An Example of Fractal Singular Homogenization

2007 ◽  
Vol 14 (1) ◽  
pp. 169-193
Author(s):  
Umberto Mosco ◽  
Maria Agostina Vivaldi
Keyword(s):  
Elliptic Operator ◽  
Divergence Form ◽  
Elliptic Operators ◽  
Second Order ◽  
Open Domain ◽  
Weighted Energy ◽  
Spectral Convergence

Abstract We construct a sequence of quadratic weighted energy forms in an open domain of the plane, that 𝑀-converges to an energy form with a singular fractal term. The weights belong to the Muckenoupt class 𝐴2 and have pointwise singularities. The result implies the spectral convergence of a sequence of second-order weighted elliptic operators in divergence form in the plane to a singular elliptic operator with a second order fractal term.

scholarly journals Commutators of Square Functions Related to Fractional Differentiation for Second-Order Elliptic Operators

2018 ◽  
Vol 2018 ◽  
pp. 1-10
Author(s):  
Xiongtao Wu ◽  
Wenyu Tao ◽  
Yanping Chen ◽  
Kai Zhu
Keyword(s):  
Elliptic Operator ◽  
Divergence Form ◽  
Elliptic Operators ◽  
Second Order ◽  
Square Function ◽  
Fractional Differentiation ◽  
Square Functions ◽  
Second Order Elliptic Operators ◽  
Complex Coefficients

Let L=-div(A∇) be a second-order divergence form elliptic operator, where A is an accretive n×n matrix with bounded measurable complex coefficients in Rn. In this paper, we mainly establish the Lp boundedness for the commutators generated by b∈Iα(BMO) and the square function related to fractional differentiation for second-order elliptic operators.

Homogeneous Calderón–Zygmund estimates for a class of second-order elliptic operators

2014 ◽  
Vol 17 (01) ◽  
pp. 1450017 ◽  
Author(s):  
G. P. Galdi ◽  
G. Metafune ◽  
C. Spina ◽  
C. Tacelli
Keyword(s):  
Elliptic Operator ◽  
Unique Solvability ◽  
Elliptic Operators ◽  
Second Order ◽  
Poisson Problem ◽  
Second Order Elliptic Operators ◽  
Uniformly Continuous

We prove unique solvability and corresponding homogeneous Lp estimates for the Poisson problem associated to the uniformly elliptic operator [Formula: see text], provided the coefficients are bounded and uniformly continuous, and admit a (non-zero) limit as |x| goes to infinity. Some important consequences are also derived.

scholarly journals Weak Uniqueness for Elliptic Operators in ℝ3 with Time-Independent Coefficients

2006 ◽  
Vol 74 (1) ◽  
pp. 91-100
Author(s):  
Cristina Giannotti
Keyword(s):  
Dirichlet Problem ◽  
Elliptic Operator ◽  
Elliptic Operators ◽  
Second Order ◽  
Weak Uniqueness

The author gives a proof with analytic means of weak uniqueness for the Dirichlet problem associated to a second order uniformly elliptic operator in ℝ3 with coefficients independent of the coordinate x3 and continuous in ℝ2 {0}.

On Variational Inequalities Driven by Elliptic Operators Not in Divergence Form

2012 ◽  
Vol 12 (3) ◽  
Author(s):  
Michele Matzeu ◽  
Raffaella Servadei
Keyword(s):  
Variational Inequalities ◽  
Elliptic Operator ◽  
Bounded Domain ◽  
Smooth Boundary ◽  
Divergence Form ◽  
Elliptic Operators

AbstractIn this paper we study semilinear variational inequalities driven by an elliptic operator not in divergence form modeled bywhere Ω is a bounded domain of RN, N ≥ 3, with smooth boundary, A is the elliptic operator, not in divergence form, given byHere a

scholarly journals Commutators with Lipschitz Functions and Nonintegral Operators

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Peizhu Xie ◽  
Ruming Gong
Keyword(s):  
Integral Representation ◽  
Fractional Power ◽  
Divergence Form ◽  
Elliptic Operators ◽  
Second Order ◽  
Lipschitz Functions ◽  
Positive Operators ◽  
Riesz Transforms ◽  
Spectral Multipliers ◽  
Size Estimates

LetTbe a singular nonintegral operator; that is, it does not have an integral representation by a kernel with size estimates, even rough. In this paper, we consider the boundedness of commutators withTand Lipschitz functions. Applications include spectral multipliers of self-adjoint, positive operators, Riesz transforms of second-order divergence form operators, and fractional power of elliptic operators.

scholarly journals Maximal regularity for elliptic operators with second-order discontinuous coefficients

2020 ◽  
Author(s):  
G. Metafune ◽  
L. Negro ◽  
C. Spina
Keyword(s):  
Elliptic Operator ◽  
Maximal Regularity ◽  
Discontinuous Coefficients ◽  
Elliptic Operators ◽  
Second Order ◽  
Parabolic Problems ◽  
Order Elliptic Operator

Abstract We prove maximal regularity for parabolic problems associated to the second-order elliptic operator $$\begin{aligned} L =\Delta +(a-1)\sum _{i,j=1}^N\frac{x_ix_j}{|x|^2}D_{ij}+c\frac{x}{|x|^2}\cdot \nabla -b|x|^{-2} \end{aligned}$$ L = Δ + ( a - 1 ) ∑ i , j = 1 N x i x j | x | 2 D ij + c x | x | 2 · ∇ - b | x | - 2 with $$a>0$$ a > 0 and $$b,\ c$$ b , c real coefficients.

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