The Lp Dirichlet and regularity problems for second order elliptic systems with application to the Lamé system

Dindoš, Martin

doi:10.1080/03605302.2021.1892131

Cited by 6 publications

(14 citation statements)

References 31 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Since A is constant and satisfies (1.4), and the measure µ defined in (1.5) is a Carleson measure in Ω, by the square function estimates and the L 2 solvability of the regularity problem for elliptic systems with drift terms [5], we know that there exists a small constant K = K(λ, d, m) > 0, such that K ≤ M 0 2 and if max{L, µ C } ≤ K, then we have 3) and (1.4), and the measure µ defined in (1.5) is a Carleson measure in Ω. Suppose that max{L, µ C } ≤ K, where K is given in Theorem 4.1.…”

Section: 1mentioning

confidence: 99%

“…Remark 2. The reason for the assumption that A satisfies (1.4) and the assumptions on Ω and B is that, under these conditions, the L 2 regularity problem for operator L + B∇ with constant leading coefficient is uniquely solvable, which was proved in [5] and will be used in the proof.…”

mentioning

confidence: 99%

“…The regularity estimates for second order elliptic equations and systems in Lipschitz domains have been studied extensively. We refer the reader to [11,4,12,1,18,7,16,8,17,15,6,5] for references. For operators not involving drift terms, W 1,p estimate for Laplace equation with Dirichlet boundary condition was established by D. Jerison and C. Kenig in [11] for 3 2 − ε < p < 3 + ε if d ≥ 3 and 4 3 − ε < p < 4 + ε when d = 2 on Lipschitz domains, and the ranges of p are sharp.…”

mentioning

confidence: 99%

“…d−2 + ε in Lipschitz graph domains with small Lipschitz constant under the assumption that the coefficients A and B satisfy certain Carleson condition. Later in [5], using the real variable method in [19], M. Dindoš improved the range to 2 − ε < p < 2(d−1) d−3 + ε by proving that the L p regularity problem is solvable for 2 − ε < p < 2 + ε, and he also proved the solvability for the L p Dirichlet problem for 2 − ε < p < 2(d−1) d−2 + ε under some weaker (oscillation-type) Carleson condition. In comparison with elliptic operators not involving drift terms, the main difficulties are caused by the presence of the drift term.…”

mentioning

confidence: 99%

See 3 more Smart Citations

$ W^{1, p} $ estimates for elliptic problems with drift terms in Lipschitz domains

Shi¹

2022

DCDS

View full text Add to dashboard Cite

In this paper, we establish the <inline-formula><tex-math id="M1">\begin{document}$ W^{1,p} $\end{document}</tex-math></inline-formula> estimates for solutions of second order elliptic problems with drift terms in bounded Lipschitz domains by using a real variable method. For scalar equations, we prove that the <inline-formula><tex-math id="M2">\begin{document}$ W^{1,p} $\end{document}</tex-math></inline-formula> estimates hold for <inline-formula><tex-math id="M3">\begin{document}$ \frac{3}{2}-\varepsilon<p<3+\varepsilon $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M4">\begin{document}$ d\geq3 $\end{document}</tex-math></inline-formula>, and the range for <inline-formula><tex-math id="M5">\begin{document}$ p $\end{document}</tex-math></inline-formula> is sharp. For elliptic systems, we prove that the <inline-formula><tex-math id="M6">\begin{document}$ W^{1,p} $\end{document}</tex-math></inline-formula> estimates hold for <inline-formula><tex-math id="M7">\begin{document}$ \frac{2d}{d+1}-\varepsilon<p<\frac{2d}{d-1}+\varepsilon $\end{document}</tex-math></inline-formula> under the assumption that the Lipschitz constant of the domain is small.

show abstract

Section: 1mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

See 2 more Smart Citations

$ W^{1, p} $ estimates for elliptic problems with drift terms in Lipschitz domains

Shi¹

2022

DCDS

View full text Add to dashboard Cite

show abstract

“…In this paper, we consider the solvability of certain boundary value problems (Regularity, Neumann) for a class of elliptic second order divergence form equations with real coefficients satisfying a natural and well-studied Carleson measure condition. Some of the extensive literature in this subject includes [2,3,5,6,9,10,[14][15][16]19].…”

Section: Introductionmentioning

confidence: 99%

Regularity and Neumann problems for operators with real coefficients satisfying Carleson condition

Dindoš¹,

Hofmann²,

Pipher³

2022

Preprint

Self Cite

View full text Add to dashboard Cite

In this paper, we continue the study of a class of second order elliptic operators of the form L = div(A∇•) in a domain above a Lipschitz graph in R n , where the coefficients of the matrix A satisfy a Carleson measure condition, expressed as a condition on the oscillation on Whitney balls. For this class of operators, it is known (since 2001) that the L q Dirichlet problem is solvable for some 1 < q < ∞. Moreover, further studies completely resolved the range of L q solvability of the Dirichlet, Regularity, Neumann problems in Lipschitz domains, when the Carleson measure norm of the oscillation is sufficiently small.We show that there exists p reg > 1 such that for all 1 1 is the number such that the L q Dirichlet problem for the adjoint operator L * is solvable for all q > q * .Additionally when n = 2, there exists p neum > 1 such that for all 1 1 is the number such that the L q Dirichlet problem for the operator L 1 = div(A 1 ∇•) with matrix A 1 = A/ det A is solvable for all q > q * .

show abstract

Boundary Value Problems for Elliptic Operators Satisfying Carleson Condition

Dindoš,

Pipher

2023

Vietnam J. Math.

View full text Add to dashboard Cite

The L^p Dirichlet and regularity problems for second order elliptic systems with application to the Lamé system

Cited by 6 publications

References 31 publications

$ W^{1, p} $ estimates for elliptic problems with drift terms in Lipschitz domains

$ W^{1, p} $ estimates for elliptic problems with drift terms in Lipschitz domains

Regularity and Neumann problems for operators with real coefficients satisfying Carleson condition

Boundary Value Problems for Elliptic Operators Satisfying Carleson Condition

Contact Info

Product

Resources

About