Some interior regularity results for solutions of Hessian equations
The paper contributes to the study of Hessian equations \(F_k[u]= \psi\) in a bounded domain \(\Omega\subset \mathbb{R}^n\), where \(F_k[u]= S_k(\lambda_1(u),\dots, \lambda_n(u))\); \(S_k\) is the \(k\)th elementary function, \(\lambda_1(u),\dots, \lambda_n(u)\) are eigenvalues of the Hessian matrix \(D^2u\) and \(\psi\) is a nonnegative function. After definining the notion of weak solution for a Hessian equation, two monotonicity formulae for such solutions are proved. These formulae are further used for the proof of the following regularity results: i) if \(u\in W^{2,q}(\Omega)\), \(q> n- 1\) (\(q\geq 2\) if \(n=2\)) is a weak solution of \(F_2[u]^{1/2}= g\in W^{2,q}(\Omega)\), then \(u\in C^{1,\alpha}(\Omega)\) for \(\alpha= 1-(n-1)/q\); ii) if \(n= 2\) and \(u\in W^{2,2}(\Omega)\) is a convex solution of the two-dimensional Monge-Ampère equation \((\text{det }D^2u)^{1/2}= g\in W^{2,2}(\Omega)\), then \(u\in C^{1,1/2}(\Omega)\); iii) if \(n= 2\) and \(u\in W^{2,2}(\Omega)\) is a convex solution of the equation of the prescribed Gaussian curvature \(\text{det }D^2u= K(1+|Du|^2)^2\), \(K\in W^{1,2}(\Omega)\cap L^\infty(\Omega)\), then \(u\in C^{1,1/2}(\Omega)\); iv) if \(u\in W^{2,p}(\Omega)\) is a weak solution of \(F_k[u]^{1/k}= g\in C^{1,1}(\Omega)\), \(k\in \{2,\dots, n\}\), \(g> 0\) in \(\Omega\) and if \(p> kn/2\), then \(u\in C^{1,1}(\Omega)\).
scientific article; zbMATH DE number 1525158