Differential and Integral Equations

Square-function estimates for singular integrals and applications to partial differential equations

Svitlana Mayboroda and Marius Mitrea

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The purpose of the present paper is to continue the program of study of elliptic boundary-value problems on Lipschitz domains with boundary data in quasi-Banach Besov spaces $B_s^{p,p}(\partial \Omega)$, initiated in [13]. Introducing a modified square function which is well-adapted for handling data with a fractional amount of smoothness, we establish the well-posedness of the Dirichlet and Neumann boundary problems for the Laplacian in Lipschitz domains, for a range of indices which includes values of $p$ less than $1$. An important ingredient in this regard is establishing suitable square-function estimates for a singular integral of potential type.

Article information

Differential Integral Equations, Volume 17, Number 7-8 (2004), 873-892.

First available in Project Euclid: 21 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35J25: Boundary value problems for second-order elliptic equations
Secondary: 31B10: Integral representations, integral operators, integral equations methods 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.) 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems


Mayboroda, Svitlana; Mitrea, Marius. Square-function estimates for singular integrals and applications to partial differential equations. Differential Integral Equations 17 (2004), no. 7-8, 873--892. https://projecteuclid.org/euclid.die/1356060334

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