Advances in Differential Equations

Mixed boundary value problems on cylindrical domains

Pascal Auscher and Moritz Egert

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We study second-order divergence-form systems on half-infinite cylindrical domains with a bounded and possibly rough base, subject to homogeneous mixed boundary conditions on the lateral boundary and square integrable Dirichlet, Neumann, or regularity data on the cylinder base. Assuming that the coefficients $A$ are close to coefficients $A_0$ that are independent of the unbounded direction with respect to the modified Carleson norm of Dahlberg, we prove a priori estimates and establish well-posedness if $A_0$ has a special structure. We obtain a complete characterization of weak solutions whose gradient either has an $L^2$-bounded non-tangential maximal function or satisfies a Lusin area bound. To this end, we combine the first-order approach to elliptic systems with the Kato square root estimate for operators with mixed boundary conditions.

Article information

Adv. Differential Equations, Volume 22, Number 1/2 (2017), 101-168.

First available in Project Euclid: 20 January 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35J55 42B25: Maximal functions, Littlewood-Paley theory 47A60: Functional calculus


Auscher, Pascal; Egert, Moritz. Mixed boundary value problems on cylindrical domains. Adv. Differential Equations 22 (2017), no. 1/2, 101--168.

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