Weighted maximal regularity estimates and solvability of non-smooth elliptic systems I
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We develop new solvability methods for divergence form second order, real and complex, elliptic systems above Lipschitz graphs, with L 2 boundary data. The coefficients A may depend on all variables, but are assumed to be close to coefficients A 0 that are independent of the coordinate transversal to the boundary, in the Carleson sense ‖A−A 0‖ C defined by Dahlberg. We obtain a number of a priori estimates and boundary behaviour results under finiteness of ‖A−A 0‖ C . Our methods yield full characterization of weak solutions, whose gradients have L 2 estimates of a non-tangential maximal function or of the square function, via an integral representation acting on the conormal gradient, with a singular operator-valued kernel. Also, the non-tangential maximal function of a weak solution is controlled in L 2 by the square function of its gradient. This estimate is new for systems in such generality, and even for real non-symmetric equations in dimension 3 or higher. The existence of a proof a priori to well-posedness, is also a new fact.
As corollaries, we obtain well-posedness of the Dirichlet, Neumann and Dirichlet regularity problems under smallness of ‖A−A 0‖ C and well-posedness for A 0, improving earlier results for real symmetric equations. Our methods build on an algebraic reduction to a first order system first made for coefficients A 0 by the two authors and A. McIntosh in order to use functional calculus related to the Kato conjecture solution, and the main analytic tool for coefficients A is an operational calculus to prove weighted maximal regularity estimates.
KeywordsDirichlet Problem Elliptic System Neumann Problem Functional Calculus Operational Calculus
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- 1.Albrecht, D.: Functional calculi of commuting unbounded operators. Ph.D. Thesis, Monash University (1994) Google Scholar
- 2.Albrecht, D., Duong, X., 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., vol. 34, pp. 77–136. Austral. Nat. Univ., Canberra (1996) Google Scholar
- 4.Alfonseca, M., Auscher, P., Axelsson, A., Hofmann, S., Kim, S.: Analyticity of layer potentials and L 2 solvability of boundary value problems for divergence form elliptic equations with complex L ∞ coefficients. Preprint at arXiv:0705.0836v1 [math.AP]
- 6.Auscher, P., Axelsson, A.: Weighted maximal regularity estimates and solvability of non-smooth elliptic systems II. Preprint Google Scholar
- 14.Barton, A.: Elliptic partial differential equations with complex coefficients. Ph.D. Thesis, University of Chicago (2010). Preprint at arXiv:0911.2513v1 [math.AP]
- 23.Dindos, M., Kenig, C., Pipher, J.: BMO solvability and the A ∞ condition for elliptic operators. Preprint at arXiv:1007.5496v1 [math.AP]
- 25.Dindos, M., Rule, D.: Elliptic equations in the plane satisfying a Carleson condition. Preprint Google Scholar
- 37.Kilty, J., Shen, Z.: The L p regularity problem on Lipschitz domains. Trans. Am. Math. Soc. (to appear) Google Scholar
- 38.Kunstmann, P., Weis, L.: Maximal L p-regularity for parabolic equations, Fourier multiplier theorems and H ∞-functional calculus. In: Functional Analytic Methods for Evolution Equations. Lecture Notes in Math., vol. 1855, pp. 65–311. Springer, Berlin (2004) Google Scholar
- 44.Šneĭberg, I.J.: Spectral properties of linear operators in interpolation families of Banach spaces. Mat. Issled. 2(32), 214–229, 254–255 (1974) Google Scholar