Boundary Harnack inequalities for parabolic operators. (Inégalités de Harnack à la frontière pour des opérateurs paraboliques.) (French) Zbl 0661.47042

Heurteaux, Yanick

C. R. Acad. Sci., Paris, Sér. I 308, No. 13, 401-404 (1989).

Let \(\Omega\) be an open set of \({\mathbb{R}}^{n+1}\) and let Q be a boundary point of \(\Omega\) having a neighborhood whose intersection with the boundary of \(\Omega\) is “Lipschitz”. For a parabolic operator, we compare the behavior of positive L-solutions in \(\Omega\) converging to zero at every point of the boundary which is sufficiently close to Q. A boundary Harnack principle is then proved and used to describe the cone of positive L-solutions converging to zero at every point of \(\partial_ p\Omega -\{Q\}\), where \(\partial_ p\Omega\) is the parabolic boundary of \(\Omega\).

MSC:

47F05	General theory of partial differential operators (should also be assigned at least one other classification number in Section 47-XX)
35K20	Initial-boundary value problems for second-order parabolic equations

Keywords:

boundary point; parabolic operator; positive L-solutions; boundary Harnack principle; parabolic boundary