On the existence and Holder regularity of solutions to some nonlinear Cauchy-Neumann problems

We prove uniform parabolic Holder estimates of De Giorgi-Nash-Moser type for sequences of minimizers of the functionals epsilon(epsilon) (w) =integral((X))(0) e(-t/epsilon) /epsilon { integral(N+1)(R+) y(a) (epsilon|partial derivative(t) W|(2)) + |del W|(2) )dX integral(N)(R)(x{0}) Phi(w) dx }dt, epsilon is an element of (0,1) where a. (-1, 1) is a fixed parameter, R-+(N+1) is the upper half-space and dX = dxdy. As a consequence, we deduce the existence and Holder regularity of weak solutions to a class of weighted nonlinear CauchyNeumann problems arising in combustion theory and fractional diffusion.