scholarly journals On Divergence Form Second-order PDEs with Growing Coefficients in W 1 p Spaces without Weights

2011 ◽  
pp. 389-414
Author(s):  
N. V. Krylov
Keyword(s):  
Divergence Form ◽  
Second Order

Perturbation and Solvability of Initial Lp Dirichlet Problems for Parabolic Equations over Non-cylindrical Domains

2014 ◽  
Vol 66 (2) ◽  
pp. 429-452 ◽  
Author(s):  
Jorge Rivera-Noriega
Keyword(s):  
Dirichlet Problem ◽  
Parabolic Equations ◽  
Elliptic Equations ◽  
Carleson Measure ◽  
Divergence Form ◽  
Second Order ◽  
Linear Operators ◽  
Dirichlet Problems ◽  
Small Perturbations ◽  
Cylindrical Domains

AbstractFor parabolic linear operators L of second order in divergence form, we prove that the solvability of initial Lp Dirichlet problems for the whole range 1 < p < ∞ is preserved under appropriate small perturbations of the coefficients of the operators involved. We also prove that if the coefficients of L satisfy a suitable controlled oscillation in the form of Carleson measure conditions, then for certain values of p > 1, the initial Lp Dirichlet problem associated with Lu = 0 over non-cylindrical domains is solvable. The results are adequate adaptations of the corresponding results for elliptic equations.

scholarly journals The Dirichlet Problem for Second-Order Divergence Form Elliptic Operators with Variable Coefficients: The Simple Layer Potential Ansatz

2015 ◽  
Vol 2015 ◽  
pp. 1-11 ◽  
Author(s):  
Alberto Cialdea ◽  
Vita Leonessa ◽  
Angelica Malaspina
Keyword(s):  
Dirichlet Problem ◽  
Differential Operators ◽  
Differential Forms ◽  
Variable Coefficients ◽  
Divergence Form ◽  
Boundary Integral ◽  
Second Order ◽  
Boundary Integral Method ◽  
Layer Potential ◽  
Partial Differential Operators

We investigate the Dirichlet problem related to linear elliptic second-order partial differential operators with smooth coefficients in divergence form in bounded connected domains ofRm(m≥3) with Lyapunov boundary. In particular, we show how to represent the solution in terms of a simple layer potential. We use an indirect boundary integral method hinging on the theory of reducible operators and the theory of differential forms.

scholarly journals The square root problem for second-order, divergence form operators with mixed boundary conditions on L p

2014 ◽  
Vol 15 (1) ◽  
pp. 165-208 ◽  
Author(s):  
Pascal Auscher ◽  
Nadine Badr ◽  
Robert Haller-Dintelmann ◽  
Joachim Rehberg
Keyword(s):  
Boundary Conditions ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Divergence Form ◽  
Second Order ◽  
Square Root

scholarly journals Affine Discontinuous Galerkin Method Approximation of Second-Order Linear Elliptic Equations in Divergence Form with Right-Hand Side in L1

2018 ◽  
Vol 2018 ◽  
pp. 1-15
Author(s):  
Abdeluaab Lidouh ◽  
Rachid Messaoudi
Keyword(s):  
Galerkin Method ◽  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Method ◽  
Divergence Form ◽  
Second Order ◽  
Discrete Problem ◽  
Renormalized Solution ◽  
Order Linear ◽  
Right Hand ◽  
The Right

We consider the standard affine discontinuous Galerkin method approximation of the second-order linear elliptic equation in divergence form with coefficients in L∞Ω and the right-hand side belongs to L1Ω; we extend the results where the case of linear finite elements approximation is considered. We prove that the unique solution of the discrete problem converges in W01,qΩ for every q with 1≤q<d/d-1 (d=2 or d=3) to the unique renormalized solution of the problem. Statements and proofs remain valid in our case, which permits obtaining a weaker result when the right-hand side is a bounded Radon measure and, when the coefficients are smooth, an error estimate in W01,qΩ when the right-hand side f belongs to LrΩ verifying Tkf∈H1Ω for every k>0, for some r>1.

Commutators with fractional differentiation for second-order elliptic operators on ℝn

2019 ◽  
Vol 22 (02) ◽  
pp. 1950010
Author(s):  
Yanping Chen ◽  
Qingquan Deng ◽  
Yong Ding
Keyword(s):  
Differential Operators ◽  
Divergence Form ◽  
Second Order ◽  
Bounded Mean Oscillation ◽  
Fractional Differentiation ◽  
Mean Oscillation ◽  
Fractional Differential ◽  
Fractional Differential Operators ◽  
Second Order Elliptic Operators ◽  
Complex Coefficients

Let [Formula: see text] be a second-order divergence form elliptic operator and [Formula: see text] an accretive, [Formula: see text] matrix with bounded measurable complex coefficients in [Formula: see text] In this paper, we establish [Formula: see text] theory for the commutators generated by the fractional differential operators related to [Formula: see text] and bounded mean oscillation (BMO)–Sobolev functions.

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