The biharmonic Neumann problem in Lipschitz domains

Acta Mathematica - Tập 194 - Trang 217-279 - 2005
Gregory C. Verchota1
1Department of Mathematics, Syracuse University, Syracuse, USA

Tài liệu tham khảo

Agmon, S., Remarks on self-adjoint and semi-bounded elliptic boundary value problems, inProc. Internat. Sympos. Linear Spaces (Jerusalem, 1960), pp. 1–13. Jerusalem Academic Press, Jerusalem, 1960. Bourlard, M. &Nicaise, S., Abstract Green formula and applications to boundary integral equations.Numer. Funct. Anal. Optim., 18 (1997), 667–689. Calderón, A. P., Boundary value problems for the Laplace equation in Lipschitzian domains, inRecent Progress in Fourier Analysis (El Escorial, 1983), pp. 33–48. North-Holland Math. Stud., 111. North-Holland, Amsterdam, 1985. Ciarlet, P. G.,Mathematical Elasticity, Vol. I. Studies Math. Appl., 20. North-Holland, Amsterdam, 1988. Coifman, R. R., McIntosh, A. &Meyer, Y., L'intégrale de Cauchy définit un opérateur borné surL 2 pour les courbes lipschitziennes.Ann. of Math., 116 (1982), 361–387. Dahlberg, B. E. J., On the Poisson integral for Lipschitz andC 1-domains.Studia Math., 66 (1979), 13–24. Dahlberg, B. E. J. &Kenig, C. E., Hardy spaces and the Neumann problem inL p for Laplace's equation in Lipschitz domains.Ann. of Math., 125 (1987), 437–465. Dahlberg, B. E. J., Kenig, C. E. &Pipher, J. &Verchota, G. C., Area integral estimates for higher order elliptic equations and systems.Ann. Inst. Fourier (Grenoble), 47 (1997), 1425–1461. Dahlberg, B. E. J., Kenig, C. E. &Verchota, G. C., The Dirichlet problem for the biharmonic equation in a Lipschitz domain.Ann. Inst. Fourier (Grenoble), 363 (1986), 109–135. —, Boundary value problems for the systems of elastostatics in Lipschitz domains.Duke Math. J., 57 (1988), 795–818. Dahlberg, B. E. J. &Verchota, G., Galerkin methods for the boundary integral equations of elliptic equations in nonsmooth domains, inHarmonic Analysis and Partial Differential Equations (Boca Raton, FL, 1988), pp. 39–60. Contemp. Math., 107. Amer. Math. Soc., Providence, RI, 1990. Fabes, E. B., Jodeit, M., Jr &Rivière, N. M., Potential techniques for boundary value problems onC 1-domains.Acta Math., 141 (1978), 165–186. Fabes, E. B. &Kenig, C. E. &Verchota, G. C., The Dirichlet problem for the Stokes system on Lipschitz domains.Duke Math. J., 57 (1988), 769–793. Fraeijs de Veubeke, B. M.,A Course in Elasticity. Appl. Math. Sci., 29. Springer, New York-Berlin, 1979. Friedrichs, K., Die Randwert- und Eigenwertprobleme aus der Theorie der elastischen Platten.Math. Ann., 98 (1928), 206–247. Gao, W. J., Layer potentials and boundary value problems for elliptic systems in Lipschitz domains.J. Funct. Anal., 95 (1991), 377–399. Giroire, J. &Nédélec, J.-C., A new system of boundary integral equations for plates with free edges.Math. Methods Appl. Sci., 18 (1995), 755–772. Gould, P. L.,Introduction to Linear Elasticity, 2nd edition. Springer, Berlin, 1994. Hille, E.,Analytic Function Theory, Vol. II. Ginn, Boston-New York-Toronto, 1962. Jaswon, M. A. &Symm, G. T.,Integral Equation Methods in Potential Theory and Elastostatics. Academic Press, London-New York, 1977. Jerison, D. S. &Kenig, C. E., The Neumann problem on Lipschitz domains.Bull. Amer. Math. Soc., 4 (1981), 203–207. — Boundary value problems on Lipschitz domains, inStudies