On the existence and Hölder regularity of solutions to some nonlinear Cauchy–Neumann problems

We prove uniform parabolic Hölder estimates of De Giorgi–Nash–Moser type for sequences of minimizers of the functionals Eε(W) = ∞ 0 e−t/ε ε RN+1 + ya ε|∂t W| 2 + |∇W| 2 dX + RN ×{0} (w) dx dt, ε ∈ (0, 1) where a ∈ (−1, 1) is a fixed parameter, RN+1 + is the upper half-space and dX = dxdy. As a consequence, we deduce the existence and Hölder regularity of weak solutions to a class of weighted nonlinear Cauchy– Neumann problems arising in combustion theory and fractional diffusion