The Dirichlet problem in weight spaces (Q1781723)

The paper concerns the solvability of the Dirichlet problem for quasilinear second order elliptic equations of non-divergence form with unbounded singularities in a domain \(\Omega\subset \mathbb R^n\). The main result is the following. Let \(1<p<\infty\), \(-\frac{1}{p}<\alpha<2-\frac{1}{p}\), and let \(\mathbb L_{\overline p,(\overline\alpha)}(\Omega)\) be the weighted space where the weight is equal to \((d(x))^{\overline\alpha}\), a power of the distance from the point \(x\) to the boundary \(\partial\Omega\). Here \(\overline p=\max\{n,p\}\) if \(p\not=n,\) \(\overline p=n+\epsilon\) if \(p=n\), \(\overline\alpha=\max\{1-\frac{n}{\overline p}-\epsilon,0\}\), \(\epsilon>0\). Let \(L u=-a^{ij}(x)D_iD_j u+b^i(x)D_i u\), where \(a^{ij}\in C(\overline\Omega),\) and the vector \((b^1,\cdots,b^n)\) belongs to \(\mathbb L_{\overline p,(\overline\alpha)}(\Omega)\). If \(f\in \mathbb L_{p,(\alpha)}(\Omega)\) then the boundary value problem \(Lu=f\) in \(\Omega\), \(u=0\) on \(\partial\Omega\) has a unique solution in a suitable weighted Sobolev space. The authors prove first some a priori estimates for the solutions and for the gradient.