A Harnack inequality for the equation \(\nabla (a \nabla u) + b \nabla u = 0\), when \(| b | \in K_{n+1}\)
This paper establishes a Harnack inequality for nonnegative solutions of the equation \(\nabla( a\nabla u)+ b \nabla u= 0\), \(x\in D\subset \mathbb{R}^n\), \(n\geq 3\), where \(b\) belongs to the Kato class \(K_{n+ 1}\), i.e. \[ \lim_{r\to 0} \sup_x \int_{|x- y|< r} {|b(y)|\over |x- y|^{n- 1}} dy= 0 \] and \(a(x)\) is a positive definite matrix (with \(C^\alpha\) coefficients) and \(D\) is a \(C^2\) bounded domain: there exist positive constants \(r_0\) and \(M\) such that for any nonnegative solution \(u\) and for any ball \(B_r\) with \(0< r\leq r_0\) and \(B_{2r}\subset D\), we have \(\sup_{B_{r/2}} u\leq M\cdot \inf_{B_{r/2}} u\). As a classical corollary it is also proved that any nonnegative or bounded weak solution is Hölder continuous. The result also applies to the parabolic equation \[ \nabla_x(A(x, t) \nabla u(x, t))+ b(x, t) \nabla u(x, t)- {\partial u\over \partial t} (x, t)= 0\quad (x, t)\in D\times [0, T], \] where the coefficients of \(A\) are Hölder continuous functions such that \(\lambda^{- 1} I\leq A\leq \lambda I\) for a \(\lambda> 1\) and \(b\) satisfies a condition which generalizes the condition \(b\in K_{n+ 1}\). The method for the proof relies on the existence of a lower and an upper bound for the Green function of the parabolic equation. The elliptic equation is treated as a special case.
scientific article; zbMATH DE number 861905