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

Barton, Ariel; Hofmann, Steve; Mayboroda, Svitlana

doi:10.1002/mana.201600116

Cited by 8 publications

(31 citation statements)

References 78 publications

(193 reference statements)

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“…The only result known to be valid for operators with variable

t

‐independent coefficients of arbitrary order is layer potential estimates for a higher order of regularity, that is, when the data

\dot{boldf}

lie in

{\overset{W}{true}}_{1}^{2} false(R^{n} false)

and

\dot{boldg}

lie in

L^{2} false(R^{n} false)

. These results were proven by the authors of the present paper in . Specifically, under the same conditions as in Theorem , we have the estimates

\begin{matrix} true \\ right \int_{R^{n}} \int_{- \infty}^{\infty} false| \nabla^{m} \partial_{t} D^{A} \dot{boldf} (x, t) |^{2} false| t false| d t d x & left ⩽ C ∥ \dot{boldf} {false∥}_{{\dot{W}}_{1}^{2} ({double-struckR}^{n})}^{2} = C {false∥ \nabla_{false∥} bold-italic \overset{f}{true} false∥}_{L^{2} ({double-struckR}^{n})}^{2}, \end{matrix}

\begin{matrix} true \\ right \int_{R^{n}} \int_{- \infty}^{\infty} false| \nabla^{m} \partial_{t} S^{L} \dot{boldg} (x, t) | \end{matrix}

…”

Section: Introductionsupporting

confidence: 67%

“…It is possible, though somewhat involved, to generalize formulas and to the higher order case, and multiple subtly different generalizations exist. We will use the potentials introduced in ; these potentials are similar to but subtly different from those used in to study the biharmonic operator

Δ^{2}

and in to study more general constant coefficient operators.…”

Section: Introductionmentioning

confidence: 99%

“…We conjecture that this theorem may be generalized from the half‐space to Lipschitz graph domains, but in contrast to the case of second‐order operators, the method of proof of requires the extra structure of

R_{+}^{n + 1}

.…”

Section: Introductionmentioning

confidence: 99%

“…The approach of the present paper is somewhat different from that of , as we may exploit the bounds and established therein. The arguments of both papers, however, rely on

T 1

‐ or

T b

‐type theorems.…”

Section: Introductionmentioning

confidence: 99%

“…(The same is true of the higher order operators

S^{L}

S_{\nabla}^{L}

given in Section 2.5.) This additional degree of smoothness is exploited in to establish the quasi‐orthogonality estimate required by the

T b

theorem of , and so the arguments of cannot be applied to bound

S_{\nabla}^{L}

; we must exploit other

T 1

‐type theorems.…”

Section: Introductionmentioning

confidence: 99%

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Bounds on layer potentials with rough inputs for higher order elliptic equations

Barton

Hofmann

Mayboroda

2019

Proc. London Math. Soc.

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“…The only result known to be valid for operators with variable

t

‐independent coefficients of arbitrary order is layer potential estimates for a higher order of regularity, that is, when the data

\dot{boldf}

lie in

{\overset{W}{true}}_{1}^{2} false(R^{n} false)

and

\dot{boldg}

lie in

L^{2} false(R^{n} false)

. These results were proven by the authors of the present paper in . Specifically, under the same conditions as in Theorem , we have the estimates

\begin{matrix} true \\ right \int_{R^{n}} \int_{- \infty}^{\infty} false| \nabla^{m} \partial_{t} D^{A} \dot{boldf} (x, t) |^{2} false| t false| d t d x & left ⩽ C ∥ \dot{boldf} {false∥}_{{\dot{W}}_{1}^{2} ({double-struckR}^{n})}^{2} = C {false∥ \nabla_{false∥} bold-italic \overset{f}{true} false∥}_{L^{2} ({double-struckR}^{n})}^{2}, \end{matrix}

\begin{matrix} true \\ right \int_{R^{n}} \int_{- \infty}^{\infty} false| \nabla^{m} \partial_{t} S^{L} \dot{boldg} (x, t) | \end{matrix}

…”

Section: Introductionsupporting

confidence: 67%

Δ^{2}

and in to study more general constant coefficient operators.…”

Section: Introductionmentioning

confidence: 99%

R_{+}^{n + 1}

.…”

Section: Introductionmentioning

confidence: 99%

“…The approach of the present paper is somewhat different from that of , as we may exploit the bounds and established therein. The arguments of both papers, however, rely on

T 1

‐ or

T b

‐type theorems.…”

Section: Introductionmentioning

confidence: 99%

“…(The same is true of the higher order operators

S^{L}

S_{\nabla}^{L}

given in Section 2.5.) This additional degree of smoothness is exploited in to establish the quasi‐orthogonality estimate required by the

T b

theorem of , and so the arguments of cannot be applied to bound

S_{\nabla}^{L}

; we must exploit other

T 1

‐type theorems.…”

Section: Introductionmentioning

confidence: 99%

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Bounds on layer potentials with rough inputs for higher order elliptic equations

Barton

Hofmann

Mayboroda

2019

Proc. London Math. Soc.

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Higher-Order Elliptic Equations in Non-Smooth Domains: a Partial Survey

Barton

Mayboroda

2016

Association for Women in Mathematics Series

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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 domains were established, followed by the higher order Wiener test. Certain boundary value problems for higher order operators with variable non-smooth coefficients were addressed, both in divergence form and in composition form, the latter being adapted to the context of Lipschitz domains. These developments brought new estimates on the fundamental solutions and the Green function, allowing for the lack of smoothness of the boundary or of the coefficients of the equation. Building on our earlier account of history of the subject in [25], this survey presents the current state of the art, emphasizing the most recent results and emerging open problems. Contents 1991 Mathematics Subject Classification. Primary 35-02; Secondary 35B60, 35B65, 35J40, 35J55.

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The Neumann problem for higher order elliptic equations with symmetric coefficients

Barton¹,

Hofmann²,

Mayboroda³

2017

Math. Ann.

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Abstract. In this paper we establish well posedness of the Neumann problem with boundary data in L 2 or the Sobolev spaceẆ 2 −1 , in the half space, for linear elliptic differential operators with coefficients that are constant in the vertical direction and in addition are self adjoint. This generalizes the well known well-posedness result of the second order case and is based on a higher order and one sided version of the classic Rellich identity, and is the first known well posedness result for a higher order operator with rough variable coefficients and boundary data in a Lebesgue or Sobolev space.

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

Cited by 8 publications

References 78 publications

Bounds on layer potentials with rough inputs for higher order elliptic equations

Bounds on layer potentials with rough inputs for higher order elliptic equations

Higher-Order Elliptic Equations in Non-Smooth Domains: a Partial Survey

The Neumann problem for higher order elliptic equations with symmetric coefficients

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