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A priori bounds for elliptic operators in weighted Sobolev spaces. (English) Zbl 1261.35047

The authors study weighted estimates for second order elliptic equations, not necessarily in divergence form, on unbounded open sets. The precise conditions on the open sets \(\Omega\) and the weights \(m\) are given in the paper. A sufficient condition that \(m\) belongs to the class \(G(\Omega)\) for which the main results are established is that \(\log m \in \text{Lip}(\Omega)\). Examples of functions in \(G(\Omega)\) include \(m(x) = e^{t| x|}\) and \(m(x) = ( 1 + | x|^2)^t, x \in \Omega, t \in R\).
When \(m \in G(\Omega)\), the Sobolev space \(W^{k,p}_s(\Omega)\) is the set of all distributions \(u\) on \(\Omega\) such that \(m^s \partial^{\alpha}u \in L^p(\Omega),|\alpha | \leq k\) equipped with the obvious norm. \(W^{\circ, k,p}_s(\Omega)\) is the closure of the \(C^{\infty}(\Omega)\) functions with compact support in \(W^{k,p}_s(\Omega)\) and \(W^{0,p}_s(\Omega) = L^p_s(\Omega)\).
The authors prove a priori estimates of the type \[ \| u \|_{W^{2, p}_s(\Omega)} \leq c \left( \| Lu \|_{L^p_s(\Omega)} + \| u \|_{L^p_s(\Omega)} \right), \forall u \in W^{2, p}_s(\Omega) \cap W^{\circ, 1,p}_s(\Omega), \] when the coefficients of the elliptic operator \(L\) are bounded and locally in \(VMO\) on \(\Omega\), in addition to other technical conditions on the coefficients. The key to their results is multiplication results on the Sobolev spaces.

MSC:

35J15 Second-order elliptic equations
35B45 A priori estimates in context of PDEs
35R05 PDEs with low regular coefficients and/or low regular data
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