The \(W^{1,2}_{( p,q )}\)-solvability for a class of fully nonlinear parabolic equations
The authors consider the following Cauchy-Dirichlet problem for fully nonlinear parabolic equations \[ \begin{cases} u_t*H(D^2u, Du, u, x, t)=0 & \text{ in } \Omega_T,\\ u=0 & \text{ on } \partial \Omega_T, \end{cases} \] where \(\Omega_T\) is a parabolic cylinder, \(\Omega\subset\mathbb{R}^d\), and \(\partial\Omega\in C^{1,1}\). The nonlinear term is supposed to be in BMO with respect to \((x,t)\). It is proved solvability in the Sobolev-Lorentz spaces \( W^{1,2}_{(p,q)}(\Omega_T) \) with \(p>d+1\), \(q>0\). The approach make use of the Fefferman-Stein inequality in the Lorentz spaces.
scientific article; zbMATH DE number 7379839
scientific article