ON WEAK SOLUTIONS OF ELLIPTIC EQUATIONS WITH SINGULAR DRIFTS

초록

We consider the Dirichlet problem for second-order linear elliptic equations with singular drift terms given by a vector field u. W-1,W-p-estimates for weak solutions are quite well known, provided that u is sufficiently regular, e.g., u is an element of L-infinity. In this paper, we establish the existence and uniqueness of weak solutions satisfying W-1,W-p- or W-1,W-2-estimates for less regular u. First, some W-1,W-p-estimates are shown for u. L-sigma(n) + L-r, where n <= r < 8 if n >= 3 and 2 < r < 8 if n = 2. Here n denotes the dimension and L-sigma(n) = {v is an element of L-n : div v = 0}. The case of more singular u is then studied. Assuming that n >= 3, u is an element of L-2, and div u is an element of L-n/2, we prove the existence, uniqueness and W-1,W-2- estimate of weak solutions. Our W-1,W-p- and W-1,W-2- results are optimal in some sense, as shown by counterexamples due to Moscariello [Adv. Calc. Var., 4 (2011), pp. 421-444].

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