## Arkiv för Matematik

• Ark. Mat.
• Volume 48, Number 2 (2010), 253-287.

### Solvability of elliptic systems with square integrable boundary data

#### Abstract

We consider second order elliptic divergence form systems with complex measurable coefficients A that are independent of the transversal coordinate, and prove that the set of A for which the boundary value problem with L2 Dirichlet or Neumann data is well posed, is an open set. Furthermore we prove that these boundary value problems are well posed when A is either Hermitean, block or constant. Our methods apply to more general systems of partial differential equations and as an example we prove perturbation results for boundary value problems for differential forms.

#### Article information

Source
Ark. Mat., Volume 48, Number 2 (2010), 253-287.

Dates
First available in Project Euclid: 31 January 2017

https://projecteuclid.org/euclid.afm/1485907113

Digital Object Identifier
doi:10.1007/s11512-009-0108-2

Mathematical Reviews number (MathSciNet)
MR2672609

Zentralblatt MATH identifier
1205.35082

Rights

#### Citation

Auscher, Pascal; Axelsson, Andreas; McIntosh, Alan. Solvability of elliptic systems with square integrable boundary data. Ark. Mat. 48 (2010), no. 2, 253--287. doi:10.1007/s11512-009-0108-2. https://projecteuclid.org/euclid.afm/1485907113

