Let \(D^2u\) be the Hessian matrix of a function \(u(x)\) defined in a domain \(\Omega\subset \mathbb{R}^n\). For \(1\leq k\leq n\), let \(S_k(\lambda [D^2u])\) be the \(k^{th}\) elementary symmetric function of the eigenvalues of the matrix \(D^2u\). Let \(\psi\) be a function of \(L^p(\Omega)\) for some \(p\geq 1\). The equation \(S_k(\lambda [D^2u])=\psi\) is investigated.
After a suitable definition of \(k\)-convexity for the domain \(\Omega\), the notion of weak solution is defined by approximation. The main result is the following. Let \(\Omega\) be a uniformly \((k-1)\)-convex domain for \(2\leq k\leq n\), \(\phi\in C^0(\overline \Omega)\) and \(\psi\) a nonnegative function in \(L^p(\Omega)\) for \(p\) greater than \(n/2k\). Then there exists a unique weak solution \(u\in C^0(\overline \Omega)\) satisfying \(u=\phi\) on \(\partial \Omega\). Furthermore, \(u\in C^\alpha(\Omega)\) for any exponent \(\alpha\) less than 1 and satisfying \(\alpha\leq 2-n/2k\). The proof uses interesting comparison estimates as well as isoperimetric inequalities for quermassintegrals.