On exponent in the Hölder condition for the first order derivatives of a solution to a linear elliptic second order equation with two variables

We study local properties of weak solutions in $W_2^2(D)$ to linear elliptic equations with measurable coefficients in the case of two independent variables. We prove that the derivatives of the weak solution to the homogeneous elliptic equation locally satisfies the Hölder condition with exponent ${\alpha_0}=\frac{{\sqrt{33 }-3}}{2}\frac{\nu }{{{\nu^2}+1}}$, where ν ∈ (0, 1] is the ellipticity constant. For inhomogeneous equations we obtain conditions under which the derivatives of a weak solution locally satisfy the Hölder condition with exponent α < α ₀. The sharpness of the results is confirmed by an example. Bibliography: 6 titles.