in Partial Differential Equations pp. 1–68. MAA Stud. Math., 23. Math. Assoc. America, Washington, DC, 1982. John, F.,Plane Waves and Spherical Means Applied to Partial Differential Equations. Interscience, New York-London, 1955. Kenig, C. E.,Harmonic Analysis Techniques for Second Order Elliptic Boundary Value Problems. CBMS Regional Conf. Ser. in Math., 83, Amer. Math. Soc., Providence, RI, 1994. Meyer, Y. &Coifman, R.,Wavelets. Cambridge Stud. Adv. Math., 48. Cambridge Univ. Press, Cambridge, 1997. Mitrea, M., The method of layer potentials in electromagnetic scattering theory on nonsmooth domains.Duke Math. J., 77 (1995), 111–133. Nadai, A.,Theory of Flow and Fracture, of Solids Vol. II. McGraw-Hill, New York, 1963. Nazaret, C., A system of boundary integral equations for polygonal plates with free edges.Math. Methods Appl. Sci., 21 (1998), 165–185. Neĉas, J., On domains of typeN.Czechoslovak Math. J., 12 (87), (1962), 274–287 (Russian). Pipher J. &Verchota, G., Area integral estimates for the biharmonic operator in Lipschitz domains.Trans. Amer. Math. Soc., 327 (1991), 903–917. —, The Dirichlet problem inL p for the biharmonic equation on Lipschitz domains.Amer. J. Math., 114 (1992), 923–972. —, A maximum principle for biharmonic functions in Lipschitz andC 1 domains.Comment. Math. Helv., 68 (1993), 385–414. —, Diation invariant estimates and the boundary Gårding inequality for higher order elliptic operators.Ann. of Math., 142 (1995), 1–38. Pleijel, Å., On the eigenvalues and eigenfunctions of elastic platesComm. Pure Appl. Math., 3 (1950), 1–10. —, On Green's functions for elastic plates with clamped, supported and free edges, inSpectral Theory and Differential Problems, pp. 413–437, Oklahoma Agricultural and Mechanical College. Stillwater, OK, 1951. Pólya, G. &Szegő, G.,Isoperimetric Inequalities in Mathematical Physics. Ann. of Math. Stud.,27. Princeton Univ. Press, Princeton, NJ, 1951. Shen, Z., TheL p Dirichlet problem for elliptic systems on Lipschitz domains. Preprint, 2004. Stein, E. M.,Singular Integrals and Differentiability Properties of Functions. Princeton Math. Ser., 30, Princeton Univ. Press, Princeton, NJ, 1970. Stein, E. M. &Weiss, G., On the theory of harmonic functions of several variables, I: The theory ofH p-spaces.Acta Math., 103 (1960), 25–62. Taylor, A. E. &Lay, D. C.,Introduction to Functional Analysis, 2nd edition, Wiley, New York-Chichester-Brisbane, 1980. Verchota, G., Layer potentials and regularity for the Dirichlet problem for Laplace's equation in Lipschitz domains.J. Funct. Anal., 59 (1984), 572–611. —, The Dirichlet problem for the biharmonic equation inC 1 domains.Indiana Univ. Math. J. 36 (1987), 867–895. —, The Dirichlet problem for the polyharmonic equation in Lipschitz domains.Indiana Univ. Math. J., 39 (1990), 671–702. —, The use of Rellich identities on certain nongraph boundaries, inHarmonic Analysis and Boundary Value Problems (Fayetteville, AR, 2000) pp. 127–138. Contemp. Math., 277. Amer. Math. Soc., Providence, RI, 2001. Verchota, G. C. & Vogel, A. L., The multidirectional Neumann problem inR 4. Preprint, 2004. Wermer, J.,Potential Theory. Lecture Notes in Math., 408. Springer, Berlin-New York, 1974. Yosida, K.,Functional Analysis, 4th edition. Grundlehren Math. Wiss., 123, Springer, New York-Heidelberg, 1974.