Regularity of data problems in edge distance weighted spaces via Hopf's uniform inequality and the duality principle.

Summary In the first part, we study the existence and nonexistence of an inequality called the Uniform Hopf Inequality (UHI), for a linear equation of the form Lv = f with measurable bounded coefficients and under homogeneous Dirichlet conditions. The IHU is a variant of the maximum principle, it was applied in the regularity proof W1.p 0 for a singular semilinear problem: Lu = F(u) where the coefficients of L are in the space vmor (evanescent mean-swinging functions) and F(u) is singular in u = 0 F(0) = +∞. Moreover, if the coefficients are lipschitzian, we prove that the optimal regularity of the gradient of the solution u is bmor (bounded mean oscillation functions i.e. Grad u in bmor).In the second part, we are interested in the regularity of the elastic system (stationary equations of elastic waves) with a singular source function in the sense that it is only integrable with respect to the distance function at the domain edge. Via duality, we show, according to ~f , that the problem admits a so-called very weak solution whose gradient is not necessarily integrable on the whole domain but only locally. We also determine the vector functions ~f for which ~u has its gradient integrable on the whole working space.