DIRICHLET AND NEUMANN PROBLEMS FOR ELLIPTIC EQUATIONS WITH SINGULAR DRIFTS ON LIPSCHITZ DOMAINS

초록

We consider the Dirichlet and Neumann problems for second-order linear elliptic equations: -Delta u+ div(ub) = f and - Delta v - b . del v = g in a bounded Lipschitz domain Omega in R-n (n >= 3), where b : Omega -> R-n is a given vector field. Under the assumption that b is an element of L-n(Omega)(n), we first establish existence and uniqueness of solutions in L-alpha(p)((Omega)) for the Dirichlet and Neumann problems. Here L-alpha(p)((Omega)) denotes the Sobolev space (or Bessel potential space) with the pair (alpha, p) satisfying certain conditions. These results extend the classical works of Jerison-Kenig [J. Funct. Anal. 130 (1995), pp. 161-219] and Fabes-Mendez-Mitrea [J. Funct. Anal. 159 (1998), pp. 323-368] for the Poisson equation. We also prove existence and uniqueness of solutions of the Dirichlet problem with boundary data in L-2(partial derivative Omega). Our results for the Dirichlet problems hold even for the case n = 2.

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