Nonlinear commutators for the fractional p-Laplacian and applications

Abstract

We prove a nonlinear commutator estimate concerning the transfer of derivatives onto testfunctions for the fractional p-Laplacian. This implies that solutions to certain degenerate nonlocal equations are higher differentiable. Also, weakly fractional p-harmonic functions which a priori are less regular than variational solutions are in fact classical. As an application we show that sequences of uniformly bounded $\frac{n}{s}$-harmonic maps converge strongly outside at most finitely many points.

Appendix A: Useful tools

The following Lemma is used to restrict the fractional p-Laplacian to smaller sets.

Lemma 8.1

(Localization lemma) Let $\Omega _1 \Subset \Omega _2 \Subset \Omega _3 \Subset \Omega \subset \mathbb {R}^n$ be open sets so that $\mathrm{dist\,}(\Omega _1,\Omega _2^c), \mathrm{dist\,}(\Omega _2,\Omega _3^c), \mathrm{dist\,}(\Omega _3,\Omega ^c) > 0$. Let $s \in (0,1)$, $p \in [2,\infty )$.

For any $u \in W^{s,p}(\Omega )$ there exists $\tilde{u} \in W^{s,p}(\mathbb {R}^n)$ so that

• (1) $\tilde{u} - u \equiv const$ in $\Omega _1$

• (2) $\mathrm{supp\,}\tilde{u} \subset \Omega _2$

• (3) $[\tilde{u}]_{W^{s,p}(\mathbb {R}^n)} \precsim \ [u]_{W^{s,p}(\Omega )}$

• (4) For any $t \in (2s-1,s)$,

  $$\begin{aligned} \Vert (-\Delta )^{s}_{p,\Omega _3} \tilde{u}\Vert _{(W^{t,p}_0(\Omega _3))^*} \precsim \Vert (-\Delta )^{s}_{p,\Omega } u\Vert _{(W^{t,p}_0(\Omega ))^*} + [u]_{W^{s,p}(\Omega )}^{p-1}. \end{aligned}$$

The constants are uniform in u and depend only on s, t, p and the sets $\Omega _1$, $\Omega _2$, $\Omega _3$, and $\Omega$.

Proof

Let $\Omega _1 \Subset \Omega$, let $\eta \equiv \eta _{\Omega _1} \in C_c^\infty (\Omega _2)$, $\eta _{\Omega _1} \equiv 1$ on $\Omega _1$. We set

$$\begin{aligned} \tilde{u} := \eta _{\Omega _1}(u-(u)_{\Omega _1}). \end{aligned}$$

Clearly $\tilde{u}$ satisfies property (1) and (2). We have property (3), too:

$$\begin{aligned}{}[\tilde{u}]_{W^{s,p}(\mathbb {R}^n)} \precsim [u]_{W^{s,p}(\Omega )}. \end{aligned}$$

We write

$$\begin{aligned} \tilde{u}(x)-\tilde{u}(y) = \underbrace{\eta (x) (u(x)-u(y))}_{a(x,y)} + \underbrace{(\eta (x)-\eta (y)) (u(y)-(u)_{\Omega _1})}_{b(x,y)}. \end{aligned}$$

Setting

$$\begin{aligned} T(a) := |a|^{p-2} a, \end{aligned}$$

observe that

$$\begin{aligned} |T(a+b) - T(a)| \precsim |b| \left( |a|^{p-2} + |b|^{p-2} \right) . \end{aligned}$$

Also note that

$$\begin{aligned} T(a(x,y)) = \eta ^{p-1}(x) |u(x)-u(y)|^{p-2}(u(x)-u(y)) \end{aligned}$$

