Explicit estimates for solutions of nonlinear radiation-type problems (Q2354172)

The author considers weak solutions of the boundary value problem \[ \begin{aligned}-\nabla\cdot(A\triangledown u)= f-\nabla\cdot\varphi\quad &\text{in }\Omega,\\ (A\triangledown u-\varphi)\cdot\nu+ b(u)= h\quad &\text{on }\Gamma,\\ (A\triangledown u-\varphi)\cdot\nu= g\quad &\text{on }\Gamma_N,\end{aligned} \] where \(\nu\) is the unit normal to the sufficiently smooth boundary \(\partial\Omega= \overline\Gamma_N\cup \overline\Gamma\) of the bounded domain \(\Omega\), the matrix \(A\in [L^\infty(\Omega)]^{n\times n}\) is uniformly elliptic and uniformly bounded, and \(b\) is a Carathéodory function with \((\ell-1)\)-growthness. Explicit expressions of the involved constants in the quantitative estimates for the solution are derived, which show the dependence on \(A\) and the size of \(\Omega\).