Abstract
This paper develops boundary integral representation formulas for the second variations of cost functionals for elliptic domain optimization problems. From the collection of all Lipschitz domains Ω which satisfy a constraint ∫ Ω g(x) dx=1, a domain is sought which maximizes either \(\mathcal{F}_{x_{0}}(\Omega )=F(x_{0},u(x_{0}))\) , fixed x 0∈Ω, or ℱ(Ω)=∫ Ω F(x,u(x)) dx, where u solves the Dirichlet problem Δu(x)=−f(x), x∈Ω, u(x)=0, x∈∂Ω. Necessary and sufficient conditions for local optimality are presented in terms of the first and second variations of the cost functionals \(\mathcal{F}_{x_{0}}\) and ℱ. The second variations are computed with respect to domain variations which preserve the constraint. After first summarizing known facts about the first variations of u and the cost functionals, a series of formulas relating various second variations of these quantities are derived. Calculating the second variations depends on finding first variations of solutions u when the data f are permitted to depend on the domain Ω.
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References
Huang, C., Miller, D.: Variations of constrained domain functionals associated with boundary value problems. J. Optim. Theory Appl. 108(3), 587–615 (2001)
Simon, J.: Second Variations for Domain Optimization Problems. International Series of Numerical Mathematics, vol. 91. Birkhauser, Basel (1989)
Simon, J.: Differentiation with respect to the domain in boundary-value problems. Numer. Funct. Anal. Optim. 2, 649–687 (1980)
Garbedian, P.R., Schiffer, M.: Convexity of domain functionals. J. Anal. Math. 2, 281–368 (1953)
Elcrat, A.R., Miller, K.G.: Variational formulas on Lipschitz domains. Trans. Am. Math. Soc. 347, 2669–2678 (1995)
Kinateder, K., McDonald, P.: Brownian functionals on hypersurfaces in Euclidean space. Proc. Am. Math. Soc. 125, 1815–1822 (1997)
Kinateder, K., McDonald, P., Miller, D.: Boundary-value problems and exit time moments for diffusions in Euclidean space. Probab. Theory Rel. Fields 111, 469–487 (1998)
Sokolowski, J., Zolesio, J.P.: Introduction to Shape Optimization; Shape Sensitivity Analysis. Springer, New York (1992)
Delfour, M.C., Zolesio, J.P.: Shapes and Geometries: Analysis, Differential Calculus, and Optimization. SIAM, Philadelphia (2001)
Miller, D.F.: Characterizations of second variations of constrained domain functionals. Technical Report, Department of Mathematics and Statistics, Wright State University (2006)
Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differentiable Equations of Second Order, 2nd edn. Springer, New York (1983)
Evans, L.C.: Partial Differential Equations. Graduate Studies in Mathematics, vol. 19. American Mathematical Society, Providence (1991)
Jerison, D.S., Kenig, C.E.: Boundary-value problems on Lipschitz domains. In: Littmann, W. (ed.) Studies in Partial Differential Equations. MAA Studies in Mathematics, vol. 23, pp. 1–68. Math. Assoc. of America, Washington (1982)
Jerison, D.S., Kenig, C.E.: The Neumann problems on Lipschitz domains. Bull. Am. Math. Soc. 4, 203–207 (1981)
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Communicated by D.A. Carlson.
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Miller, D.F. Second-Order Optimality Conditions for Constrained Domain Optimization. J Optim Theory Appl 134, 413–432 (2007). https://doi.org/10.1007/s10957-007-9218-9
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DOI: https://doi.org/10.1007/s10957-007-9218-9