Very weak solutions of parabolic systems of \(p\)-Laplacian type
The authors study very weak solutions of a class of second-order parabolic systems \[ \frac{\partial u_i}{\partial t} = \operatorname {div} A_i(x,t,\nabla u) + B_i(x,t,\nabla u) \] including systems of \(p\)-Laplacian type. The term ``very weak solutions'' denotes functions which lie in a Sobolev space with exponent less than the natural Sobolev exponent and which satisfy the equation in the sense of distributions. The main result states that very weak solutions which are defined using a Sobolev space with appropriate exponent \(r\) are in fact weak solutions. This allows to get the usual regularity results under weaker assumptions. The proof of this result involves the construction of suitable test functions and the application of reverse Hölder inequalities.
scientific article; zbMATH DE number 1887463