Newton Space, A Class Of Functional Minimal Regularity Problems

The Newton space is a generalization of the Sobolev space in a metric measure space,where the modulus of the gradient is replaced by a notion of the upper gradient.In this article we examine the regular problem of functions in Newton space that minimize the functional F(u,g_u) = ∫ f(u,g_u)dμ with for some c > 0. We prove that the minimizer belongs to De Giorgi class and then prove that the minimizer is local bounded and locally Holder continuous using the De Giorgi method.