Variational Methods in Shape Optimization Problems

By Dorin Bucur, Giuseppe M. Buttazzo. Progress in Nonlinear Differential Equations and Their Applications, 2005

525 citations recorded by the source corpus.

Subfields: Civil and Structural Engineering, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/a1d30e104b6b5e75ef8ec85a2b9a701eff38ea9a

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In Civil and Structural Engineering

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Citations in the corpus

Optimal Shape and Location of Sensors for Parabolic Equations with Random Initial Data (2014, background): “This is what is usually referred to as relaxation (see, for example, [10]).”
Optimal Observation of the One-dimensional Wave Equation (2013, background): “To overcome this non-existence result, it is usual in shape optimization problems to consider convexiﬁed formulations permitting to derive existence results (see e.g. [2]).”
Optimal location of controllers for the one-dimensional wave equation (2013, background): “This procedure is very usual in shape optimization problems (see e.g. [3]).”
Estimates for the Torsion Function and Sobolev Constants (2012, background): “with u = 0 on the regular points of ∂� . The torsion function u arises in many areas of mathematics, for example in elasticity theory [1], heat conduction [2], the definition of gamma convergence [ 6 ], and the study of minimal submanifolds [11].”
On Pólya’s Inequality for Torsional Rigidity and First Dirichlet Eigenvalue (2016, background): “It also arises in the calculation of the heat content of sets with time-dependent boundary conditions [3], in the definition of gamma convergence [6], and in the study of minimal submanifolds [11].”
A weighted eigenvalue problem for the p-Laplacian plus a potential (2009, background): “For other characterizations and examples of p-stable sets, we refer to [6,16].”
Hardy inequality and Lp estimates for the torsion function (2009, background): “(5) The torsion function and torsional rigidity arise in many areas of mathematics, for example, in elasticity theory [3, 10], heat conduction [6], the definition of gamma convergence [8], and the study of minimal submanifolds [9].”
The Faber–Krahn inequality for the short-time Fourier transform (2021, unknown): “The maximization of Φ Ω among all sets Ω of prescribed measure can be regarded as a shape optimization problem (see [14]) and, in this respect, Theorem 1.1 shares many analogies with the celebrated Faber-Krahn inequality (beyond the fact that both problems have the ball as a solution).”
Regularity of the optimal sets for some spectral functionals (2016, background): “Moreover, while in the theory of minimal surfaces it is well-known that the critical dimension is precisely 8 (thanks to the works of Simons [36] and Bombieri, De Giorgi, Giusti [4]), up to our knowledge (see, for example, [19] and the recent [26]) it is only known that d ∈ [5, 7].”
Domain Perturbation for Linear and Semi-Linear Boundary Value Problems (2008, background): “The property that Rn(λ) f → R(λ) f , at least in the case of the Laplace operator, is often called the γ -convergence of the solutions (see for instance [27]).”