We thus have for any $\varphi \in C_c^\infty (\Omega _3)$,

$$\begin{aligned}&(-\Delta )^{s}_{p,\Omega }\tilde{u}[\varphi ]\\&\quad =\int _{\Omega } \int _{\Omega } \frac{|\tilde{u}(x)-\tilde{u}(y)|^{p-2}(\tilde{u}(x)-\tilde{u}(y))\ (\varphi (x)-\varphi (y))}{|x-y|^{n+sp}}\ dx\ dy\\&\quad = \int _{\Omega } \int _{\Omega } \frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))\ \eta ^{p-1}(x)\ (\varphi (x)-\varphi (y))}{|x-y|^{n+sp}}\ dx\ dy\\&\qquad +\int _{\Omega } \int _{\Omega } \frac{(T(a+b) - T(a))\ (\varphi (x)-\varphi (y))}{|x-y|^{n+sp}}\ dx\ dy\\&\quad = \int _{\Omega } \int _{\Omega } \frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))\ ( \eta ^{p-1}(x) \varphi (x)- \eta ^{p-1}(y) \varphi (y))}{|x-y|^{n+sp}}\ dx\ dy\\&\qquad -\int _{\Omega } \int _{\Omega } \frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))\ (\eta ^{p-1}(x)-\eta ^{p-1}(y))\varphi (y)}{|x-y|^{n+sp}}\ dx\ dy\\&\qquad +\int _{\Omega } \int _{\Omega } \frac{(T(a+b) - T(a))\ (\varphi (x)-\varphi (y))}{|x-y|^{n+sp}}\ dx\ dy\\&\quad = (-\Delta )^{s}_{p,\Omega }u[\eta ^{p-1}\ \varphi ]\\&\quad \quad -\int _{\Omega } \int _{\Omega } \frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))\ (\eta ^{p-1}(x)-\eta ^{p-1}(y))\varphi (y)}{|x-y|^{n+sp}}\ dx\ dy\\&\quad \quad +\int _{\Omega } \int _{\Omega } \frac{(T(a+b) - T(a))\ (\varphi (x)-\varphi (y))}{|x-y|^{n+sp}}\ dx\ dy. \end{aligned}$$

Consequently,

$$\begin{aligned}&| (-\Delta )^{s}_{p,\Omega } \tilde{u}[\varphi ]|\\&\quad \precsim \Vert (-\Delta )^{s}_{p,\Omega }u\Vert _{(W^{t,p}_0(\Omega ))^*}\ [\eta ^{p-1} \varphi ]_{W^{t,p}(\Omega )}\\&\quad \quad + \int _{\Omega } \int _{\Omega } \frac{|u(x)-u(y)|^{p-1}\ |\eta ^{p-1}(x)-\eta ^{p-1}(y)|\ |\varphi (y)|}{|x-y|^{n+sp}}\ dx\ dy\\&\quad \quad +\int _{\Omega } \int _{\Omega } \frac{|\eta (x)-\eta (y)|\ |u(y)-(u)_{\Omega _1}|\ \eta (x)^{p-2}\ |u(x)-u(y)|^{p-2}\ |\varphi (x)-\varphi (y)|}{|x-y|^{n+sp}}\ dx\ dy\\&\quad \quad +\int _{\Omega } \int _{\Omega } \frac{|\eta (x)-\eta (y)|^{p-1}\ |u(y)-(u)_{\Omega _1}|^{p-1}\ |\varphi (x)-\varphi (y)|}{|x-y|^{n+sp}}\ dx\ dy. \end{aligned}$$

