On a decomposition of regular domains into John domains with uniform constants

Friedrich, Manuel

ESAIM: Control, Optimisation and Calculus of Variations, Tome 24 (2018) no. 4, pp. 1541-1583

We derive a decomposition result for regular, two-dimensional domains into John domains with uniform constants. We prove that for every simply connected domain $\Omega \subset {ℝ}^2$ with $C^1$-boundary there is a corresponding partition $\Omega ={\Omega }_1\cup ...\cup {\Omega }_N$ with ${\Sigma }_{j=1}^N{\mathscr{H}}^1(\partial {\Omega }_j\setminus \partial \Omega )\le \theta$ such that each component is a John domain with a John constant only depending on $\theta$. The result implies that many inequalities in Sobolev spaces such as Poincaré’s or Korn’s inequality hold on the partition of $\Omega$ for uniform constants, which are independent of $\Omega$.

Classification : 26D10, 70G75, 46E35
Keywords: John domains, Korn’s inequality, free discontinuity problems, shape optimization problems

[1] G. Acosta, R.G. Durán and M.A. Muschietti, Solutions of the divergence operator on John Domains. Adv. Math. 206 (2006) 373–401 | Zbl | MR | DOI

[2] L Ambrosio, A. Coscia and G. Dal Maso, Fine properties of functions with bounded deformation. Arch. Ration. Mech. Anal. 139 (1997) 201–238 | Zbl | MR | DOI

[3] L. Ambrosio, N. Fusco and D. Pallara, Functions of bounded variation and free discontinuity problems. Oxford University Press, Oxford (2000) | Zbl | MR | DOI

[4] G. Bellettini, A. Coscia and G. Dal Maso Compactness and lower semicontinuity properties in SBD(Ω). Math. Z. 228 (1998) 337–351 | Zbl | MR | DOI

[5] B. Bojarski, Remarks on Sobolev imbedding inequalities. In Vol. 1351 of Lecture Notes in Math. Springer, Berlin (1989) 52–68 | Zbl | MR

[6] S. Buckley and P. Koskela, Sobolev-Poincaré implies John. Math. Res. Lett. 2 (1995) 577–593 | Zbl | MR | DOI

[7] D. Bucur and G. Buttazzo, Variational Methods in Shape Optimization Problems. In Vol. 65 of Progress in Nonlinear Differential Equations, Birkhäuser Verlag, Basel (2005) | Zbl | MR

[8] D. Bucur and N. Varchon, A duality approach for the boundary variation of Neumann problems. SIAM J. Math. Anal. 34 (2002) 460–477 | Zbl | MR | DOI

[9] A. Chambolle, A density result in two-dimensional linearized elasticity, and applications. Arch. Ration. Mech. Anal. 167 (2003) 167–211 | Zbl | MR | DOI

[10] A. Chambolle, An approximation result for special functions with bounded deformation. J. Math. Pures Appl. 83 (2004) 929–954 | Zbl | MR | DOI

[11] A. Chambolle, A. Giacomini and M. Ponsiglione, Piecewise rigidity. J. Funct. Anal. 244 (2007) 134–153 | Zbl | MR | DOI

[12] G. Cortesani and R. Toader, A density result in SBV with respect to non-isotropic energies. Nonl. Anal. 38 (1999) 585–604 | Zbl | MR | DOI

[13] G. Dal Maso and R. Toader, A model for the quasi-static growth of brittle fractures: existence and approximation results. Arch. Ration. Mech. Anal. 162 (2002) 101–135 | Zbl | MR | DOI

[14] E. De Giorgi and L. Ambrosio, Un nuovo funzionale del calcolo delle variazioni. Acc. Naz. Lincei, Rend. Cl. Sci. Fis. Mat. Natur. 82 (1988) 199–210 | Zbl

[15] L. Diening, M. Růzicka and K. Schumacher, A decomposition technique for John domains. Ann. Acad. Sci. Fenn. Math. 35 (2010) 87–114 | Zbl | MR | DOI

[16] R. Durán, An elementary proof of the continuity from $L_0^2\left(\omega \right)$ to $H_0^1{\left(\omega \right)}^n$ of Bogovskii’s right inverse of the divergence. Rev. Union Mat. Argentina 53 (2012) 59–78 | Zbl | MR

