Advances in Differential Equations

Elliptic operators with unbounded diffusion and drift coefficients in $L^p$ spaces

G. Metafune, C. Spina, and C. Tacelli

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We prove that, for $\alpha > 2$, $\frac{N}{N-2+c} < p <\infty$, the operator $Lu=(1+|x|^\alpha)\Delta u+c|x|^{\alpha-1}\tfrac{x}{|x|}\cdot\nabla$ generates an analytic semigroup in $L^p$, which is contractive if and only if $p \ge \frac{N+\alpha-2}{N-2+c}$. Moreover, for $\alpha <\frac{N}{p'}+c$, we provide an explicit description of the domain. Spectral properties of the operator $L$ and kernel estimates are also obtained.

Article information

Adv. Differential Equations Volume 19, Number 5/6 (2014), 473-526.

First available in Project Euclid: 3 April 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 47D07: Markov semigroups and applications to diffusion processes {For Markov processes, see 60Jxx} 35B50: Maximum principles 35J25: Boundary value problems for second-order elliptic equations 35J70: Degenerate elliptic equations


Metafune, G.; Spina, C.; Tacelli, C. Elliptic operators with unbounded diffusion and drift coefficients in $L^p$ spaces. Adv. Differential Equations 19 (2014), no. 5/6, 473--526.

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