Smoothness of solutions to the Dirichlet problem for a second-order elliptic equation with a square integrable boundary function

The global smoothness properties of weak solutions to the Dirichlet problem for the second-order linear elliptic equation with a integrable boundary function have been considered. It was assumed that the solutions to the equation have a stronger internal smoothness property that covers Hölder continuity, membership in the Sobolev space, and intermediate properties. The weak solutions to this Dirichlet problem are Hölder continuous inside a given bounded domain with an exponent depending only on space dimension and the ellipticity constant. The solutions to the Dirichlet problem with a square integrable boundary function were found to have properties that do not follow from Hölder continuity.