Rad HAZU, Matematičke znanosti, Vol. 20 (2016), 109-123.

BOUNDARY PERTURBATION FOR THE DIRICHLET BOUNDARY VALUE PROBLEM

Tomislav Fratrović and Eduard Marušić-Paloka

Department of Mathematics, University of Zagreb, Bijenička cesta 30, HR-10000 Zagreb, Croatia
e-mail: emarusic@math.hr

Abstract.   We study the effects of small boundary perturbations on the solutions of the boundary value problems posed in such domains. We start from the domain Ω and then perturb its boundary by the product of a small parameter and some smooth function. The zeroth order approximation is simply the same boundary value problem posed in domain Ω, but the first order corrector is also found, containing some interesting effects.

2010 Mathematics Subject Classification.   76D05, 76D07, 35B40, 35J05, 35Q30.

Key words and phrases.   Boundary perturbation, asymptotic expansion, fluid flow, Poisson equation, Poiseuille flow.

References:

1. H. Ammari, H. Kang, M. Lim and H. Zribi, Conductivity interface problems. Part I: Small perturbation of an interface, Trans. Amer. Math. Soc. 362 (1) (2010), 2435-2449.
   MathSciNet     CrossRef

2. H. Ammari, H. Kang, H. Lee and J. Lim, Boundary perturbations due to the presence of small linear cracks in an elastic body, J. Elasticity 113 (1) (2013), 75-91.
   MathSciNet     CrossRef

3. G. K. Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press, 2000.
   MathSciNet

4. E. Beretta and E. Francini, An asymptotic formula for the displacement field in the presence of thin elastic inhomogeneities, SIAM J. Math. Anal. 38 (4) (2006), 1249-1261.
   MathSciNet     CrossRef

5. D. Daners, Domain perturbation for linear and nonlinear parabolic equations, J. Differ. Equ. 129 (2) (1996), pp. 358-402.
   MathSciNet     CrossRef

6. D. Daners, Dirichlet problems on varying domains, J. Differ. Equ. 188 (2) (2003), 591-624.
   MathSciNet     CrossRef

7. D. Henry, Perturbation of the Boundary in Boundary-Value Problems of Partial Differential Equations, London Math. Soc. Lecture Note Ser. 318, Cambridge University Press, 2005.
   MathSciNet     CrossRef

8. M. H. Holmes, Introduction to Perturbation Methods, Springer Science and Business Media, 2012.
   MathSciNet     CrossRef

9. A. K. Kerimov, On the dependence of solutions of boundary value problems on perturbation of the domain by vector fields, Math. USSR Sb. 69 (2) (1991), 393-416.
   MathSciNet

10. T. Kobayashi, Time periodic solutions of the Navier-Stokes equations with the time periodic Poiseuille flow in two and three dimensional perturbed channels, Tohoku Math. J. 66 (1) (2014), 119-135.
    MathSciNet     CrossRef

11. T. H. C. Luong and C. Daveau, Asymptotic formula for the solution of the Stokes problem with a small perturbation of the domain in two and three dimensions, Complex Var. Elliptic Equ. 59 (9) (2014), 1269-1282.
    MathSciNet     CrossRef

12. P. B. Mucha and T. Piasecki, Compressible perturbation of Poiseuille type flow, J. Math. Pures Appl. 102 (2014), 338-363.
    MathSciNet     CrossRef

13. M. Van Dyke, Perturbation Methods in Fluid Mechanics, Parabolic Press, 1975.
    MathSciNet

14. M. S. Vogelius and D. Volkov, Asymptotic formulas for perturbations in the electromagnetic fields due to the presence of inhomogeneities of small diameter, ESAIM: M2AN 34 (4) (2000), 723-748.
    MathSciNet     CrossRef