Differential and Integral Equations

On the domains of elliptic operators in $L^1$

Alessandra Lunardi and Giorgio Metafune

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Abstract

We prove optimal embedding estimates for the domains of second-order elliptic operators in $L^1$ spaces. Our procedure relies on general semigroup theory and interpolation arguments, and on estimates for $\nabla T(t)f$ in $L^1$, in $L^\infty$, and possibly in fractional Sobolev spaces, for $f\in L^1$. It is applied to a number of examples, including some degenerate hypoelliptic operators, and operators with unbounded coefficients.

Article information

Source
Differential Integral Equations Volume 17, Number 1-2 (2004), 73-97.

Dates
First available in Project Euclid: 21 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356060473

Mathematical Reviews number (MathSciNet)
MR2035496

Zentralblatt MATH identifier
1164.35365

Subjects
Primary: 35J70: Degenerate elliptic equations
Secondary: 26D15: Inequalities for sums, series and integrals 35H10: Hypoelliptic equations 35K65: Degenerate parabolic equations 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems 46F05: Topological linear spaces of test functions, distributions and ultradistributions [See also 46E10, 46E35] 47D06: One-parameter semigroups and linear evolution equations [See also 34G10, 34K30]

Citation

Lunardi, Alessandra; Metafune, Giorgio. On the domains of elliptic operators in $L^1$. Differential Integral Equations 17 (2004), no. 1-2, 73--97. https://projecteuclid.org/euclid.die/1356060473.


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