Parabolic Q-minima and minimal solutions to variational flow
The purpose of the paper is to provide a unified version of some regularity results on parabolic systems of the form \[ u'-\sum D_{x_ i}(D_{p_ i}F(x,u,Du))+D_ uF(x,u,Du)=0. \] To this aim, the notion of parabolic Q-minimum is introduced, where Q is a real number greater than or equal to 1. It is an adaptation to the parabolic case of the notion of (elliptic) Q-minimum introduced in \textit{M. Giaquinta} and \textit{E. Giusti} [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 1, 79-104 (1984; Zbl 0541.49008)]. Results on \(L^ p\)-summability of the spatial derivatives of u, Hölder continuity of u (in the scalar case) and partial Hölder continuity of u (in the vector case) are given for parabolic Q-minima. More special results are proved for the so-called minimal solutions, namely Q-minima with \(Q=1\). A characterization of the existence of such minimal solutions is given in terms of the convexity of the associated functional \(\int F(x,u,Du)dx\).
scientific article; zbMATH DE number 4103588
scientific article