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W1, p-estimates for elliptic equations with lower order terms

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The authors were supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MOE) (No. 2013R1A1A2007290).
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  • We consider the Neumann and Dirichlet problems for second-order linear elliptic equations

    $ - {\rm{div}}{\mkern 1mu} (A\nabla u) - b \cdot \nabla u + \lambda u = f + {\rm{div}}{\mkern 1mu} F,\quad - {\rm{div}}{\mkern 1mu} ({A^t}\nabla v) + {\rm{div}}{\mkern 1mu} (vb) + \lambda v = g + {\rm{div}}{\mkern 1mu} G$

    in a bounded Lipschitz domain $\Omega \subset \mathbb{R}^n$, $n \geq 2$, where $A: \mathbb{R}^n \to \mathbb{R}^{n^2}$, $b: \Omega \to \mathbb{R}^n$ and $\lambda \geq 0$ are given. Some $W^{1, 2}$-estimates have been already known, provided that $A \in L^\infty(\Omega)^{n^2}$ and $b \in L^r(\Omega)^n$, where $n \leq r < \infty$ if $n \geq 3$ and $2 < r < \infty$ if $n=2$. Under more regularity assumptions on $A$ and $\Omega$, we establish the existence and uniqueness of weak solutions satisfying $W^{1, p}$-estimates. Our $W^{1, p}$-estimates are uniform on $\lambda \geq 0$ for the case of the Dirichlet problems. For the Neumann problems, the $W^{1, p}$-estimates are uniform with respect to $\lambda \geq 0$ if $f$ and $g$ satisfy some compatibility conditions. These uniform estimates allow us to obtain strong stability results in $W^{1, p}$ with respect to $\lambda $ for the Neumann and Dirichlet problems.

    Mathematics Subject Classification: Primary: 35J25.

    Citation:

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