Universal inequalities on domains in Euclidean space and their applications

In domains in Euclidean spaces, for test functions, we construct and prove several new Gagliardo-Nirenberg type inequalities with explicit constants. These inequalities are true in any domain, they are nonlinear, integrand functions involve the powers of the absolute values of the gradient and the Laplacian of a test function $u$, as well as factors of type $f(|u(x)|)$, $f'(|u(x)|)$, where $f$ is a continuously differentiable non-decaying function, $f(0)=0$. As weight functions, the powers of the distance from a point to the boundary of the domain serve as well as the powers of the varying hyperbolic (conformal) radius.
As applications of universal inequalities of Gagliardo-Nirenberg type we obtain new integral Rellich type inequalities in planar domains with uniformly perfect boundaries. For these Rellich type $L_p$-inequalities we establish criteria of the positivity of the constants, obtain two-sided estimates for these constants depending on the Euclidean maximal modulus of the domain and on the parameter $p\geq 2$. In the proof we use several scalar characteristics for domains with uniformly perfect boundaries.