Abstract
In this note, we show that, for domains satisfying the separation property, certain weighted Korn inequality is equivalent to the John condition. Our result generalizes previous result from Jiang–Kauranen [Calc. Var. Partial Differential Equations, 56, Art. 109, (2017)] to weighted settings.
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Acosta, G., Durán, R. G., Lombardi, A. L.: Weighted Poincaré and Korn inequalities for Holder a domains. Math. Methods Appl. Sci., 29, 387–400 (2006)
Acosta, G., Durán, R. G., Muschietti, M. A.: Solutions of the divergence operator on John domains. Adv. Math. 206, 373–401 (2006)
Astala, K., Gehring, F. W.: Quasiconformal analogues of theorems of theorems of Koebe and Hardy-Littlewood. Michigan Math. J., 32(1), 99–107 (1985)
Buckley, S., Koskela, P.: Sobolev-Poincare implies John. Math. Res. Lett., 2, 577–593 (1995)
Chua, S. K., Wheeden, R. L.: Self-improveing properties of inequalities of Poincare type on s-John domains. Pacific J. Math., 250, 67–108 (2011)
Cianchi, A.: Korn type inequalities in Orlicz spaces. J. Funct. Anal., 267, 2313–2352 (2014)
Diening, L., Ružicka, M., Schumacher K.: A decomposition technique for John domains. Ann. Acad. Sci. Fenn. Math., 35, 87–114 (2010)
Duran, R. G., López García, F.: Solution of the divergence and Korn inequalities on domains with an external cusp. Ann. Acad. Sci. Fenn. Math., 35, 421–438 (2010)
Duvaut, G., Lions, J. L.: Inequalities in Mechanics and Physics, Springer-Verlag, Berlin, Heidelberg, 1976
Friedrichs, K. O.: On the doundary-value problems of the thoery of elasticity and Korn’s inequality. Ann. of Math., 48(2), 441–471 (1947)
Hajlasz, P., Koskela, P.: Isoperimetric inequalities and imbeding theroems in irregular domains. J. Lond. Math. Soc., 58(2), 425–450 (1998)
Horgan, C. O.: Korn’s inequalities and their applications in continuum mechanics. SIAM Rev., 37, 491–511 (1995)
Jiang, R., Kauranen, A.: Korn’s inequality and John domain. Calc. Var. Partial Differential Equations, 56, Art. 109, 18 pp (2017)
Jiang, R., Kauranen, A.: Korn inequality on irregular domains. J. Math. Anal. Appl., 423, 41–59 (2015)
John, F.: Rotation and strain. Comm. Pure Appl. Math., 14, 391–413 (1961)
Kondratiev, V. A., Oleinik, O. A.: On Korn’s inequalities. C. R. Acad. Sci. Paris Ser. I Math., 308, 483–487 (1989)
López García, F.: Weighted Korn inequalities on John domains. Studia Math., 241, 17–39 (2018)
Martio, O., Sarvas, J.: Injectivity theorems in plane and space. Ann. Acad. Sci. Fenn. Ser. A I Math., 4, 383–401 (1979)
Nečas, J.: Les methodes directes en theorie des equations elliptiques, (French) Masson et Cie (Eds.), Paris, Academia, Editeurs, Prague, 1967
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Jiang, M.R., Jiang, R.J. A Note on Weighted Korn Inequality. Acta. Math. Sin.-English Ser. 34, 691–698 (2018). https://doi.org/10.1007/s10114-018-7310-8
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DOI: https://doi.org/10.1007/s10114-018-7310-8