Gradient estimates and the fundamental solution for higher-order elliptic systems with rough coefficients

Barton, Ariel

doi:10.1007/s00229-016-0839-x

Cited by 32 publications

(58 citation statements)

References 37 publications

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“…The main result of the paper was a construction of the fundamental solution

E^{L}

in the case of higher‐order operators.

E^{L}

was constructed as an order‐ m antiderivative of the kernel to the operator

Π^{L}

, the Newton potential for L , defined as follows.…”

Section: Definitionsmentioning

confidence: 99%

“…We will need two additional properties of the Newton potential from . First, we will need the symmetry relation

{(〈〉 bold-italic \overset{G}{true}, \nabla^{m} {normalΠ}^{L} bold-italic \overset{H}{true})}_{R^{n + 1}} = {(〈〉 \nabla^{m} {normalΠ}^{L^{*}} bold-italic \overset{G}{true}, bold-italic \overset{H}{true})}_{R^{n + 1}}

for all

bold-italic \overset{H}{true} \in L^{2} false(R^{n} false)

and all

bold-italic \overset{G}{true} \in L^{2} false(R^{n} false)

.…”

Section: Definitionsmentioning

confidence: 99%

“…The main result of may be stated as follows. Theorem Let L be an operator of order 2 m that satisfies the bounds and .…”

Section: Definitionsmentioning

confidence: 99%

“…(The

\partial_{X}^{α}

derivative is included in formula because

{normalΠ}^{L} bold-italic \overset{H}{true} \in {\overset{W}{true}}_{m}^{2} (R^{n + 1})

, and so

{normalΠ}^{L} bold-italic \overset{H}{true}

is also defined only up to adding polynomials.) If q and s are small enough, then there is a unique normalization of the derivatives

\nabla_{X}^{m - q} \nabla_{Y}^{m - s} E^{L} false(X, Y false)

that satisfies the bound or ; in this normalization was found using the Gagliardo–Nirenberg–Sobolev inequality. However, if

2 m \geq n + 1

then

E^{L}

itself (and possibly some of its derivatives) are still not well‐defined.…”

Section: Definitionsmentioning

confidence: 99%

“…We begin with some bounds on solutions to elliptic equations. Specifically, we begin with the following higher‐order generalization of the Caccioppoli inequality; in its full generality it was proven in , but the

j = m

case was proven in and an intriguing version appears in . Lemma Suppose that L is a divergence form elliptic operator associated to coefficients

bold-italicA

satisfying the ellipticity conditions and .…”

Section: Preliminary Argumentsmentioning

confidence: 99%

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

Barton

Hofmann

Mayboroda

2017

Mathematische Nachrichten

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“…The main result of the paper was a construction of the fundamental solution

E^{L}

in the case of higher‐order operators.

E^{L}

was constructed as an order‐ m antiderivative of the kernel to the operator

Π^{L}

, the Newton potential for L , defined as follows.…”

Section: Definitionsmentioning

confidence: 99%

“…We will need two additional properties of the Newton potential from . First, we will need the symmetry relation

{(〈〉 bold-italic \overset{G}{true}, \nabla^{m} {normalΠ}^{L} bold-italic \overset{H}{true})}_{R^{n + 1}} = {(〈〉 \nabla^{m} {normalΠ}^{L^{*}} bold-italic \overset{G}{true}, bold-italic \overset{H}{true})}_{R^{n + 1}}

for all

bold-italic \overset{H}{true} \in L^{2} false(R^{n} false)

and all

bold-italic \overset{G}{true} \in L^{2} false(R^{n} false)

.…”

Section: Definitionsmentioning

confidence: 99%

“…The main result of may be stated as follows. Theorem Let L be an operator of order 2 m that satisfies the bounds and .…”

Section: Definitionsmentioning

confidence: 99%

“…(The

\partial_{X}^{α}

derivative is included in formula because

{normalΠ}^{L} bold-italic \overset{H}{true} \in {\overset{W}{true}}_{m}^{2} (R^{n + 1})

, and so

{normalΠ}^{L} bold-italic \overset{H}{true}

is also defined only up to adding polynomials.) If q and s are small enough, then there is a unique normalization of the derivatives

\nabla_{X}^{m - q} \nabla_{Y}^{m - s} E^{L} false(X, Y false)

that satisfies the bound or ; in this normalization was found using the Gagliardo–Nirenberg–Sobolev inequality. However, if

2 m \geq n + 1

then

E^{L}

itself (and possibly some of its derivatives) are still not well‐defined.…”

Section: Definitionsmentioning

confidence: 99%

j = m

case was proven in and an intriguing version appears in . Lemma Suppose that L is a divergence form elliptic operator associated to coefficients

bold-italicA

satisfying the ellipticity conditions and .…”

Section: Preliminary Argumentsmentioning

confidence: 99%

See 3 more Smart Citations

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

Barton

Hofmann

Mayboroda

2017

Mathematische Nachrichten

Self Cite

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show abstract

Basic Properties of Weak Solutions

Auscher¹,

Egert²

2023

Progress in Mathematics

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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.

show abstract

Gradient estimates and the fundamental solution for higher-order elliptic systems with rough coefficients

Cited by 32 publications

References 37 publications

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

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

Basic Properties of Weak Solutions

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

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