The regularity problem for second order elliptic operators with complex-valued bounded measurable coefficients

Mathematische Nachrichten

2017

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“…This method has been used in , , , , in the case of harmonic functions (that is, the case

A = I

and

L = - Δ

). This method has also been used to study more general second order problems in , , , , , , under various assumptions on the coefficients

bold-italicA

. Layer potentials have been used in other ways in , , , , , .…”

Section: Introductionmentioning

confidence: 99%

“…A common starting regularity condition is t ‐independence, that is,

A (x, t) = A (x, s) = A (x) for all x \in R^{n} and all s, t \in double-struckR .

Boundary value problems for such coefficients have been investigated extensively in domains Ω where the distinguished t ‐direction is always transverse to the boundary, that is,

Ω = {(x, t) : t > φ (x)}

Section: Introductionmentioning

confidence: 99%

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

Barton

Mathematische Nachrichten

2017

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“…A substantial class of such operators is provided by the results of [1] and [27]. We note, furthermore, that for any t-independent operator L 0 with real, possibly non-symmetric, coefficients, there exists 1 < p < 2 + ε such that the (D) p and (R) p are solvable, accompanied by the invertibility results for the single layer potential at the boundary (see [25,26]). …”

Section: Introduction and Statements Of Resultsmentioning

confidence: 98%

Layer potentials and boundary value problems for elliptic equations with complex L∞ coefficients satisfying the small Carleson measure norm condition

Advances in Mathematics

Mourgoglou

2015

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We consider divergence form elliptic equations Lu := ∇ · (A∇u) = 0 in the half space R n+1 + := {(x, t) ∈ R n × (0, ∞)}, whose coefficient matrix A is complex elliptic, bounded and measurable. In addition, we suppose that A satisfies some additional regularity in the direction transverse to the boundary, namely that the discrepancy A(x, t) − A(x, 0) satisfies a Carleson measure condition of Fefferman-Kenig-Pipher type, with small Carleson norm. Under these conditions, we establish a full range of boundedness results for double and single layer potentials in L p , Hardy, Sobolev, BMO and Hölder spaces. Furthermore, we prove solvability of the Dirichlet problem for L, with data in L p (R n ), BMO(R n ), and C α (R n ), and solvability of the Neumann and Regularity problems, with data in the spaces L p (R n )/H p (R n ) and L p 1 (R n )/H 1,p (R n ) respectively, with the appropriate restrictions on indices, assuming invertibility of layer potentials for the t-independent operator L 0 := −∇ · (A(·, 0)∇).

“…There are many ways to use layer potentials to study boundary value problems; see in the case of harmonic functions (that is, the case

A = I

and

L = - Δ

) and in the case of more general second‐order problems. In particular, the second‐order double and single layer potentials have been used to study higher order differential equations in .…”

Section: Introductionmentioning

confidence: 99%

“…Boundary value problems for such coefficients have been investigated extensively. See, for example, .…”

Section: Introductionmentioning

confidence: 99%

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

Barton

Proc. London Math. Soc.

2019

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