Calderón-Zygmund estimates for generalized double phase problems
Let \(\Omega\subset \mathbb R^n\) be a bounded open domain with \(n\geq 2\). The paper deals with distributional solutions of the equation \[ -\operatorname{div}A(x,Du)=-\operatorname{div}B(x,F) \text{ in } \Omega , \tag{1} \] where \(A:\Omega\times\mathbb R^n\to\mathbb R^n\) is continuous, differentiable with respect to the second variable \(z\in \mathbb R^n\setminus \{0\}\), and \(B:\Omega \times \mathbb R^n\to\mathbb R^n\) is a Carathéodory vector field. To (1) is associated the model equation \[ -\operatorname{div}\Big(G^{'}(|Du|)\frac{Du}{|Du|}+a(x) H^{'}(|Du|)\frac{Du}{|Du|}\Big)= -\operatorname{div}\Big(G^{'}(|F|)\frac{F}{|F|}+a(x) H^{'}(|F|)\frac{F}{|F|}\Big) \tag{2} \] in \(\Omega\), where \(F:\Omega\to\mathbb R^n\) is a given vector-field. Sharp conditions are imposed on \(G\), \(H\) and \(a\), namely: \[ \sup_{t\geq 0}\frac{H(t)}{G(t)+G^{1+\frac{\alpha}{n}}(t)}<+\infty, \quad a\in C^{0,\alpha}(\Omega), \quad\alpha\in (0,1],\quad a(x)\geq 0. \] The notation \(\Psi (x,z)=G(|z|)+a(x) H(|z|)\) is introduced. In (1) sharp conditions are imposed on \(A\), \(B\), in terms of \(G\), \(H\) and \(a\). Then Calderon-Zygmund type estimates for distributional solutions \(u\) of (1) satisfying \[ \Psi(x,Du),\Psi(x,F)\in L^1(\Omega) \] are established.
scientific article; zbMATH DE number 7233281
scientific article