On Some Interpolation Inequalities Due to Olga~Ladyzhenskaya and Nonlinear Partial Differential Equations

We consider some multiplicative interpolation inequalities between the H"older space and the Le\-bes\-gue space. Multiplicative interpolation inequalities of the Gagliardo--Nirenberg type are used in the investigations of partial differential equations. Several such inequalities involving the H"older norm (seminorm) were already proved and applied. In the present paper we generalise previous results to the anisotropic ``parabolic'' case with another simple proof due to idea of Olga~Ladyzhenskaya. The manuscript also contains an application of such Gagliardo--Nirenberg type inequality with the H"older norm. Some integral estimate and this inequality give a priori estimate of the solution to quasilinear parabolic problem in the smooth H"older classes. Moreover, using this a priori estimate, we establish the existence of solution of the quasilinear parabolic problem. In order to prove multiplicative inequality of the Gagliardo--Nirenberg type with the H"older norm we use an equivalent normalization of the higher order H"older spaces over higher order finite differences. The key technical tool is the representation of a function $u(x,t)$ at an arbitrary fixed point $(x,t)$ over a higher order finite difference at this point and the corresponding additional sum of values at neighboring points. After that we integrate with respect to the neighboring points over the balls $B_{r}((x,t))$ of small radius $r$. Estimating the finite difference over the corresponding H"older seminorm, we obtain an additive inequality with the parameter $r$, involving the H"older and integral norms. Optimizing this inequality over $r$ we get the multiplicative estimate of the Gagliardo--Nirenberg type with the H"older norm and the Le\-bes\-gue norm.