[17] R. Durán and M.A. Muschietti, The Korn inequality for Jones domains. Electron. J. Diff. Eqs. 127 (2004) 1–10 | Zbl | MR

[18] M. Friedrich, A piecewise Korn inequality in SBD and applications to embedding and density results. Preprint (2016) | arXiv | MR

[19] M. Friedrich and F. Solombrino, Quasistatic crack growth in linearized elasticity. Ann. Inst. Henri Poincaré Anal. Non Linéaire, to appear | MR

[20] K.O. Friedrichs, On certain inequalities and characteristic value problems for analytic functions and for functions of two variables. Trans. Amer. Math. Soc. 41 (1937) 321–364 | JFM | MR | DOI

[21] K.O. Friedrichs, On the boundary-value problems of the theory of elasticity and Korn’s inequality. Ann. Math. 48 (1947) 441–471 | Zbl | MR | DOI

[22] G Geymonat and G. Gilardi, Contre-exemples à l’inégalité de Korn et au Lemme de Lions dans des domaines irréguliers. Equations aux Dérivées Partielles et Applications. Gauthiers-Villars (1998) 541–548 | Zbl | MR

[23] D. Harutyunyan, New asymptotically sharp Korn and Korn-like inequalities in thin domains. J. Elasticity 117 (2014) 95–109 | Zbl | MR | DOI

[24] C.O. Horgan, Korn’s inequalities and their applications in continuum mechanics. SIAM Rev. 37 (1995) 491–511 | Zbl | MR | DOI

[25] F. John, Rotation and strain. Commun. Pure Appl. Math. 14 (1961) 391–413 | Zbl | MR | DOI

[26] P. Jones, Quasiconformal mappings and extendability of functions in Sobolev spaces. Acta Math. 147 (1981) 71–88 | Zbl | MR | DOI

[27] V.A. Kondratiev and O.A. Oleinik, On Korn’s inequalities. C. R. Acad. Sci. Paris 308 (1989) 483–487 | Zbl | MR

[28] M. Lassak, Approximation of convex bodies by rectangles. Geom. Dedicata 47 (1993) 111–117 | Zbl | MR | DOI

[29] G. Lazzaroni and R. Toader, Energy release rate and stress intensity factor in antiplane elasticity. J. Math. Pures Appl. 95 (2011) 565–584 | Zbl | MR | DOI

[30] M. Lewicka and S. Müller, The uniform Korn−Poincaré inequality in thin domains. Ann. Inst. Henri Poincaré Anal. Non Linéaire 28 (2011) 443–469 | Zbl | Numdam | MR | DOI

[31] O. Martio and J. Sarvas, Injectivity theorems in plane and space. Ann. Acad. Sci. Fenn. Math. 4 (1978) 383–401 | Zbl | MR | DOI

[32] F. Murat, The Neumann sieve. Nonlinear Variational Problems, Edited by A. Marino et al. In vol. 127 of Research Notes in Math. Pitman, London (1985) 24–32 | Zbl | MR

[33] R. Näkki and J. Väisälä, John disks. Exposition. Math. 9 (1991) 3–43 | Zbl | MR

[34] M. Negri and R. Toader, Scaling in fracture mechanics by Bažant’s law: from finite to linearized elasticity. Math. Models Methods Appl. Sci. 25 (2015) 1389–1420 | Zbl | MR | DOI

[35] J.A. Nitsche, On Korn’s second inequality. RAIRO Anal. Numér. 15 (1981) 237–248 | Zbl | Numdam | MR | DOI

[36] J. O’Rourke, Computational Geometry in C. Cambridge University Press, Cambridge (1994) | Zbl | MR

[37] L. Rondi, Reconstruction in the inverse crack problem by variational methods. Eur. J. Appl. Math. 19 (2008) 635–660 | Zbl | MR | DOI

[38] V. Šverák, On optimal shape design. J. Math. Pures Appl. 72 (1993) 537–551 | Zbl | MR

[39] J. Väisälä, Unions of John domains. Proc. Amer. Math. Soc. 128 (2000) 1135–1140 | Zbl | MR | DOI

[40] N. Weck, Local compactness for linear elasticity in irregular domains. Math. Methods Appl. Sci. 17 (1994) 107–113 | Zbl | MR | DOI