Abstract
We consider the Cauchy problem for non-autonomous forms inducing elliptic operators in divergence form with Dirichlet, Neumann, or mixed boundary conditions on an open subset \({\Omega \subseteq \mathbb{R}^n}\). We obtain maximal regularity in \({L^2(\Omega)}\) if the coefficients are bounded, uniformly elliptic, and satisfy a scale invariant bound on their fractional time-derivative of order one-half. Previous results even for such forms required control on a time-derivative of order larger than one-half.
Similar content being viewed by others
References
R. Dautray and J.L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology – Volume 5 Evolution Problems I, Springer-Verlag, Berlin, 1992.
David G., Journé J.L.: A boundedness criterion for generalized Calderón-Zygmund operators. Ann. of Math. (2) 120, 371–397 (1984)
D. Dier and R. Zacher, Non-autonomous maximal regularity in Hilbert spaces, to appear in J. Evol. Equ. Available at http://arxiv.org/abs/1601.05213.
S. Fackler, J.L. Lions’ problem concerning maximal regularity of equations governed by non-autonomous forms, to appear in Ann. Henri Poincaré. Available at http://arxiv.org/abs/1601.08012.
Haak B.H., Ouhabaz E.M.: Maximal regularity for non-autonomous evolution equations. Math. Ann. 363, 1117–1145 (2015)
J.L. Lions, Équations différentielles opérationnelles et problèmes aux limites, Die Grundlehren der mathematischen Wissenschaften, vol. 111, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1961.
Murray M.A.M.: Commutators with fractional differentiation and BMO Sobolev spaces. Indiana Univ. Math. J. 34, 205–215 (1985)
Nyström K.: Square functions estimates and the Kato problem for second order parabolic operators in \({\mathbb{R}^{n+1}}\). Adv. Math. 293, 1–36 (2016)
Ouhabaz E.M., Spina C.: Maximal regularity for non-autonomous Schrödinger type equations. J. Differential Equations 248, 1668–1683 (2010)
de Simon L.: Un’applicazione della teoria degli integrali singolari allo studio delle equazioni differenziali lineari astratte del primo ordine. Rend. Sem. Mat. Univ. Padova 34, 205–223 (1964)
Stein E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton Univ. Press, Princeton (1970)
Strichartz R.S.: Bounded mean oscillation and Sobolev spaces. Indiana Univ. Math. J. 29, 539–558 (1980)
Vogt H.: Equivalence of pointwise and global ellipticity estimates. Math. Nachr. 237, 125–128 (2002)
Author information
Authors and Affiliations
Corresponding author
Additional information
Pascal Auscher and Moritz Egert were partially supported by the ANR project “Harmonic Analysis at its Boundaries”, ANR-12-BS01-0013. Moritz Egert was supported by a public grant as part of the FMJH.
Rights and permissions
About this article
Cite this article
Auscher, P., Egert, M. On non-autonomous maximal regularity for elliptic operators in divergence form. Arch. Math. 107, 271–284 (2016). https://doi.org/10.1007/s00013-016-0934-y
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00013-016-0934-y