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Γ-CONVERGENCE FOR AN OPTIMAL DESIGN PROBLEM WITH VARIABLE EXPONENT
HAMDI ZORGATI 한국산업응용수학회 2023 Journal of the Korean Society for Industrial and A Vol.27 No.4
In this paper, we derive the Γ-limit of functionals pertaining to some optimal material distribution problems that involve a variable exponent, as the exponent goes to infinity. In addition, we prove a relaxation result for supremal optimal design functionals with respect to the weak-∗ L∞(Ω; [0, 1]) × W 1,p0 (Ω; Rm ) weak topology.
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COMPACTNESS AND DIRICHLET’S PRINCIPLE
JIN KEUN SEO,HAMDI ZORGATI 한국산업응용수학회 2014 Journal of the Korean Society for Industrial and A Vol.18 No.2
In this paper we explore the emergence of the notion of compactness within its historical beginning through rigor versus intuition modes in the treatment of Dirichlet‘s principle. We emphasize on the intuition in Riemann‘s statement on the principle criticized byWeierstrass‘ requirement of rigor followed by Hilbert‘s restatement again criticized by Hadamard, which pushed the ascension of the notion of compactness in the analysis of PDEs. A brief overview of some techniques and problems involving compactness is presented illustrating the importance of this notion. Compactness is discussed here to raise educational issues regarding rigor vs intuition in mathematical studies. The concept of compactness advanced rapidly afterWeierstrass’s famous criticism of Riemann’s use of the Dirichlet principle. The rigor of Weierstrass contributed to establishment of the concept of compactness, but such a focus on rigor blinded mathematicians to big pictures. Fortunately, Poincar´e and Hilbert defended Riemann’s use of the Dirichlet principle and found a balance between rigor and intuition. There is no theorem without rigor, but we should not be a slave of rigor. Rigor (highly detailed examination with toy models) and intuition (broader view with real models) are essentially complementary to each other.
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