The Initial Dirichlet Boundary Value Problem for General Second Order Parabolic Systems in Nonsmooth Manifolds

In a series of recent papers [1], [2], [3], [4], [5] we have initiated the study of boundary value problems for (variable coefficients, second order, strongly) elliptic PDE’s in nonsmooth subdomains of Riemannian manifolds via integral equation methods. Here we take the first steps in the direction of extending this theory to initial boundary value problems (IBV P ’s) for variable coefficient (strongly) parabolic systems in non-smooth cylinders. Problems as such have a long history and the field remains a very active area of research. For work in the context of smooth manifolds the reader is referred to [6], [7], [8]. See also [9], [10], [11], [12], [13], [14], [15], [16], for IBV P ’s associated with PDE’s of parabolic type in the smooth Euclidean setting. With the work of E.B. Fabes, N. Rivieré and their collaborators starting in the 1960’s, a new direction of research has emerged, emphasizing L-boundary data and less regular domains and/or coefficients. In this vein see [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27]. The papers cited above deal with domains exibiting a limited amount of smoothness and new techniques needed to be developed in order to treat the nonsmooth case. After the breakthrough in the elliptic case, cf. [28], [29], [30], [31], [32] and the references therein, there has been a substantial amount of work in the direction of solving IBV P ’s for parabolic PDE’s in minimally smooth domains. In flat-space, Euclidean Lipschitz cylinders, ∂t −∆ was treated via caloric measure estimates in [33] (inspired by the work in [28]) and via integral methods in [34], [35], [36] (after the pioneering work in [22] and by adapting the approach from [30]). The latter work has been further extended to include second-order constant coefficient PDE’s such as the parabolic versions of the Lamé system, the linearized Navier-Stokes system and the Maxwell system in [37], [38], [39]. Higher order, homogeneous, constant coefficient, parabolic PDE’s have been treated in [40], [41], [42], following the work in elliptic case from [43]. The Dirichlet problem for more general, scalar, divergence-form parabolic PDE in Lipschitz cylinders has been considered in [44]. Extensions to time-varying domains have been developed in [45], [46], [47], [48], [49], [50]. ∗1991 Mathematics Subject Classification: Primary 35K50, 42B20; Secondary 58G20.