Equivalence between regularity theorems and heat kernel estimates for higher-order elliptic operators and systems under divergence form
The authors study the heat kernel of higher-order elliptic operators under divergence form of the type \[ L=\sum_{|\alpha|\leq m,|\beta|\leq m}(-1)^{|\alpha|}D^\alpha(a_{\alpha\beta}D^\beta) \] whose principal part satisfies the Gårding inequality. It is shown that for homogeneous operators Gaussian upper bounds and Hölder regularity of the heat kernel is equivalent to local regularity of weak solutions. In addition, it is studied the stability of such bounds under \(L^\infty\)-perturbations of the coefficients or under perturbation with bounded coefficients lower-order terms. There are obtained heat kernel bounds for operators with uniformly continuous or \(VMO\) coefficients. Analogous questions are discussed also for elliptic, in sense of the Gårding inequality, systems.
scientific article; zbMATH DE number 1556911
scientific article