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.

Asymptotic behaviour for elliptic operators with second-order discontinuous coefficients

2020 ◽  
Vol 32 (2) ◽  
pp. 399-415 ◽  
Author(s):  
Luigi Negro ◽  
Chiara Spina
Keyword(s):  
Asymptotic Behaviour ◽  
Elliptic Operator ◽  
Discontinuous Coefficients ◽  
Elliptic Operators ◽  
Second Order ◽  
Parabolic Problems ◽  
Order Elliptic Operator

AbstractWe study the behaviour at infinity, in suitable weighted {L^{p}}-norms, of solutions of parabolic problems associated to the second order elliptic operatorL=\Delta+(a-1)\sum_{i,j=1}^{N}\frac{x_{i}x_{j}}{|x|^{2}}D_{ij}+c\frac{x}{|x|^{% 2}}\cdot\nabla-b|x|^{-2},where {a>0} and {b,c\in\mathbb{R}}.

scholarly journals Some results and examples about the behavior of harmonic functions and Green’s functions with respect to second order elliptic operators

2002 ◽  
Vol 165 ◽  
pp. 123-158 ◽  
Author(s):  
Alano Ancona
Keyword(s):  
Harmonic Function ◽  
Elliptic Operator ◽  
Critical Points ◽  
Harmonic Functions ◽  
Elliptic Operators ◽  
Second Order ◽  
Small Perturbations ◽  
Order Elliptic Operator ◽  
Symmetry Assumption ◽  
Second Order Elliptic Operators

Let M be a manifold and let L be a sufficiently smooth second order elliptic operator in M such that (M, L) is a transient pair. It is first shown that if L is symmetric with respect to some density in M, there exists a positive L-harmonic function in M which dominates L-Green’s function at infinity. Other classes of elliptic operators are investigated and examples are constructed showing that this property may fail if the symmetry assumption is removed. Another part of the paper deals with the existence of critical points for certain L-harmonic functions with periodicity properties. A class of small perturbations of second order elliptic operators is also described.

Partial avaraging of the nondivergence second order elliptic operator

2016 ◽  
Vol 31 (3) ◽  
pp. 47-53
Author(s):  
M.M. Sirazhudinov ◽  
◽  
S.P. Dzhamaludinova ◽  
M.E. Mahmudova ◽  
◽  
...  
Keyword(s):  
Elliptic Operator ◽  
Second Order ◽  
Order Elliptic Operator

scholarly journals Green’s Function Estimates for Time-Fractional Evolution Equations

2019 ◽  
Vol 3 (2) ◽  
pp. 36
Author(s):  
Ifan Johnston ◽  
Vassili Kolokoltsov
Keyword(s):  
Green’S Function ◽  
Elliptic Operator ◽  
Green's Function ◽  
Green's Functions ◽  
Evolution Equations ◽  
Green’S Functions ◽  
Variable Coefficients ◽  
Second Order ◽  
Fractional Evolution Equations ◽  
Order Elliptic Operator

We look at estimates for the Green’s function of time-fractional evolution equations of the form D 0 + * ν u = L u , where D 0 + * ν is a Caputo-type time-fractional derivative, depending on a Lévy kernel ν with variable coefficients, which is comparable to y - 1 - β for β ∈ ( 0 , 1 ) , and L is an operator acting on the spatial variable. First, we obtain global two-sided estimates for the Green’s function of D 0 β u = L u in the case that L is a second order elliptic operator in divergence form. Secondly, we obtain global upper bounds for the Green’s function of D 0 β u = Ψ ( - i ∇ ) u where Ψ is a pseudo-differential operator with constant coefficients that is homogeneous of order α . Thirdly, we obtain local two-sided estimates for the Green’s function of D 0 β u = L u where L is a more general non-degenerate second order elliptic operator. Finally we look at the case of stable-like operator, extending the second result from a constant coefficient to variable coefficients. In each case, we also estimate the spatial derivatives of the Green’s functions. To obtain these bounds we use a particular form of the Mittag-Leffler functions, which allow us to use directly known estimates for the Green’s functions associated with L and Ψ , as well as estimates for stable densities. These estimates then allow us to estimate the solutions to a wide class of problems of the form D 0 ( ν , t ) u = L u , where D ( ν , t ) is a Caputo-type operator with variable coefficients.

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.

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