#### References

• Albrecht, D., Duong, X. and McIntosh, A., Operator theory and harmonic analysis, in Instructional Workshop on Analysis and Geometry, Part III (Canberra, 1995), Proc. Centre Math. Appl. Austral. Nat. Univ. 34, pp. 77–136, Austral. Nat. Univ., Canberra, 1996.
• Alfonseca, M., Auscher, P., Axelsson, A., Hofmann, S. and Kim, S., Analyticity of layer potentials and L2 solvability of boundary value problems for divergence form elliptic equations with complex L coefficients. Preprint, 2007.
• Auscher, P., Axelsson, A. and Hofmann, S., Functional calculus of Dirac operators and complex perturbations of Neumann and Dirichlet problems, J. Funct. Anal. 255 (2008), 374–448.
• Auscher, P., Axelsson, A. and McIntosh, A., On a quadratic estimate related to the Kato conjecture and boundary value problems, to appear in Proceedings of the 8th International Conference on Harmonic Analysis and PDEs (El Escorial).
• Auscher, P., Hofmann, S., Lacey, M., McIntosh, A. and Tchamitchian, P., The solution of the Kato square root problem for second order elliptic operators on Rn, Ann. of Math. 156 (2002), 633–654.
• Auscher, P., Hofmann, S., McIntosh, A. and Tchamitchian, P., The Kato square root problem for higher order elliptic operators and systems on $\Bbb{R}\sp n$, J. Evol. Equ. 1 (2001), 361–385.
• Axelsson, A., Transmission problems for Dirac’s and Maxwell’s equations with Lipschitz interfaces. Ph.D. thesis, The Australian National University, Canberra, 2003.
• Axelsson, A., Non-unique solutions to boundary value problems for non-symmetric divergence form equations, to appear in Trans. Amer. Math. Soc.
• Axelsson, A., Keith, S. and McIntosh, A., The Kato square root problem for mixed boundary value problems, J. London Math. Soc. 74 (2006), 113–130.
• Axelsson, A., Keith, S. and McIntosh, A., Quadratic estimates and functional calculi of perturbed Dirac operators, Invent. Math. 163 (2006), 455–497.
• Caffarelli, L., Fabes, E. and Kenig, C., Completely singular elliptic-harmonic measures, Indiana Univ. Math. J. 30 (1981), 917–924.
• Calderón, A. P., Cauchy integrals on Lipschitz curves and related operators, Proc. Natl. Acad. Sci. USA 74 (1977), 1324–1327.
• Coifman, R. R., McIntosh, A. and Meyer, Y., L’intégrale de Cauchy définit un opérateur borné sur L2 pour les courbes lipschitziennes, Ann. of Math. 116 (1982), 361–387.
• Cowling, M., Doust, I., McIntosh, A. and Yagi, A., Banach space operators with a bounded H functional calculus, J. Austral. Math. Soc. Ser. A 60 (1996), 51–89.
• Dahlberg, B., Estimates of harmonic measure, Arch. Ration. Mech. Anal. 65 (1977), 275–288.
• Dahlberg, B., Jerison, D. and Kenig, C., Area integral estimates for elliptic differential operators with nonsmooth coefficients, Ark. Mat. 22 (1984), 97–108.
• Dahlberg, B., Kenig, C., Pipher, J. and Verchota, G., Area integral estimates for higher order elliptic equations and systems, Ann. Inst. Fourier (Grenoble) 47 (1997), 1425–1461.
• Dahlberg, B., Kenig, C. and Verchota, G., Boundary value problems for the systems of elastostatics in Lipschitz domains, Duke Math. J. 57 (1988), 795–818.
• Fabes, E., Layer potential methods for boundary value problems on Lipschitz domains, in Potential Theory—Surveys and Problems (Prague, 1987 ), Lectures Notes in Math. 1344, pp. 55–80, Springer, Berlin–Heidelberg, 1988.
• Fabes, E., Jerison, D. and Kenig, C., Necessary and sufficient conditions for absolute continuity of elliptic-harmonic measure, Ann. of Math. 119 (1984), 121–141.
• Fabes, E., Kenig, C. and Verchota, G., The Dirichlet problem for the Stokes system on Lipschitz domains, Duke Math. J. 57 (1988), 769–793.
• Gao, W., Layer potentials and boundary value problems for elliptic systems in Lipschitz domains, J. Funct. Anal. 95 (1991), 377–399.
• Jerison, D. and Kenig, C., The Neumann problem on Lipschitz domains, Bull. Amer. Math. Soc. 4 (1981), 203–207.
• Jerison, D. and Kenig, C., The Dirichlet problem in nonsmooth domains, Ann. of Math. 113 (1981), 367–382.
• Kenig, C., Harmonic Analysis Techniques for Second Order Elliptic Boundary Value Problems, Amer. Math. Soc., Providence, RI, 1994.
• Kenig, C., Koch, H., Pipher, J. and Toro, T., A new approach to absolute continuity of elliptic measure, with applications to non-symmetric equations, Adv. Math. 153 (2000), 231–298.
• Kenig, C. and Pipher, J., The Neumann problem for elliptic equations with nonsmooth coefficients, Invent. Math. 113 (1993), 447–509.
• Kenig, C. and Rule, D., The regularity and Neumann problem for non-symmetric elliptic operators. Preprint.
• McIntosh, A., Second-order properly elliptic boundary value problems on irregular plane domains, J. Differential Equations 34 (1979), 361–392.
• McIntosh, A. and Mitrea, M., Clifford algebras and Maxwell’s equations in Lipschitz domains, Math. Methods Appl. Sci. 2 (1999), 1599–1620.
• McIntosh, A. and Qian, T., Convolution singular integral operators on Lipschitz curves, in Harmonic Analysis (Tianjin, 1988 ), Lecture Notes in Math. 1494, pp. 142–162, Springer, Berlin–Heidelberg, 1991.
• Mitrea, M., The method of layer potentials in electromagnetic scattering theory on nonsmooth domains, Duke Math. J. 77 (1995), 111–133.
• Moser, J., On Harnack’s theorem for elliptic differential equations, Comm. Pure Appl. Math. 14 (1961), 577–591.
• Verchota, G., Layer potentials and regularity for the Dirichlet problem for Laplace’s equation in Lipschitz domains, J. Funct. Anal. 59 (1984), 572–611.
• Verchota, G. and Vogel, A., Nonsymmetric systems on nonsmooth planar domains, Trans. Amer. Math. Soc. 349 (1997), 4501–4535.
• Verchota, G. and Vogel, A., Nonsymmetric systems and area integral estimates, Proc. Amer. Math. Soc. 128 (2000), 453–462.