The authors asked us to use the interpolation inequality and improve the interior Holder estimate. Here $d=\text{dist}(U, \partial \Omega)$. $$[Du]_{\alpha; U} \leq C d^{-\alpha}$$ to the following form $$[Du]_{\alpha; U} \leq C(d^{-1-\alpha} \sup_{\Omega} \vert u \vert + 1)$$ Could someone tell me how to apply the interpolation inequality in this context? To me it seems quite bizarre because interpolation usually improve the space sandwiched between two spaces, but the authors asked for same $\alpha$.
2026-03-26 14:16:16.1774534576
Gilbarg Trudinger Problem $13.1$
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