We prove uniform parabolic Hölder estimates of De Giorgi–Nash–Moser type for sequences of minimizers of the functionals

$$\begin{aligned} {\mathcal {E}}_\varepsilon (W) = \int _0^\infty \frac{e^{- t/\varepsilon }}{\varepsilon } \bigg \{ \int _{\mathbb {R}_+^{N+1}} y^a \left( \varepsilon |\partial _t W|^2 + |\nabla W|^2 \right) \textrm{d}X + \int _{\mathbb {R}^N \times \{0\}} \Phi (w) \,\textrm{d}x\bigg \}\,\textrm{d}t, \qquad \varepsilon \in (0,1) \end{aligned}$$

where \(a \in (-1,1)\) is a fixed parameter, \(\mathbb {R}_+^{N+1}\) is the upper half-space and \(\textrm{d}X = \textrm{d}x \textrm{d}y\). 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.

中文翻译：

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一些非线性柯西-诺依曼问题解的存在性及其霍尔德正则性

我们证明了泛函极小值序列的 De Giorgi-Nash-Moser 类型的均匀抛物线 Hölder 估计

$$\begin{对齐} {\mathcal {E}}_\varepsilon (W) = \int _0^\infty \frac{e^{- t/\varepsilon }}{\varepsilon } \bigg \{ \int _{\mathbb {R}_+^{N+1}} y^a \left( \varepsilon |\partial _t W|^2 + |\nabla W|^2 \right) \textrm{d}X + \int _{\mathbb {R}^N \times \{0\}} \Phi (w) \,\textrm{d}x\bigg \}\,\textrm{d}t, \qquad \varepsilon \在 (0,1) \end{对齐}$$

其中\(a \in (-1,1)\)是固定参数，\(\mathbb {R}_+^{N+1}\)是上半空间，\(\textrm{d} X = \textrm{d}x \textrm{d}y\)。因此，我们推导出燃烧理论和分数扩散中出现的一类加权非线性柯西-诺依曼问题弱解的存在性和霍尔德正则性。