Potential theory and harmonic analysis methods for quasilinear and Hessian equations

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Potential theory and harmonic analysis methods for quasilinear and Hessian equations

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Title: Potential theory and harmonic analysis methods for quasilinear and Hessian equations
Author: Nguyen, Phuc Cong, 1976-
Date: 2006
Publisher: University of Missouri--Columbia
Abstract: The existence problem is solved, and global pointwise estimates of solutions are obtained for quasilinear and Hessian equations of Lane-Emden type, including the following two model problems:-[delta]pu = uq + [mu], Fk[-u] = uq + [mu], u [greater than or equal to] 0, on Rn, or on a bounded domain [omega] [subset of or implied by] Rn. Here [delta]p is the p-Laplacian defined by [delta]pu = div ([delta]u [delta]u p-2), and Fk[u] is the k-Hessian defined as the sum of k x k principal minors of the Hessian matrix D2u (k = 1, 2, . . . , n); [mu] is a nonnegative measurable function (or measure) on [omega]. The solvability of these classes of equations in the renormalized (entropy) or viscosity sense has been an open problem even for good data [mu] [such that element of] Ls([omega]), s [greater than] 1. Such results are deduced from our existence criteria with the sharp exponents s = (n(q-p+1)) / pq for the first equation, and s = (n(q-k) q-k) / 2kq for the second one. Furthermore, a complete characterization of removable singularities for each corresponding homogeneous equation is given as a consequence of our solvability results.
URI: http://hdl.handle.net/10355/4402
Other Identifiers: NguyenP-042806-D4581

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