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

DOI10.1007/978-3-319-30961-3_4zbMath1403.35096OpenAlexW2520718820MaRDI QIDQ4555485

Ariel Barton, Svitlana Mayboroda

Publication date: 20 November 2018

Published in: Harmonic Analysis, Partial Differential Equations, Complex Analysis, Banach Spaces, and Operator Theory (Volume 1) (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1007/978-3-319-30961-3_4

zbMATH Keywords

boundary value problemsLipschitz domainspolyharmonic equationhigher-order elliptic equations

Mathematics Subject Classification ID

Biharmonic and polyharmonic equations and functions in higher dimensions (31B30) Research exposition (monographs, survey articles) pertaining to partial differential equations (35-02) Higher-order elliptic equations (35J30)

Related Items (7)

Gradient estimates and the fundamental solution for higher-order elliptic systems with lower-order terms ⋮ A \(C^1\) virtual element method on polyhedral meshes ⋮ The Neumann problem for higher order elliptic equations with symmetric coefficients ⋮ The Ẇ−1,p Neumann problem for higher order elliptic equations ⋮ Trace and extension theorems relating Besov spaces to weighted averaged Sobolev spaces ⋮ On the explicit representation of the trace space \(H^{\frac{3}{2}}\) and of the solutions to biharmonic Dirichlet problems on Lipschitz domains via multi-parameter Steklov problems ⋮ Dirichlet and Neumann boundary values of solutions to higher order elliptic equations

