Second order regularity for degenerate nonlinear elliptic equations
The authors study the second regularity of weak solutions of degenerate nonlinear elliptic equations of the form \[ -\text{div}\left(A\left(\left |\nabla u\right|\right)\nabla u\right)+b(x)\left| \nabla u\right|^q=f(x)\quad\text{in }\Omega, \] where \(\Omega\subseteq \mathbb{R}^N\) is an open set. The real function \(A: \mathbb{R}^+ \to \mathbb{R}^+\) is of class \(C^1\) such that \[ \text{lim\,sup}_{t\to 0^+} tA(t)<\infty \] and \[ -1<\inf_{t>0}\frac{tA'(t)}{A(t)}:=m_A \leq M_A:= \sup_{t>0} \frac{tA'(t)}{A(t)}<\infty. \] Furthermore, it is assumed that \[ A(s) \geq Ks^{\tilde{\vartheta}} \quad\text{for some }\tilde{\vartheta}\geq 0 \text{ and for all }s>0. \] When \(b(x)\) is not identically zero, then \(q>\frac{\tilde{\vartheta}+1}{2}\). The main aim of the authors is to study the summability of the second derivatives of the solutions which is stated in several theorems under different conditions.
scientific article; zbMATH DE number 6919416
scientific article