About an uncommon theorem of Morrey

Question

I am searching for a reference with the following result: Let $\Omega \subset R^n$ an open bounded domain with smooth boundary . Let $2\leq p < n$ and $u \in W^{1,p}(\Omega)\cap L^{\infty}(\Omega)$. Suppose that for all ball $B$ $ \subset \Omega $ we have

$$ || \nabla u||_{L^p (B)} \leq M r^{n/p}$$ where $r$ is the radius of $B$ and $M$ is a constant that doesn't depend on $B$. Then $u$ is in some Holder space.

I only know some results of Morrey in the case $n<p$. I never saw the above result and there is a paper that I am reading a paper where the authors use this property without say a reference .

Someone could point me a reference with the result please?

Thanks in advance

Answer 1

Ok, here is it. We will exactly follow the proof of Morrey's theorem.

I will replace $$ \|\nabla u\|_{L^p(B)}\leq Mr^{N/p} $$ by $$ \|\nabla u\|_{L^p(Q)}\leq Mr^{N/p} \tag 1$$ where $Q$ is a cube with length $r$. I do this only intend to match the proof of Morry.

Now, for any $x\in Q$ where $Q$ is a cube with length $r$ and centered at origin, we have $$ \lvert u(x)-u(0)\rvert\leq \left\lvert\int_0^1 \nabla u(tx)dt\cdot x\right\rvert \leq r\int_0^1\lvert \nabla u(tx)\rvert dx$$ Next, we define $$ \bar u:=\frac{1}{|Q|}\int_0^1 u(z)\,dz $$ and we have $$\lvert \bar u-u(0)\rvert\leq \frac{r}{|Q|}\int_Q\int_0^1 \lvert \nabla u(tx)\rvert\,dtdx=\frac{1}{r^{N-1}}\int_0^1\int_Q\lvert \nabla u(tx)\rvert\,dxdt =\frac{1}{r^{N-1}}\int_0^1\int_{tQ}\lvert \nabla u(y)\rvert\,dy\frac{1}{t^N}dt $$ Now calling $(1)$, we have $$ \int_{tQ}\lvert \nabla u(y)\rvert\,dy\leq |tQ|^{\frac{1}{p'}}\left(\int_{tQ}\lvert \nabla u(y)\rvert^p\,dy \right)^{\frac{1}{p}}\leq |tQ|^{\frac{1}{p'}} (tr)^{\frac{N}{p}}$$ Hence, we have $$\lvert \bar u-u(0)\rvert\leq C\frac{1}{r^{N-1}}r^{\frac{N}{p'}}r^{\frac{N}{p}}\int_0^1 \frac{1}{t^{N}}t^{\frac{N}{p'}}t^{\frac{N}{p}}dt=C\frac{1}{r^{N-1}}r^{\frac{N}{p'}}r^{\frac{N}{p}}\int_0^1 1dt\leq C'r$$ where $C$ in an constant does not depends on $r$.

The rest will go exactly as the Morrey's proof. I suggest you to read Haïm Brezis' book, Page 282, theorem $9.12$

In the end, we actually conclude that you have Lipschitz continuous function, which is way more strong then Holder continuous.

Answer 2

Another way of looking at this is through the Lebesgue Differentiation theorem, that says: If $v\in L^p_{loc}(\Omega)$ then for a.e. $x\in \Omega$ we have $$ g_v(x,r):= \frac{1}{|B(x,r)|}\int_{B(x,r)} |v|^p dy \to |v(x)|^p \qquad \text{ as } r\to 0. $$ Your hypothesis implies that $g_{\nabla u} (x,r)\leq M$ and so $|\nabla u(x)|^p \leq M$ for a.e. $x$. Therefore $\nabla u \in L^\infty_{loc}(\Omega)$ and so $u$ is locally Lipschitz continuous (and therefore in every Hölder space of order $\alpha<1$).