Cites Work

• Boundary value problems and integral operators for the bi-Laplacian in non-smooth domains
• The Dirichlet problem for higher order equations in composition form
• Multi-layer potentials and boundary problems for higher-order elliptic systems in Lipschitz domains
• Weighted maximal regularity estimates and solvability of non-smooth elliptic systems. I
• Optimal estimates for the inhomogeneous problem for the bi-Laplacian in three-dimensional Lipschitz domains
• A bilinear estimate for biharmonic functions in Lipschitz domains
• Analyticity of layer potentials and \(L^{2}\) solvability of boundary value problems for divergence form elliptic equations with complex \(L^{\infty }\) coefficients
• Estimates for Green function and Poisson kernels of higher-order Dirichlet boundary value problems
• Solvability of elliptic systems with square integrable boundary data
• Gradient estimates and the fundamental solution for higher-order elliptic systems with rough coefficients
• Teorema del massimo modulo e teorema di esistenza e di unicità per il problema di Dirichlet relativo alle equazioni ellittiche in due variabili
• Area integral estimates for elliptic differential operators with non- smooth coefficients
• The \(L^p\) Dirichlet problem for elliptic systems on Lipschitz domains
• Wiener type regularity of a boundary point for the 3D Lamé system
• Wiener's criterion for divergence form parabolic operators with \(C^ 1\)- Dini continuous coefficients
• Functional calculus of Dirac operators and complex perturbations of Neumann and Dirichlet problems
• To the theory of the Dirichlet and Neumann problems for strongly elliptic systems in Lipschitz domains
• An irregular complex valued solution to a scalar uniformly elliptic equation
• The Dirichlet problem in Lipschitz domains for higher order elliptic systems with rough coefficients
• Boundedness of the gradient of a solution and Wiener test of order one for the biharmonic equation
• The Dirichlet problem for the biharmonic equation in a Lipschitz domain
• Adjoint boundary value problems for the biharmonic equation on \(C^ 1\) domains in the plane
• The vertex of a cone can be nonregular in the Wiener sense for a fourth- order elliptic equation
• The Green matrix for strongly elliptic systems of second order with continuous coefficients
• Wiener criteria and energy decay for relaxed Dirichlet problems
• The Dirichlet problem in non-smooth domains
• The Green function for uniformly elliptic equations
• The Wiener test for degenerate elliptic equations
• On the Agmon-Miranda maximum principle for solutions of elliptic equations in polyhedral and polygonal domains
• On the Agmon-Miranda maximum principle for solutions of strongly elliptic equations in domains of \(\mathbb{R}{}^ n\) with conical points
• A priori estimates in uniform norms for strongly elliptic systems
• Potential techniques for boundary value problems on \(C^1\)-domains
• The inhomogeneous Dirichlet problem for \(\Delta^2\) in Lipschitz domains
• A maximum principle for biharmonic functions in Lipschitz and \(C^ 1\) domains
• The Neumann problem for elliptic equations with non-smooth coefficients
• Area integral estimates for higher order elliptic equations and systems
• On \(L^p\) estimates for square roots of second order elliptic operators on \(\mathbb R^n\)
• A new approach to absolute continuity of elliptic measure, with applications to non-symmetric equations
• Equivalence between regularity theorems and heat kernel estimates for higher-order elliptic operators and systems under divergence form
• The Kato square root problem for higher order elliptic operators and systems on \(\mathbb{R}^n\)
• Weak type \((p,p)\) estimates for Riesz transforms
• Wiener's criterion for the heat equation
• Potential estimates for a class of fully nonlinear elliptic equations.
• The Wiener test for higher order elliptic equations.
• Dilation invariant estimates and the boundary Gårding inequality for higher order elliptic operators
• Maximum principles for the polyharmonic equation on Lipschitz domains
• Estimates for Green's matrices of elliptic systems by \(L^ p\) theory
• A family of fundamental solutions of elliptic partial differential operators with real constant coefficients
• The connections between Dirichlet, regularity and Neumann problems for second order elliptic operators with complex bounded measurable coefficients
• The regularity problem for second order elliptic operators with complex-valued bounded measurable coefficients
• Necessary and sufficient conditions for the solvability of the \(L^p\) Dirichlet problem on Lipschitz domains
• Regularity of solutions to the polyharmonic equation in general domains
• The \(L^{p}\) boundary value problems on Lipschitz domains
• A relationship between the Dirichlet and regularity problems for elliptic equations
• The Green function estimates for strongly elliptic systems of second order
• The biharmonic Neumann problem in Lipschitz domains
• On estimates of biharmonic functions on Lipschitz and convex domains
• Layer potentials beyond singular integral operators
• A partial differential equation arising in conformal mapping
• The 𝐿^{𝑝} regularity problem on Lipschitz domains
• On the Regularity of Green Functions in Lipschitz Domains
• Multiple layer potentials and the dirichlet problem for higher order elliptic equations in the plane I
• Maximum theorems for solutions of higher order elliptic equations
• On necessary and sufficient conditions for 𝐿^{𝑝}-estimates of Riesz transforms associated to elliptic operators on ℝⁿ and related estimates
• Boundedness of the Hessian of a Biharmonic Function in a Convex Domain
• On the Dirichlet and Regularity Problems for the Bi-Laplacian in Lipschitz Domains
• The regularity and Neumann problem for non-symmetric elliptic operators
• A survey on boundary conditions for the biharmonic
• Pointwise estimates for the polyharmonic Green function in general domains
• On Harnack's theorem for elliptic differential equations
• On the Poisson integral for Lipschitz and $C^{1}$-domains
• Weighted norm inequalities for the Lusin area integral and the nontangential maximal functions for functions harmonic in a Lipschitz domain
• The Neumann problem on Lipschitz domains
• Area Integral Estimates for the Biharmonic Operator in Lipschitz Domains
• The Dirichlet Problem in L p for the Biharmonic Equation on Lipschitz Domains
• An Elementary Proof That the Biharmonic Green Function of an Eccentric Ellipse Changes Sign
• On the wiener type regularity of a boundary point for the polyharmonic operator
• On the weak continuity of elliptic operators and applications to potential theory
• Square function estimates on layer potentials for higher‐order elliptic equations
• Boundary Coerciveness and the Neumann Problem for 4th Order Linear Partial Differential Operators
• Boundary-value Problems for Higher-order Elliptic Equations in Non-smooth Domains
• Layer Potentials and Boundary-Value Problems for Second Order Elliptic Operators with Data in Besov Spaces
• Conformal invariants and partial differential equations
• Square function/non-tangential maximal function estimates and the Dirichlet problem for non-symmetric elliptic operators
• On a Question of Hadamard Concerning Super‐Biharmonic Functions
• Nonlinear problems of elasticity
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item

This page was built for publication: Higher-Order Elliptic Equations in Non-Smooth Domains: a Partial Survey