Asymptotics for Solutions of Elliptic Equations in Double Divergence Form |
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Authors: | Vladimir Maz'ya |
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Affiliation: | 1. Department of Mathematics , Ohio State University ,;2. Department of Mathematical Sciences , University of Liverpool ,;3. Department of Mathematics , Link?ping University , |
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Abstract: | We consider weak solutions of an elliptic equation of the form ? i ? i (a ij u) = 0 and their asymptotic properties at an interior point. We assume that the coefficients are bounded, measurable, complex-valued functions that stabilize as x → 0 in that the norm of the matrix (a ij (x) ? δ ij ) on the annulus B 2r B r is bounded by a function Ω(r), where Ω2(r) satisfies the Dini condition at r = 0, as well as some technical monotonicity conditions; under these assumptions, solutions need not be continuous. Our main result is an explicit formula for the leading asymptotic term for solutions with at most a mild singularity at x = 0. As a consequence, we obtain upper and lower estimates for the L p -norm of solutions, as well as necessary and sufficient conditions for solutions to be bounded or tend to zero in L p -mean as r → 0. |
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Keywords: | Asymptotics of solutions Isolated singularity 2nd order elliptic equations |
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