We will discuss certain variant of strongly nonlinear inequality of multiplicative type, which involves the elliptic operator P and its first order derivatives and can be considered as the nonlinear variant of the Gagliardo-Nirenberg interpolation inequality between derivatives. We will present some applications of such inequalities to the existence regularity theory for the nonlinear PDE’s of elliptic type, its relations with the coercive inequalities, and focus on certain related open problems in the theory of function spaces.

The talk will be based on the the chain of my joint works obtained together with Katarzyna Pietruska-Pałuba, Jan Peszek, Katarzyna Mazowiecka, Tomasz Choczewski, Alberto Fiorenza and Claudia Capogne, Tomas Roskovec and Dalmil Pesa.