EXISTENCE, UNIQUENESS, AND REGULARITY RESULTS FOR ELLIPTIC EQUATIONS WITH DRIFT TERMS IN CRITICAL WEAK SPACES

초록

We consider Dirichlet problems for linear elliptic equations of second order in divergence form on a bounded or exterior smooth domain Omega in R-n, n >= 3, with drifts b in the critical weak L-n-space L-n,L-infinity(Omega;R-n). First, assuming that the drift b has nonnegative weak divergence in L-n/2,(infinity)(Omega), we establish existence and uniqueness of weak solutions in W-1,W-p(Omega) or D-1,D-p(Omega) for any p with n' = n/(n- 1) < p < n. By duality, a similar result also holds for the dual problem. Next, we prove W-1,W-n+epsilon- or W-2,n(/2)+delta-regularity of weak solutions of the dual problem for some epsilon, delta > 0 when the domain Omega is bounded. These gradient estimates go beyond the borderlines and are based on the global Holder regularity by the De Giorgi-Nash-Moser methods as well as the Miranda-Nirenberg interpolation inequalities. By duality, these results enable us to obtain a quite general uniqueness result as well as an existence result for weak solutions belonging to boolean AND W-p<n' (1,P)(Omega). Finally, we prove a uniqueness result for exterior problems, which implies in particular that (very weak) solutions are unique in both L-n/((n-2)),infinity(Omega) and L-n(,infinity) (Omega).

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