We prove uniform parabolic H\"older estimates of De Giorgi-Nash-Moser type for sequences of minimizers of the functionals \[ \mathcal{ E}_\varepsilon(W) = \int_0^\infty \frac{ e^{ - t/\varepsilon}}{ \varepsilon} \bigg\{ \int_{ \mathbb{ R}_+^{ N+1}} y^a \left( \varepsilon|\partial_t W|^2 + |\nabla W|^2 \right) dX + \int_{ \mathbb{ R}^N \times\{ 0\}} \Phi(w) dx \bigg\}dt, \qquad \varepsilon \in (0,1) \] where $a \in (-1,1)$ is a fixed parameter, $\mathbb{ R}_+^{ N+1}$ is the upper half-space and $dX = dxdy$. As a consequence, we deduce the existence and H\"older regularity of weak solutions to a class of weighted nonlinear Cauchy-Neumann problems arising in combustion theory and fractional diffusion.
Categories: math.AP