In this article, we consider the existence of local and global solution to the Cauchy problem of a doubly nonlinear equation. By introducing the norms |||f||| h and 〈f〉h, we give the sufficient and necessary conditions on the initial value to the existence of local solution of doubly nonlinear equation. Moreover some results on the global existence and nonexistence of solutions are considered.
In this article, we consider nonlinear elliptic systems of divergence type with Dini continuous coefficients. The authors use a new method introduced by Duzaar and Grotowski, to prove partial regularity for weak solutions, based on a generalization of the technique of harmonic approximation and directly establish the optimal HSlder exponent for the derivative of a weak solution on its regular set.
This paper is devoted to the Tricomi problem for a mixed type equation of second order. The coeffcients are assumed to be discontinuous on the line where the type is changed. The unique existence of the solution to the problem is proved if the domain is small enough. Correspondingly, some estimates on the solution is also established.
In this paper, we are concerned with the partial regularity for the weak solutions of energy minimizing p-harmonic maps under the controllable growth condition. We get the interior partial regularity by the p-harmonic approximation method together with the technique used to get the decay estimation on some Degenerate elliptic equations and the obstacle problem by Tan and Yan. In particular, we directly get the optimal regularity.
