Let $B \subset \mathbb R^2$ be the unit ball and $T>0.$ Let $u \in W^{2,1}_p(B \times [0,T]),$ that is $u \in L^p(B \times [0,T])$ and we also have, $$ \partial_t u, \nabla u, \nabla^2 u \in L^p(B \times [0,T]). $$ Here $\nabla$ denotes differentiation in the spacial direction only.

I am looking for a proof of the following result:

If $p>4,$ then $\nabla u \in C^{\alpha,\alpha/2}(\overline B \times [0,T])$ and there is $C_p > 0$ such that $$ \sup_{\overline B \times [0,T]} |\nabla u| + \sup_{(x,t) \neq (y,s) \in \overline B \times [0,T]} \frac{|\nabla u(x,t) - \nabla u(y,s)|}{|x-y|^{\alpha} + |t-s|^{\alpha/2}} \leq C_p \left( \lVert u \rVert_{L^p(B \times [0,T])} + \lVert \nabla u \rVert_{L^p(B \times [0,T])} + \lVert \nabla^2 u \rVert_{L^p(B \times [0,T])} + \lVert \partial_t u \rVert_{L^p(B \times [0,T])} \right), $$ or in more compact (but possibly non-standard) notation $$\lVert \nabla u \rVert_{C^{\alpha,\alpha/2}(\overline B \times [0,T])} \leq C_p \lVert u \rVert_{W^{2,1}_p(B \times [0,T])},$$ where $\alpha = \left(1-\frac4p\right).$

This result was stated without a proof or reference as lemma 3.1 in the paper "The existence of heat flow of $H$-systems" by Chen and Levine. I presume it's well-known, but I've been unable to find a reference for it.

Some thoughts: My initial idea was to try to adapt one of the proofs of the usual Morrey-Sobolev embedding $W^{1,p} \hookrightarrow C^{1-n/p}$ separately in $x$ and $t,$ perhaps by breaking it up as, $$ |\nabla u(x,t) - \nabla u(y,s)| = |\nabla u(x,t) - \nabla u(x,s)| + |\nabla u(x,s) - \nabla u(y,s)|. $$ The exponent $\alpha$ suggests we apply Sobolev embedding in the $x$ variable with exponent $p/2,$ but it's not clear to do this uniformly in $t.$ Moreover this naiive approach obviously fails because we only have information about $\partial_t u$ and not its gradient. So some interpolation argument would be needed, which is where I'm stuck.

  • $\begingroup$ Maybe the following part could be useful for you. I have a similar problem and I was looking for some classic reference. Appendix E of Deterministic and Stochastic optimal control, W.H. Fleming & R. Rishel have a part on Holder estimates of parabolic type. $\endgroup$ – Cuoredicervo Jun 14 at 8:33
  • $\begingroup$ @Cuoredicervo Thank you for the reference, I will take a look when I have the time. I did end up finding a proof based on establishing a parabolic Poincare-Sobolev inequality (using the standard Rellich compactness theorem) and verifying a Camanato-type characterisation of Holder continuity which I've been meaning to post as an answer, but I haden't gotten around to doing so yet. $\endgroup$ – ktoi Jun 21 at 13:30

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