W<SUP>1P</SUP>-ESTIMATES FOR ELLIPTIC EQUATIONS WITH LOWER ORDER TERMS

초록

We consider the Neumann and Dirichlet problems for second-order linear elliptic equations -div (A del u)-b.del u + lambda u = f +div F, -div(A(t)del u)+div (ub) +lambda u = g +div G in a bounded Lipschitz domain Omega subset of R-n, n >= 2, where A : R-n -> R-n2 b : Omega -> R-n and lambda >= 0 are given. Some W-1,W-2 -estimates have been already known, provided that A is an element of L-infinity(Omega)(n2) and b is an element of L-r(Omega)(n), where n <= r < infinity if n > 3 and 2 < r < Do if n = 2. Under more regularity assumptions on A and Q, we establish the existence and uniqueness of weak solutions satisfying W--1,W-p -estimates. Our W--1,W-p -estimates are uniform on lambda >= 0 for the case of the Dirichlet problems. For the Neumann problems, the W--1,W-p -estimates are uniform with respect to A > 0 if f and g satisfy some compatibility conditions. These uniform estimates allow us to obtain strong stability results in W--1,W-p with respect to lambda for the Neumann and Dirichlet problems.

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