That is for any $t < s$

$$\begin{aligned}&| (-\Delta )^{s}_{p,\Omega } \tilde{u}[\varphi ]|\\&\quad \precsim \Vert (-\Delta )^{s}_{p,\Omega }u\Vert _{(W^{t,p}_0(\Omega ))^*}\ [\eta ^{p-1} \varphi ]_{W^{t,p}(\Omega )}\\&\quad \quad + [u]_{W^{s,p}(\Omega )}^{p-1}\ \left( \int _{\Omega } \int _{\Omega } \frac{|\eta ^{p-1}(x)-\eta ^{p-1}(y)|^p\ |\varphi (y)|^p}{|x-y|^{n+sp}}\ dx\ dy \right) ^{\frac{1}{p}}\\&\quad \quad +[\varphi ]_{W^{t,p}(\Omega )}\ [u]_{W^{s,p}(\Omega )}^{p-2} \left( \int _{\Omega } \int _{\Omega _2} \frac{|\eta (x)-\eta (y)|^{p}\ |u(y)-(u)_{\Omega _1}|^p}{|x-y|^{n+(2s-t)p}}\ dx\ dy \right) ^{\frac{1}{p}}\\&\quad \quad +[\varphi ]_{W^{t,p}(\Omega )} \left( \int _{\Omega } \int _{\Omega _2} \frac{|\eta (x)-\eta (y)|^{p}\ |u(y)-(u)_{\Omega _1}|^p}{|x-y|^{n+(2s-t)p}}\ dx\ dy \right) ^{\frac{p-1}{p}}. \end{aligned}$$

Since $\eta$ is bounded and Lipschitz, $\mathrm{supp\,}\eta \subset \Omega _2$, and $\varphi \in C_c^\infty (\Omega _3)$ we have that

$$\begin{aligned}{}[\eta ^{p-1} \varphi ]_{W^{t,p}(\Omega )} \precsim [\varphi ]_{W^{t,p}(\mathbb {R}^n)}. \end{aligned}$$

Also, choosing some bounded $\Omega _4 \Subset \Omega$ so that $\Omega _3 \Subset \Omega _4$,

$$\begin{aligned}&\int _{\Omega } \int _{\Omega } \frac{|\eta ^{p-1}(x)-\eta ^{p-1}(y)|^p\ |\varphi (y)|^p}{|x-y|^{n+sp}}\ dx\ dy\\&\quad \precsim \int _{\Omega _3} \int _{\Omega _4} |x-y|^{(1-s)p-n}\ \ dx\ |\varphi (y)|^p dy\\&\quad \quad +\int _{\Omega _3} \int _{\mathbb {R}^n\backslash \Omega _4} |x-y|^{-n-sp}\ \ dx\ |\varphi (y)|^p dy\\&\quad \precsim \Vert \varphi \Vert _{p}^p \precsim [\varphi ]_{W^{t,p}(\mathbb {R}^n)}^p. \end{aligned}$$

Finally, using Lipschitz continuity of $\eta$ and that $2s-1 < t < s$

$$\begin{aligned}&\int _{\Omega } \int _{\Omega _2} \frac{|\eta (x)-\eta (y)|^{p}\ |u(y)-(u)_{\Omega _1}|^p}{|x-y|^{n+(2s-t)p}}\ dx\ dy\\&\quad \precsim \int _{\Omega _3} |u(y)-(u)_{\Omega _1}|^p \int _{\Omega _2} |x-y|^{-n+(t+1-2s)p}\ dx\ dy\\&\quad \quad +\int _{\Omega \backslash \Omega _3} |u(y)-(u)_{\Omega _1}|^p \int _{\Omega _2} \frac{1 }{|x-y|^{n+sp}}\ dx\ dy\\&\quad \precsim \int _{\Omega _1} \int _{\Omega _3} |u(y)-u(z)|^p\ dy\ dz\\&\quad \quad +\int _{\Omega _1} \int _{\Omega \backslash \Omega _3} |u(y)-u(z)|^p \int _{\Omega _2} \frac{1 }{|x-y|^{n+sp}}\ dx\ dy\ dz\\ \end{aligned}$$

Note that for $x,z \in \Omega _2$ and $y \in \Omega _3^c$ we have that $|x-y| \approx |y-z|$, and since $\Omega _1,\Omega _2,\Omega _3$ are bounded we then have

$$\begin{aligned} \int _{\Omega } \int _{\Omega _2} \frac{|\eta (x)-\eta (y)|^{p}\ |u(y)-(u)_{\Omega _1}|^p}{|x-y|^{n+(2s-t)p}}\ dx\ dy \precsim [u]_{W^{s,p}(\Omega )}. \end{aligned}$$

