Bibliographic Details
| Title: |
Relaxation and optimization for linear-growth convex integral functionals under PDE constraints. |
| Authors: |
Arroyo-Rabasa, Adolfo1 adolfo.arroyo.rabasa@hcm.uni-bonn.de |
| Source: |
Journal of Functional Analysis. Oct2017, Vol. 273 Issue 7, p2388-2427. 40p. |
| Subjects: |
Integral functions, Saddlepoint approximations, Integrable functions, Partial differential equations, Radon measures |
| Abstract: |
We give necessary and sufficient conditions for the minimality of generalized minimizers of linear-growth integral functionals of the form F [ u ] = ∫ Ω f ( x , u ( x ) ) d x , u : Ω ⊂ R d → R N , where f : Ω × R N → R is a convex integrand and u is an integrable function satisfying a general PDE constraint. Our analysis is based on two ideas: a relaxation argument into a subspace of the space of bounded vector-valued Radon measures M ( Ω ; R N ) , and the introduction of a set-valued pairing on M ( Ω ; R N ) × L ∞ ( Ω ; R N ) . By these means we are able to show an intrinsic relation between minimizers of the relaxed problem and maximizers of its dual formulation also known as the saddle-point conditions. In particular, our results can be applied to relaxation and minimization problems in BV, BD and divergence-free spaces. [ABSTRACT FROM AUTHOR] |
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| Database: |
Engineering Source |
