variational problem with recursive integral functionals in Sobolev spaces: Existence and representation” [J. Math. Anal. Appl. 327 (1) (2007) 203–219]

We study some properties of a nonconvex variational problem. We fail to attain the infimum of the functional that has to be minimized. Instead, minimizing sequences develop gradient oscillations which allow them to reduce the value of the functional. We show an existence result for a perturbed nonconvex version of the problem, and we study the qualitative properties of the corresponding minimizer. The pattern of the gradient oscillations for the original nonperturbed problem is analyzed numerically.

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Second order differentiability of integral functionals on Sobolev spaces and L2-spaces.

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crll.1993.436.1 ◽

1993 ◽

Vol 1993 (436) ◽

pp. 1-18 ◽

Cited By ~ 2

Keyword(s):

Sobolev Spaces ◽

Second Order ◽

Integral Functionals

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Minimization of nonsmooth integral functionals

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171292000899 ◽

1992 ◽

Vol 15 (4) ◽

pp. 673-679

Author(s):

Nikolaos S. Papageorgiou ◽

Apostolos S. Papageorgiou

Keyword(s):

Sobolev Spaces ◽

Convex Analysis ◽

Optimization Problems ◽

Sufficient Conditions ◽

Integral Functionals ◽

Infinite Dimensional ◽

Finite Dimensional ◽

Dimensional State Space ◽

Sufficient Conditions For Optimality ◽

Necessary And Sufficient

In this paper we examine optimization problems involving multidimensional nonsmooth integral functionals defined on Sobolev spaces. We obtain necessary and sufficient conditions for optimality in convex, finite dimensional problems using techniques from convex analysis and in nonconvex, finite dimensional problems, using the subdifferential of Clarke. We also consider problems with infinite dimensional state space and we finally present two examples.

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Coercivity of integral functionals with non-everywhere superlinear lagrangians

Advances in Calculus of Variations ◽

10.1515/acv-2017-0014 ◽

2019 ◽

Vol 12 (4) ◽

pp. 447-458

Author(s):

Cristina Marcelli

Keyword(s):

Variational Problem ◽

Sufficient Conditions ◽

Integral Functionals ◽

Superlinear Growth

AbstractWe consider the classical non-autonomous variational problem\text{minimize }\biggl{\{}F(v)=\int_{a}^{b}f(x,v(x),v^{\prime}(x))\,\mathrm{d}% x:v\in\Omega\biggr{\}},where {\Omega:=\{v\in W^{1,1}(a,b),\,v(a)=A,\,v(b)=B,\,v(x)\in I\}}, when the lagrangian f has non-everywhere superlinear growth, in the sense that it can vanish at some {x_{0}\in[a,b]}, or {s_{0}\in I}. We prove some sufficient conditions ensuring the coercivity of the functional F. As a consequence, when f is convex with respect to the last variable, the existence of the minimum can be immediately derived.

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