Thus we have shown that for any $\varphi \in C_c^\infty (\Omega _3)$,

$$\begin{aligned} | (-\Delta )^{s}_{p,\Omega } \tilde{u}[\varphi ]| \precsim \left( \Vert (-\Delta )^{s}_{p,\Omega }u\Vert _{(W^{t,p}_0(\Omega ))^*} +[u]_{W^{s,p}(\Omega )}^{p-1} \right) \ [\varphi ]_{W^{t,p}(\mathbb {R}^n)}. \end{aligned}$$

Since moreover, $\mathrm{supp\,}\tilde{u} \subset \Omega _2$, for any $\varphi \in C_c^\infty (\Omega _3)$,

$$\begin{aligned} | (-\Delta )^{s}_{p,\Omega _3} \tilde{u}[\varphi ]| \precsim | (-\Delta )^{s}_{p,\Omega } \tilde{u}[\varphi ]| + [u]_{W^{s,p}(\Omega )}^{p-1}\ [\varphi ]_{W^{t,p}(\mathbb {R}^n)}, \end{aligned}$$

we get the claim. $\square$

The next Lemma estimates the $W^{s,p}$-norm in terms of the fractional p-Laplacian.

Lemma 8.2

Let $B \subset \mathbb {R}^n$ be a ball and 4B the concentric ball with four times the radius. Then for any $\delta > 0$, $[u]_{W^{s,p}(B)}^{p}$ can be estimated by

$$\begin{aligned}&\delta ^p [u]_{W^{s,p}(4B)}^{p}\\&\quad + \frac{C}{\delta ^{p'}} \left( \sup _{\varphi } \int _{4B}\int _{4B} \frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))\ (\varphi (x)-\varphi (y))}{|x-y|^{n+sp}}\ dx \ dy \right) ^{\frac{p}{p-1}}\\&\quad + \frac{C}{\delta ^{p'}}\ \mathrm{diam\,}(B)^{-sp}\ \int \limits _{4B} |u(x)-(u)_B|^p\ dx \end{aligned}$$

where the supremum is over all $\varphi \in C_c^\infty (2B)$ and $[\varphi ]_{W^{s,p}(\mathbb {R}^n)} \le 1$.

Proof

Let $\eta \in C_c^\infty (2B)$, $\eta \equiv 1$ in B be the usual cutoff function in 2B.

$$\begin{aligned} \psi (x) := \eta (x) (u(x)-(u)_{B}), \quad \text{ and } \quad \varphi (x) := \eta ^2(x) (u(x)-(u)_{B}). \end{aligned}$$

Then,

$$\begin{aligned}{}[\psi ]_{W^{s,p}(\mathbb {R}^n)} + [\varphi ]_{W^{s,p}(\mathbb {R}^n)} \precsim [u]_{W^{s,p}(2B)}. \end{aligned}$$

(8.1)

We have

$$\begin{aligned}{}[u]_{W^{s,p}(B)}^{p} \le \int \limits _{4B}\int \limits _{4B} \frac{|u(x)-u(y)|^{p-2} (\psi (x)-\psi (y))\ (\psi (x)-\psi (y))}{|x-y|^{n+sp}} dx\ dy. \end{aligned}$$

Now we observe

$$\begin{aligned} (\psi (x) - \psi (y))^2= & {} (\psi (x) - \psi (y)) (\eta (x) - \eta (y))(u(x)-(u)_{B})\\&+ \psi (x)(\eta (y)-\eta (x))\ (u(x)-u(y))\\&+ (\varphi (x) - \varphi (y))(u(x)-u(y)). \end{aligned}$$

