Article
Homogenization of convex functionals which are weakly coercive and not equi-bounded from above
Author/s | Briane, Marc
Casado Díaz, Juan |
Department | Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico |
Publication Date | 2013-08-01 |
Deposit Date | 2022-11-10 |
Published in |
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Abstract | This paper deals with the homogenization of nonlinear convex energies defined in W_{0}^{1,1}(\Omega )W01,1(Ω), for a regular bounded open set Ω of \mathbb{R}^{N}RN, the densities of which are not equi-bounded from above, ... This paper deals with the homogenization of nonlinear convex energies defined in W_{0}^{1,1}(\Omega )W01,1(Ω), for a regular bounded open set Ω of \mathbb{R}^{N}RN, the densities of which are not equi-bounded from above, and which satisfy the following weak coercivity condition: There exists q > N−1q>N−1 if N > 2N>2, and q⩾1q⩾1 if N = 2N=2, such that any sequence of bounded energy is compact in W_{0}^{1,q}(\Omega )W01,q(Ω). Under this assumption the Γ-convergence of the functionals for the strong topology of L^{\infty }(\Omega )L∞(Ω) is proved to agree with the Γ-convergence for the strong topology of L^{1}(\Omega )L1(Ω). This leads to an integral representation of the Γ-limit in C_{0}^{1}(\Omega )C01(Ω) thanks to a local convex density. An example based on a thin cylinder with very low and very large energy densities, which concentrates to a line shows that the loss of the weak coercivity condition can induce nonlocal effects. |
Citation | Briane, M. y Casado Díaz, J. (2013). Homogenization of convex functionals which are weakly coercive and not equi-bounded from above. Annales de l'Institut Henri Poincaré. Analyse non linéaire, 30 (4), 547-571. https://doi.org/10.1016/J.ANIHPC.2012.10.005. |
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