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We prove uniform parabolic H\"older estimates of De Giorgi-Nash-Moser typefor 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 ofweighted nonlinear Cauchy-Neumann problems arising in combustion theory andfractional diffusion.

On the existence and H\"older regularity of solutions to some nonlinear Cauchy-Neumann problems