That is,

$$\begin{aligned}{}[u]_{W^{s,p}(B)}^{p} \precsim I + II + III, \end{aligned}$$

with

$$\begin{aligned} I:= & {} \int \limits _{4B}\int \limits _{4B} \frac{|u(x)-u(y)|^{p-2} (u(x)-u(y))(\varphi (x)-\varphi (y))}{|x-y|^{n+sp}} dx\ dy,\\ II:= & {} \int \limits _{4B}\int \limits _{4B} \frac{|u(x)-u(y)|^{p-2} |\eta (x)-\eta (y)|\ |\psi (x)-\psi (y)|}{|x-y|^{n+sp}}\ |u(x)-(u)_B|\ dx\ dy,\\ III:= & {} \int \limits _{4B}\int \limits _{4B} \frac{|u(x)-u(y)|^{p-1} |\eta (x)-\eta (y)|}{|x-y|^{n+sp}}\ |\psi (x)| dx\ dy. \end{aligned}$$

With (8.1),

$$\begin{aligned} I \le [u]_{W^{s,p}(4B)}\ \sup _{[\varphi ]_{W^{s,p}(\mathbb {R}^n)} \le 1} \int \limits _{4B}\int \limits _{4B} \frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))\ (\varphi (x)-\varphi (y))}{|x-y|^{n+sp}}\ dx \ dy. \end{aligned}$$

As for II,

$$\begin{aligned} II \precsim \Vert \nabla \eta \Vert _{\infty }\ \int \limits _{4B}\int \limits _{4B} \frac{|u(x)-u(y)|^{p-2} |\psi (x)-\psi (y)|\ |u(x)-(u)_B| }{|x-y|^{n+sp-1}}\ \ dx\ dy. \end{aligned}$$

For any $t_2 > 0$ so that $t_2 = 1-s$, we have with Hölder’s inequality

Since $t_2 > 0$,

$$\begin{aligned} \int \limits _{4B}\int \limits _{4B} \frac{ |u(x)-(u)_B|^p }{|x-y|^{n- t_2p}}\ \ dx\ dy \precsim (\mathrm{diam\,}B) ^{t_2p}\ \int \limits _{4B} |u(x)-(u)_B|^p\ dx. \end{aligned}$$

So using again (8.1), we arrive at

$$\begin{aligned} II \precsim \mathrm{diam\,}(B)^{-s}\ [u]_{W^{s,p}(4B)}^{p-1}\ \left( \int \limits _{4B} |u(x)-(u)_B|^p\ dx \right) ^{\frac{1}{p}}. \end{aligned}$$

III can be estimated the same way as II, and we have the following estimate for $[u]_{W^{s,p}(B)}^{p}$

$$\begin{aligned}&[u]_{W^{s,p}(4B)}\ \sup _{\varphi } \int \limits _{4B}\int \limits _{4B} \frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))\ (\varphi (x)-\varphi (y))}{|x-y|^{n+sp}}\ dx \ dy\\&\quad +[u]_{W^{s,p}(4B)}^{p-1}\ \mathrm{diam\,}(B)^{-s}\ \left( \int \limits _{4B} |u(x)-(u)_B|^p\ dx \right) ^{\frac{1}{p}}. \end{aligned}$$

We conclude with Young’s inequality. $\square$

The next Proposition follows immediately from Jensen’s inequality and the definition of $[u]_{W^{t,p}(\lambda B)}^p$.

Proposition 8.3

(A Poincaré type inequality) Let B be a ball and for $\lambda \ge 1$ let $\lambda B$ be the concentric ball with $\lambda$ times the radius. Then for any $t \in (0,1)$, $p \in (1,\infty )$,

$$\begin{aligned} \int \limits _{\lambda B} |u(x)-(u)_B|^p\ dx \precsim \lambda ^{n+tp} \mathrm{diam\,}(B)^{t p} \ [u]_{W^{t,p}(\lambda B)}^p. \end{aligned}$$