Dirichlet and Neumann boundary values of solutions to higher order elliptic equations

Barton, Ariel; Hofmann, Steve; Mayboroda, Svitlana

doi:10.5802/aif.3278

Cited by 7 publications

(20 citation statements)

References 18 publications

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“…We conclude this section by establishing the bound ; this estimate will be of use in . Recall that we have square‐function estimates on

R_{t}^{2}

, where

\begin{matrix} true \\ right R_{t}^{2} (bold-italic \overset{b}{true}, f) (x) & left = t^{k + 1} \partial_{⊥}^{m + k} S_{\nabla}^{L} (bold-italic \overset{b}{true} {scriptQ}_{t} f) (x, t) - t^{k + 1} \partial_{t}^{m + k} S_{\nabla}^{L} \dot{boldb} (x, t) Q_{t} f (x), \end{matrix}

and where

\partial_{⊥}^{m + k} {scriptS}_{\nabla}^{L} false(\dot{boldb} Q_{t} f false) false(x, t false) = \partial_{t}^{m + k} {scriptS}_{\nabla}^{L} false(\dot{boldb} Q_{s} f false) false(x, t false) {false|}_{s = t}

.…”

Section: Vertical Derivatives Of S∇l: Estimates For P>2 and Other Boundsmentioning

confidence: 92%

“…In this section, we prove the bounds , and on

S_{\nabla}^{L}

of Theorem . These estimates will be used in our paper to establish certain Fatou‐type theorems. Lemma Suppose that

L

is an operator of the form of order

2 m

m ⩾ 1

, acting on functions defined in

R^{n + 1}

n ⩾ 1

, associated to coefficients

bold-italicA

that are

t

‐independent in the sense of formula and satisfy the bounds and .…”

Section: Vertical Derivatives Of S∇l: Estimates For P>2 and Other Boundsmentioning

confidence: 99%

“…These two facts shall be established in . We remark that we will use the estimates and in ; it is this intended use that makes these estimates of immediate interest to us.…”

Section: Introductionmentioning

confidence: 96%

“…We will also establish some bounds on the modified single layer potential

S_{\nabla}^{L}

. We will apply these bounds in the paper .…”

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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“…We conclude this section by establishing the bound ; this estimate will be of use in . Recall that we have square‐function estimates on

R_{t}^{2}

, where

\begin{matrix} true \\ right R_{t}^{2} (bold-italic \overset{b}{true}, f) (x) & left = t^{k + 1} \partial_{⊥}^{m + k} S_{\nabla}^{L} (bold-italic \overset{b}{true} {scriptQ}_{t} f) (x, t) - t^{k + 1} \partial_{t}^{m + k} S_{\nabla}^{L} \dot{boldb} (x, t) Q_{t} f (x), \end{matrix}

and where

\partial_{⊥}^{m + k} {scriptS}_{\nabla}^{L} false(\dot{boldb} Q_{t} f false) false(x, t false) = \partial_{t}^{m + k} {scriptS}_{\nabla}^{L} false(\dot{boldb} Q_{s} f false) false(x, t false) {false|}_{s = t}

.…”

Section: Vertical Derivatives Of S∇l: Estimates For P>2 and Other Boundsmentioning

confidence: 92%

“…In this section, we prove the bounds , and on

S_{\nabla}^{L}

of Theorem . These estimates will be used in our paper to establish certain Fatou‐type theorems. Lemma Suppose that

L

is an operator of the form of order

2 m

m ⩾ 1

, acting on functions defined in

R^{n + 1}

n ⩾ 1

, associated to coefficients

bold-italicA

that are

t

‐independent in the sense of formula and satisfy the bounds and .…”

Section: Vertical Derivatives Of S∇l: Estimates For P>2 and Other Boundsmentioning

confidence: 99%

“…These two facts shall be established in . We remark that we will use the estimates and in ; it is this intended use that makes these estimates of immediate interest to us.…”

Section: Introductionmentioning

confidence: 96%

“…We will also establish some bounds on the modified single layer potential

S_{\nabla}^{L}

. We will apply these bounds in the paper .…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

Barton

Hofmann

Mayboroda

2019

Proc. London Math. Soc.

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“…A v formulated in [21] rather than formula (1.6). One of the main results of [21] is that if Lu = 0 in R n+1 + and A + 2 (t∇ m v) ∈ L p (R n ) for some p with 1 < p ≤ 2, then this formulation ofṀ +…”

mentioning

confidence: 99%

Nontangential Estimates on Layer Potentials and the Neumann Problem for Higher-Order Elliptic Equations

Barton

Hofmann

Mayboroda

2020

International Mathematics Research Notices

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We solve the Neumann problem in the half space R n+1 + , for higher order elliptic differential equations with variable self-adjoint t-independent coefficients, and with boundary data in the negative smoothness spaceẆ −1,p , where max(0, 1 2 − 1 n −ε) < 1 p < 1 2 . Our arguments are inspired by an argument of Shen and build on known well posedness results in the case p = 2.We use the same techniques to establish nontangential and square function estimates on layer potentials with inputs in L p orẆ ±1,p for a similar range of p, based on known bounds for p near 2; in this case we may relax the requirement of self-adjointess.

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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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Dirichlet and Neumann boundary values of solutions to higher order elliptic equations

Cited by 7 publications

References 18 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

Nontangential Estimates on Layer Potentials and the Neumann Problem for Higher-Order Elliptic Equations

The Neumann problem for higher order elliptic equations with symmetric coefficients

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