A counter-example in G-convergence of non-divergence elliptic operators

2003 ◽  
Vol 133 (6) ◽  
pp. 1299-1310 ◽  
Author(s):  
Luigi D'Onofrio ◽  
Luigi Greco
Keyword(s):  
Differential Operator ◽  
Suitable Condition ◽  
Elliptic Operators ◽  
Planar Domain ◽  
Elliptic Differential Operator ◽  
Counter Example

In a planar domain, to each function w ∈ W2,2 satisfying a suitable condition we associate a non-divergence elliptic differential operator 𝔏 such that 𝔏w = 0. For a given converging sequence {wk}, we study G-convergence of the corresponding sequence {𝔏k} of operators.

Continuity Points Via Riesz Potentials for ℂ-Elliptic Operators

2020 ◽  
Vol 71 (4) ◽  
pp. 1201-1218
Author(s):  
Lars Diening ◽  
Franz Gmeineder
Keyword(s):  
Differential Operator ◽  
Arbitrary Order ◽  
Riesz Potential ◽  
Elliptic Operators ◽  
Gradient Estimates ◽  
Elliptic Differential Operator ◽  
Riesz Potentials

Abstract We establish a Riesz potential criterion for Lebesgue continuity points of functions of bounded $\mathbb{A}$-variation, where $\mathbb{A}$ is a $\mathbb{C}$-elliptic differential operator of arbitrary order. This result generalizes a potential criterion that is known for full gradients to the case where full gradient estimates are not available by virtue of Ornstein’s non-inequality.

Lp-Theory of degenerate-elliptic and parabolic operators of second order

1983 ◽  
Vol 95 (1-2) ◽  
pp. 95-113 ◽  
Author(s):  
Baoswan Wong-Dzung
Keyword(s):  
Banach Space ◽  
Differential Operator ◽  
Positive Semidefinite ◽  
Second Order ◽  
Minimal Realization ◽  
Elliptic Differential Operator ◽  
Second Derivatives ◽  
Degenerate Elliptic ◽  
Image Position ◽  
Lp Theory

SynopsisWe consider the formal operator given byin the Banach space X = LP(Rn), 1<p<∞. The coefficients ajk(x), aj(x), and a(x) are real-valued functions, ajk ε C2(Rn) has bounded second derivatives, aj ε Cl(Rn) has bounded first derivatives, and aεL∞(Rn). Furthermore, we assume that the n × n matrix (ajk(x)) is symmetric and positive semidefinite (i.e. ajk(x)ξjξk≧0 for all (ξ1,…,ξn)ε Rn and x ε Rn). We prove that the degenerate-elliptic differential operator given by –A and restricted to , the minimal realization of –A, is essentially quasi-m-dispersive in Lp(Rn), (hence that the minimal realization of +A is quasi-m-accretive) and that its closure coincides with the maximal realization of –A.

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