ON THE STABILITY OF THE FRIEDRICHS INEQUALITY

We will discuss improvements to the classical Friedrichs inequality, which describes the continuity of the embedding of the Sobolev space $W_0^{1,p}(\Omega)$ into the Lebesgue space $L^q(\Omega)$, in a bounded domain $\Omega$. Namely, in [Bobkov, Kolonitskii, 2023], we obtained several independent versions of the improved Friedrichs inequality. One of them is based on the analysis of quadratic forms associated with the linearization of the p-Laplace operator. Another improvement is based on the stability of the so-called principle of hidden convexity. In both versions, the property of the new improving term is to describe the measure of the distance between a given function and the space of minimizers of the classical Friedrichs inequality. The obtained results are applied to study a nonlinear Fredholm alternative, in particular, to prove the existence of solutions in the resonant case. The talk is based on the work [Bobkov, Kolonitskii, 2023].