Authors’ summary: “For the admissible solutions \(u:\overline Q\to \mathbb{R}^1\) of the first initial-boundary value problem for nonlinear equations of the form \[ -u_t+ \sqrt{1+ u^2_x} S_m^{1/m} \bigl(k(u))= g\sqrt{1+u^2_x},\;m=2, \dots,n, \] a majorant for the absolute values of the second derivatives with respect to the spatial variables is evaluated on the parabolic boundary of the domain \(Q=\Omega \times(0,T)\), \(\Omega\subset \mathbb{R}^n\). In these equations, \(S_m\) is the \(m\)th elementary symmetric function, \(k(u)(x,t)= (k_1(u), \dots, k_n(u))(x,t)\) are the principal curvatures of the surfaces \({\mathcal T}_t(u): x_{n+1}= u(x,t)\), \(x\in \Omega\), in \(\mathbb{R}^{n+1}\), and \(g:\overline Q\to \mathbb{R}^1\) is a given function. A majorant \(M_1\) for \(\sup_Q | u_x|\) and two positive constants \(\underline M\) and \(\overline M\) in the inequalities \(0<\underline M\leq (u_t+g \sqrt{1+u^2_x}) (x,t)\leq \overline M\), \((x,t)\in \overline Q\), are viewed as known”.