On differentiability of solutions to homogeneous elliptic equations of divergence type
A function \(u:\Omega\to\mathbb{R}\) is differentiable in the sense of convergence in \(W^1_2\) at some point \(a\in\Omega\) if there exists a linear map \(L:\mathbb{R}^n\to\mathbb{R}\) such that (for \(h\) small enough) the functions \(r_h(x):={1\over h}\{u(a+hx)-u(a)-hL(x)\}\) are of class \(W^1_2(B_1)\) and \(|r_h|_{W^1_2}\to 0\) as \(h\to 0\). Let \(u\in W^1_2(\Omega)\) now denote a weak solution of the elliptic equation \({\partial\over\partial x_i}(a_{ij}(x){\partial\over\partial x_j}u)=0\) on \(\Omega\). Suppose that the coefficients are bounded and elliptic and that, for some \(x_0\in\Omega\), we have (with constants \(k\), \(\alpha>0\)) \[ \int^k_0{\omega(t)\over t^{1+\alpha}} dt<\infty,\quad\omega(t):=\sup_{|x-x_0|\leq t} \sum^n_{i,j=1} |a_{ij}-a_{ij}(x_0)|. \] Then the solution is differentiable in the sense of convergence in \(W^1_2\) at \(x_0\). As a corollary, this implies differentiability of \(u\) at \(x_0\) in the classical sense.
differentiable in the sense of convergence in \(W^ 1_ 2\)
scientific article; zbMATH DE number 917136