Fully nonlinear parabolic equations and the Dini condition
Let \(\Omega\) be a domain in \(\mathbb{R}^n\) and \(I=(a,b)\). This paper is concerned with viscosity solutions of the fully nonlinear parabolic equation \(u_t-F(D^2u,x,t)=g(x,t)\) in \(Q=\Omega\times I\), where \(F(M,x,t)\) is a real function defined on \(S\times Q\), \(S\) being the space of \(n\times n\) symmetric matrices, \(F(M,x,t)\) is uniformly elliptic in the usual meaning and satisfies \(F(0,0,0)=0\). The following function plays an important role: \(\theta(x,t)=\sup_{M\in S}|F(M,x,t)-F(M,0,0)|/(||M||+1)\). The function \(g(x,t)\) is supposed to be continuous and such that \(g(0,0)=0\). The following conditions are assumed: \((|Q_r|^{-1}\int_{Q_r}\theta^{n+1})^{1/(n+1)}\leq C_1\omega(r)\) and \((|Q_r|^{-1}\int_{Q_r}|g|^{n+1})^{1/(n+1)}\leq C_2\omega(r)\), where \(Q_r=\{|x|<r\}\times(-r^2,0)\) and \(\omega(r)\) is a Dini modulus of continuity satisfying \(\omega(1)=1\). Assuming that any solution of the equation \(\eta_t-F(D^2\eta+C,0,0)=D\), where \(-F(C,0,0)=D\), has a \(C^{2,\beta}\) interior estimates, the authors prove that \(u(x,t)\) is \(C^{2,\psi}\) at the origin, with \(\psi(t)=\omega(t)+\int_0^t\frac{\omega(r)}{r} dr\). More precisely, there is a polynomial \(P\) of degree 2 such that \(||u-P||_{L^\infty(Q_r)}\leq C_3r^2\psi(r).\)
scientific article; zbMATH DE number 1885846