Estimates of the integral modulus of continuity for the derivatives of solutions of elliptic equations. (Q2755971)

The author considers the linear equation \(Lu=f\) in \(G\), where \(Lu=\sum_{|k|\leq 2m}a_k(x)D^ku\) is a differential operator in general form with \(m\geq 1\), \(G\) is a bounded domain in \(\mathbb R^N\), the coefficients \(a_k(x)\) are Hölder continuous and \(f\in L^p(G)\) with \(p>1\) and satisfies an integral Hölder condition with exponent \(0<\alpha<1\) in \(L^p(G)\). He obtains a priori interior estimates for the moduli of continuity of derivatives of weak solutions (in the Sobolev space \(W^{2m,p}(G)\)) of this equation. An interesting fact is that, in contrast to some previous related results, the constant arising in the estimates does not depend on \(p\).