The Dirichlet problem for parabolic operators with singular drift terms (Q2718314)

Steve Hofmann and John Lewis prove existence and uniqueness for solutions to the Dirichlet problem on the right half space for special parabolic and elliptic operators of the form \(\partial /\partial t-\nabla (A\nabla )\pm \nabla B\). The drift term can become singular. They also establish mutual absolute continuity of the associated parabolic measure \( d\omega \) with Lebesgue measure \(dx^{\prime }dt\) on the boundary of the right half space by proving a reverse Hölder condition for the Radon-Nikodým derivative \(d\omega /(dx^{\prime }dt)\). The special nature of the operators under consideration is that several Carleson-measure conditions (as well as boundedness) are assumed for weak derivatives of the coefficient matrix \(A\) and for the drift term \(B\). The model situation that gives rise to these assumptions is that they are all satisfied by the pullback of the heat operator, \(\partial /\partial t-\Delta \), from a domain whose boundary is Lipschitz in space and has one half a time derivative in the space of bounded mean oscillation (BMO). NEWLINENEWLINEIn previous work, \textit{J. L. Lewis} and \textit{M. A. M. Murray} [Mem. Am. Math. Soc. 545 (1995; Zbl 0826.35041)] had proved absolute continuity for caloric measure and surface measure on such domains, and Hofmann and Lewis had proved solvability of the L\(^{2}\)-Dirichlet problem for domains with small BMO norm. NEWLINENEWLINENEWLINEThe memoir is divided into three parts; each part has its own list of references. In Part I, existence and uniqueness for the continuous Dirichlet problem and the above-mentioned reverse Hölder condition are proved for the matrix \(A\) being a perturbation from a constant coefficient matrix,\(A_{0}\). The \(L^{\infty }\) norms of \(A-A_{0}\), distance \(( x,t) \) to \(\mathbb{R}^{d}\times B\) and several Carleson-measure norms are assumed to be uniformly small. The smallness criterion is used to establish certain basic estimates for solutions, such as Hölder continuity, and a doubling condition for parabolic measure. The main result is then proved in three stages: first for \(B=0\), then for \(A\) and \(B\) being smooth and finally with the smoothness assumption being dropped. In Part II the smallness assumption is removed at the expense of re-introducing some smoothness. To circumvent the fact that parabolic measure may not satisfy a center-doubling condition, the authors prove comparison results for parabolic measure and Lebesgue measure. They establish a key estimate for parabolic measure by confining their operator to a sawtooth domain, and then extending this operator to one on the right half space that satisfies the smallness criterion of Part I. Finally they use extrapolation to obtain the general result. NEWLINENEWLINEIn Part III of their paper the authors establish \(L^{q}\) solvability of the Dirichlet problem and obtain \(L^{q}\) norm estimates for the nontangential maximal function of solutions to both elliptic and parabolic equations. In fact, they prove versions of the main theorem of \textit{R. A. Fefferman, C. E. Kenig} and \textit{J. Pipher} [Ann. Math. (2) 134, No. 1, 65--124 (1991; Zbl 0770.35014)], for special operators with singular drift term, assuming they are perturbations of ``good'' operators. The proofs differ from prior work in that no doubling condition is assumed for parabolic measure; they build on the results and methods of Part II. NEWLINENEWLINEThe results of the memoir are proved, often in detail, using a wide range of standard tools for estimating solutions to parabolic PDE, as well as techniques from singular integral theory and \(A^{p}\) weights. The authors discuss the work of many major contributors to the area of elliptic and parabolic functions on nonsmooth domains; they do not mention any work on non-doubling measures other than their own.