Regularity of the solutions of degenerate elliptic equations in divergent form
The paper under review is concerned with the Dirichlet problem \[ Lu=f_0 +\sum^n_{i=1} \frac{\partial f_i}{\partial x_i} \quad \text{in }D, \qquad u=0 \quad \text{on }\partial D, \] where \(D\subset \mathbb{R}^n\) is an arbitrary domain, and \[ Lu:= \sum^n_{i=1} \dfrac{\partial}{\partial x_i} (A_i(x,u,\nabla u)), \] \[ \sum^n_{i=1} A_i(x,u,\eta)\eta_i\geq \omega(x) | \eta| ^p \quad \forall\;x\in D, \;\forall u\in\mathbb{R}, \;\forall \eta \in \mathbb{R}^n \quad (1<p<+\infty). \] Here, \(\omega\) is a measurable, almost everywhere bounded and positive function in \(\mathbb{R}^n\) which is assumed to satisfy the condition \[ \Bigg(\int_{B_\rho(x)} \omega \,dy\Bigg) \Bigg( \int_{B_\rho(x)} \omega^{1-p'} \,dy\Bigg)^{p-1} \leq C_p \rho^{np} \quad \forall x\in \mathbb{R}^n, \quad \forall \rho>0 \quad (C_p = \text{const}). \] The authors consider weak solutions \(u\) to (1) within the weighted Sobolev space \(W^{1,p}_0(D,\omega)\) equipped with the norm \[ \| u\| _{W^{1,p}(D,\omega)} := \Bigg(\int_D | u| ^p \omega \,dx\Bigg)^{\frac{1}{p}} + \sum^n_{i=1} \Bigg( \int_\Omega \bigg| \frac{\partial u}{\partial x_i}\bigg| ^p \omega \,dx\Bigg)^{\frac{1}{p}}. \] The main results of the paper consist in establishing sufficient conditions on \(f_0\), \(f:= \sum^n_{i=1} | f_i| \), \(v\) and \(\omega\) for estimates on \[ \int_D | u| ^q \omega \,dx \quad \text{resp.} \quad \operatorname{ess\,sup} {D} | u| \] in terms of integrals over \(f_0, f, v\) and \(\omega\), where \(u\) is a weak solution to (1). Here, \(v\) is a measurable, almost everywhere bounded and positive function in \(\mathbb{R}^n\) such that \[ \rho^{1-n} \Bigg(\int_{B_\rho(x)} v\,dy\Bigg)^{1/s} \;\Bigg(\int_{B_\rho(x)} \omega^{1-p'} \,dy\Bigg)^{1/p'} \leq C_{p,s} \quad(=\text{const}). \]
scientific article; zbMATH DE number 5352311