Square function estimates on layer potentials for higher‐order elliptic equations

@article{Barton2015SquareFE,
  title={Square function estimates on layer potentials for higher‐order elliptic equations},
  author={Ariel Barton and Steve Hofmann and Svitlana Mayboroda},
  journal={Mathematische Nachrichten},
  year={2015},
  volume={290}
}
In this paper we establish square‐function estimates on the double and single layer potentials for divergence form elliptic operators, of arbitrary even order 2m, with variable t‐independent coefficients in the upper half‐space. This generalizes known results for variable‐coefficient second‐order operators, and also for constant‐coefficient higher‐order operators. 
Bounds on layer potentials with rough inputs for higher order elliptic equations
In this paper, we establish square‐function estimates on the double and single layer potentials with rough inputs for divergence form elliptic operators, of arbitrary even order 2m , with variable t
Nontangential Estimates on Layer Potentials and the Neumann Problem for Higher-Order Elliptic Equations
We solve the Neumann problem, with nontangential estimates, for higher order divergence form elliptic operators with variable $t$-independent coefficients. Our results are accompanied by
The Neumann problem for higher order elliptic equations
Abstract We solve the Neumann problem in the half space for higher order elliptic differential equations with variable self-adjoint t-independent coefficients, and with boundary data in the negative
Layer potentials for general linear elliptic systems
In this paper we construct layer potentials for elliptic differential operators using the Lax-Milgram theorem, without recourse to the fundamental solution; this allows layer potentials to be
THE L NEUMANN PROBLEM FOR HIGHER ORDER ELLIPTIC EQUATIONS
We solve the Neumann problem in the half space R + , for higher order elliptic differential equations with variable self-adjoint t-independent coefficients, and with boundary data in Lp, where max (
Gradient estimates and the fundamental solution for higher-order elliptic systems with rough coefficients
This paper considers the theory of higher-order divergence-form elliptic differential equations. In particular, we provide new generalizations of several well-known tools from the theory of
The Neumann problem for higher order elliptic equations with symmetric coefficients
In this paper we establish well posedness of the Neumann problem with boundary data in $$L^2$$L2 or the Sobolev space $$\dot{W}^2_{-1}$$W˙-12, in the half space, for linear elliptic differential
THE Ẇ−1,p NEUMANN PROBLEM FOR HIGHER ORDER ELLIPTIC EQUATIONS
We solve the Neumann problem in the half space R + , for higher order elliptic differential equations with variable self-adjoint t-independent coefficients, and with boundary data in the negative
Higher-order elliptic equations in non-smooth domains: history and recent results
Recent years have brought significant advances in the theory of higher order elliptic equations in non-smooth domains. Sharp pointwise estimates on derivatives of polyharmonic functions in arbitrary
Higher-Order Elliptic Equations in Non-Smooth Domains: a Partial Survey
Recent years have brought significant advances in the theory of higher-order elliptic equations in non-smooth domains. Sharp pointwise estimates on derivatives of polyharmonic functions in arbitrary
...
...

References

SHOWING 1-10 OF 109 REFERENCES
Multi-Layer Potentials and Boundary Problems: for Higher-Order Elliptic Systems in Lipschitz Domains
1 Introduction.- 2 Smoothness scales and Caldeon-Zygmund theory in the scalar-valued case.- 3 Function spaces of Whitney arrays.- 4 The double multi-layer potential operator.- 5 The single
Gradient estimates and the fundamental solution for higher-order elliptic systems with rough coefficients
This paper considers the theory of higher-order divergence-form elliptic differential equations. In particular, we provide new generalizations of several well-known tools from the theory of
The regularity and Neumann problem for non-symmetric elliptic operators
We establish optimal Lp bounds for the non-tangential maximal function of the gradient of the solution to a second-order elliptic operator in divergence form, possibly non-symmetric, with bounded
The regularity and Neumann problem for non-symmetric elliptic operators
We establish optimal L^p bounds for the nontangential maximal function of the gradient of the solution to a second order elliptic operator in divergence form, possibly non-symmetric, with bounded
Layer Potentials and Boundary-Value Problems for Second Order Elliptic Operators with Data in Besov Spaces
This monograph presents a comprehensive treatment of second order divergence form elliptic operators with bounded measurable t-independent coefficients in spaces of fractional smoothness, in Besov
The dirichlet problem in lipschitz domains for higher order elliptic systems with rough coefficients
We study the Dirichlet problem, in Lipschitz domains and with boundary data in Besov spaces, for divergence form strongly elliptic systems of arbitrary order with bounded, complex-valued
The method of layer potentials in Lp and endpoint spaces for elliptic operators with L∞ coefficients
We consider layer potentials associated to elliptic operators Lu=−div(A∇u) acting in the upper half‐space R+n+1 for n⩾2 , or more generally, in a Lipschitz graph domain, where the coefficient matrix
The Dirichlet problem for higher order equations in composition form
Boundary value problems and integral operators for the bi-Laplacian in non-smooth domains
— In this paper we explore the e¤ectiveness of the classical method of layer potentials in the treatment of boundary value problems for the bi-Laplacian formulated in arbitrary Lipschitz domains,
Layer potentials beyond singular integral operators
We prove that the double layer potential operator and the gradient of the single layer potential operator are L_2 bounded for general second order divergence form systems. As compared to earlier
...
...