The weak maximum principle for second-order elliptic and parabolic conormal derivative problems
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Bibliographic record
Abstract
<p style='text-indent:20px;'>We prove the weak maximum principle for second-order elliptic and parabolic equations in divergence form with the conormal derivative boundary conditions when the lower-order coefficients are unbounded and domains are beyond Lipschitz boundary regularity. In the elliptic case we consider John domains and lower-order coefficients in <inline-formula><tex-math id="M1">\begin{document}$ L_n $\end{document}</tex-math></inline-formula> spaces (<inline-formula><tex-math id="M2">\begin{document}$ a^i, b^i \in L_q $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ c \in L_{q/2} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ q = n $\end{document}</tex-math></inline-formula> if <inline-formula><tex-math id="M5">\begin{document}$ n \geq 3 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$ q > 2 $\end{document}</tex-math></inline-formula> if <inline-formula><tex-math id="M7">\begin{document}$ n = 2 $\end{document}</tex-math></inline-formula>). For the parabolic case, the lower-order coefficients <inline-formula><tex-math id="M8">\begin{document}$ a^i $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M9">\begin{document}$ b^i $\end{document}</tex-math></inline-formula>, and <inline-formula><tex-math id="M10">\begin{document}$ c $\end{document}</tex-math></inline-formula> belong to <inline-formula><tex-math id="M11">\begin{document}$ L_{q,r} $\end{document}</tex-math></inline-formula> spaces (<inline-formula><tex-math id="M12">\begin{document}$ a^i,b^i, |c|^{1/2} \in L_{q,r} $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M13">\begin{document}$ n/q+2/r \leq 1 $\end{document}</tex-math></inline-formula>), <inline-formula><tex-math id="M14">\begin{document}$ q \in (n,\infty] $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M15">\begin{document}$ r \in [2,\infty] $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M16">\begin{document}$ n\ge 2 $\end{document}</tex-math></inline-formula>. We also consider coefficients in <inline-formula><tex-math id="M17">\begin{document}$ L_{n,\infty} $\end{document}</tex-math></inline-formula> with a smallness condition for parabolic equations.
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|---|---|---|
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