Abstract
We show that W 2,p weak solutions of the k-Hessian equation F k (D 2 u) = g(x) with k≥ 2 can be approximated by smooth k-convex solutions v j of similar equations with the right hands sides controlled uniformly in C 0,1 norm, and so that the quantities \( \int_{B_r} (\Delta v_j)^{p-k+1} F_{k-1}(D^2v_j) \) are bounded independently of j. This result simplifies the proof of previous interior regularity results for solutions of such equations. It also permits us to extend certain estimates for smooth solutions of degenerate two dimensional Monge–Ampère equations to W 2,p solutions.
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Supported by an Australian Research Council Senior Fellowship.
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Urbas, J. An approximation result for solutions of Hessian equations. Calc. Var. 29, 219–230 (2007). https://doi.org/10.1007/s00526-006-0064-7
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DOI: https://doi.org/10.1007/s00526-006-0064-7