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Research Report MRR99-046

An interior second derivative bound for solutions of Hessian equations

John Urbas

Abstract: In previous work we showed that weak solutions in $W^{2,p}(\Omega)$ of the $k$-Hessian equation $F_k[u]=g(x)$ have locally bounded second derivatives if $g$ is positive and sufficiently smooth and $p>kn/2$. Here we improve this result to $p>k(n-1)/2$, which is known to be sharp in the Monge-Amp\`ere case $k=n>2$.

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