# zbMATH — the first resource for mathematics

Boundary value problems for the systems of elastostatics in Lipschitz domains. (English) Zbl 0699.35073
The authors give a nice extension of their previous results for the Laplace and biharmonic operators in Lipschitz domains to the study of the solvability of the Dirichlet and traction problems (with boundary data in $$L^ 2)$$ for the linear elastostatics. The boundary conditions are, as always in the works of the authors, taken in the sense of nontangential convergence.
Reviewer: G.Geymonat

##### MSC:
 35J55 Systems of elliptic equations, boundary value problems (MSC2000) 35C15 Integral representations of solutions to PDEs 31A30 Biharmonic, polyharmonic functions and equations, Poisson’s equation in two dimensions 76D07 Stokes and related (Oseen, etc.) flows 74B99 Elastic materials
Full Text:
##### References:
 [1] A. P. Calderón, Boundary value problems for the Laplace equation in Lipschitzian domains , Recent Progress in Fourier Analysis (El Escorial, 1983), North Holland Mathematical Studies, vol. 111, North-Holland, Amsterdam, 1985, pp. 33-48. · Zbl 0608.31001 [2] R. R. Coifman, A. McIntosh, and Y. Meyer, L’intégrale de Cauchy définit un opérateur borné sur $$L^2$$ pour les courbes lipschitziennes , Ann. of Math. (2) 116 (1982), no. 2, 361-387. JSTOR: · Zbl 0497.42012 [3] B. E. J. Dahlberg, On the Poisson integral for Lipschitz and $$C^1$$-domains , Studia Math. 66 (1979), no. 1, 13-24. · Zbl 0422.31008 [4] B. E. J. Dahlberg and C. E. Kenig, Hardy spaces and the Neumann problem in $$L^ p$$ for Laplace’s equation in Lipschitz domains , Ann. of Math. (2) 125 (1987), no. 3, 437-465. JSTOR: · Zbl 0658.35027 [5] B. E. J. Dahlberg, C. E. Kenig, and G. C. Verchota, The Dirichlet problem for the biharmonic equation in a Lipschitz domain , Ann. Inst. Fourier (Grenoble) 36 (1986), no. 3, 109-135. · Zbl 0589.35040 [6] G. David, Noyau de Cauchy et opérateurs de Calderón-Zygmund , Thèse, Université Paris Sud, 1986. [7] E. B. Fabes, Boundary value problems of linear elastostatics and hydrostatics on Lipschitz domains , Miniconference on linear analysis and function spaces (Canberra, 1984), Proc. Centre Math. Anal. Austral. Nat. Univ., vol. 9, Austral. Nat. Univ., Canberra, 1985, pp. 27-45. · Zbl 0608.35014 [8] E. B. Fabes, M. Jodeit, Jr., and N. M. Rivière, Potential techniques for boundary value problems on $$C^1$$-domains , Acta Math. 141 (1978), no. 3-4, 165-186. · Zbl 0402.31009 [9] E. B. Fabes, C. E. Kenig, and G. C. Verchota, The Dirichlet problem for the Stokes system on Lipschitz domains , Duke Math. J. 57 (1988), no. 3, 769-793. · Zbl 0685.35085 [10] K. O. Friedrichs, On the boundary-value problems of the theory of elasticity and Korn’s inequality , Ann. of Math. (2) 48 (1947), 441-471. JSTOR: · Zbl 0029.17002 [11] D. S. Jerison and C. E. Kenig, The Neumann problem on Lipschitz domains , Bull. Amer. Math. Soc. (N.S.) 4 (1981), no. 2, 203-207. · Zbl 0471.35026 [12] C. E. Kenig, Boundary value problems of linear elastostatics and hydrostatics on Lipschitz domains , Seminaire Goulaouic-Meyer-Schwartz, 1983-1984, École Polytechnique, Palaiseau, France, 1984, Exposé No. XXI. · Zbl 0547.73007 [13] C. E. Kenig, Recent progress on boundary value problems on Lipschitz domains , Pseudodifferential operators and applications (Notre Dame, Ind., 1984), Proc. of Symps. in Pure Math., vol. 43, Amer. Math. Soc., Providence, RI, 1985, pp. 175-205. · Zbl 0593.35038 [14] C. E. Kenig, Elliptic boundary value problems on Lipschitz domains , Beijing Lectures in Harmonic Analysis (Beijing, 1984), Annals of Math. Studies, vol. 112, Princeton Univ. Press, Princeton, NJ, 1986, pp. 131-183. · Zbl 0624.35029 [15] V. D. Kupradze, Potential Methods in the Theory of Elasticity , Translated from the Russian by H. Gutfreund. Translation edited by I. Meroz, Israel Program for Scientific Translations, Jerusalem, 1965. · Zbl 0188.56901 [16] V. D. Kupradze, T. G. Gegelia, M. O. Basheleĭ shvili, and T. V. Burchuladze, Three dimensional problems of the mathematical theory of elasticity and thermoelasticity , North-Holland Series in Applied Mathematics and Mechanics, vol. 25, North Holland, New York, 1979. · Zbl 0406.73001 [17] J. Nečas, Les méthodes directes en théorie des équations elliptiques , Academia, Prague, 1967. · Zbl 1225.35003 [18] L. E. Payne and H. F. Weinberger, New bounds for solutions of second order elliptic partial differential equations , Pacific J. Math. 8 (1958), 551-573. · Zbl 0093.10901 [19] F. Rellich, Darstellung der Eigenwerte von $$\Delta u+\lambda u=0$$ durch ein Randintegral , Math. Z. 46 (1940), 635-636. · Zbl 0023.04204 [20] G. C. Verchota, Layer potentials and regularity for the Dirichlet problem for Laplace’s equation in Lipschitz domains , J. Funct. Anal. 59 (1984), no. 3, 572-611. · Zbl 0589.31005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.