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We study the Hölder continuity of the homogeneous gradient of the weak solutions of double obstacle problems involving nonlinear elliptic operators in the Heisenberg group.

In this paper, we are interested in the parabolic equation u_t − ∆u = λu^q + u^p in a bounded domain of IR^N, with the Dirichlet boundary condition and the parameters 0 < q < 1 < p and λ > 0. We study the initial value problem and the global behavior of the the positive solutions. We are mainly interested in the relations between the global (in time) solutions of the parabolic equation and the solutions of the stationary, elliptic problem. We show in particular that there exists a global solution if and only if there exists a weak solution of the stationary equation.

In this paper we construct a weak solution for the heat-type equation of harmonic maps from a noncompact manifold into a sphere using the so-called minimizing movement method with the aid of the relative energy.