A centennial of the Zaremba-Hopf-Oleinik lemma (Q2882324)

The author proposes a detailed discussion on the Hopf-Oleinik lemma which is one of the most important tools in studying solutions to elliptic and parabolic equations. Recall that, for the Laplace operator, the result has been known for 100 years and goes back to \textit{M.S.~Zaremba} [Krakau Anz. (A), 313--344 (1910; JFM 41.0854.12)], and reads as follows. Let \(\partial\Omega\in C^2.\) If \(0\in\partial\Omega\) then NEWLINENEWLINE\[NEWLINE -\Delta u=f\geq0 \mathrm{ in }\Omega; \quad u(x)>u(0)\mathrm{ in }\Omega \Longrightarrow NEWLINE\frac{\partial u}{\partial\mathbf{n}}(0) < 0, NEWLINE\]NEWLINE NEWLINEwith \(\mathbf{n}\) being the outward normal to \(\partial\Omega.\)NEWLINENEWLINEFor general operators in nondivergence form with bounded measurable coefficients, this result was established independently by \textit{E.~Hopf} [Proc. Am. Math. Soc. 3, 791--793 (1952; Zbl 0048.07802)] and \textit{O.~Oleinik} [Mat. Sb., N. Ser. 30(72), 695--702 (1952; Zbl 0046.10401] in the elliptic case, and by \textit{L.~Nirenberg} [Commun. Pure Appl. Math. 6, 167--177 (1953; Zbl 0050.09601)] in the parabolic one.NEWLINENEWLINEThe author considers strong solutions to the nondivergence form elliptic/parabolic equations with measurable coefficients of the type NEWLINE\[NEWLINE -a_{ij}(x)D_iD_j u + b_i(x)D_iu=f(x), NEWLINE\]NEWLINE NEWLINE\[NEWLINE \partial_tu-a_{ij}(x,t)D_iD_j u + b_i(x,t)D_iu=f(x,t), NEWLINE\]NEWLINE assuming uniform ellipticity/parabolicity of the operator considered. Sharp conditions on the lower-order coefficients \(b_i\) are provided, ensuring the validity of the Hopf-Oleinik lemma.