Abstract
We have learned that the p-harmonic functions are Hölder continuous. In fact, much more regularity is valid. Even the gradients are locally Hölder continuous. In symbols, the function is of class \(C_{\text {loc}}^{1,\alpha }(\Omega )\).
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Notes
- 1.
The second Russian edition of the book [LU] by Ladyzhenskaya and Uraltseva includes the proof.
- 2.
For the equation
$$\sum a_{ij}(x, u,\nabla u)\frac{\partial ^2 u}{\partial x_{i} \partial x_{j}}\,=\, a_0(x, u,\nabla u)$$the Cordes Condition reads
$$\Bigl (\sum _{j=1}^{n} a_{jj}\Bigr )^2\,\ge \,(n-1+\delta )\sum _{i, j=1}^{n} a_{ij}^2.$$
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Lindqvist, P. (2019). Differentiability. In: Notes on the Stationary p-Laplace Equation. SpringerBriefs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-030-14501-9_4
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DOI: https://doi.org/10.1007/978-3-030-14501-9_4
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