Regularity results for nonlocal equations and applications

Abstract

We introduce the concept of $C^{m,\alpha }$-nonlocal operators, extending the notion of second order elliptic operator in divergence form with $C^{m,\alpha }$-coefficients. We then derive the nonlocal analogue of the key existing results for elliptic equations in divergence form, notably the Hölder continuity of the gradient of the solutions in the case of $C^{0,\alpha }$-coefficients and the classical Schauder estimates for $C^{m+1,\alpha }$-coefficients. We further apply the regularity results for $C^{m,\alpha }$-nonlocal operators to derive optimal higher order regularity estimates of Lipschitz graphs with prescribed Nonlocal Mean Curvature. Applications to nonlocal equation on manifolds are also provided.

Appendix

Proof of Lemma 5.1

Case $2s+\alpha <2$. For simplicity, recalling (1.25) and (1.26), we assume that

$$\begin{aligned} \Vert A\Vert _{C^{k+2s+\alpha }(Q_\infty )\times L^\infty (S^{N-1} )}+ \Vert B\Vert _{{{\mathcal {C}}}^{k}_{\tau _s}(Q_\infty )\times L^\infty (S^{N-1} )} \le 1, \end{aligned}$$

where $\tau _s:=\alpha +(2s-1)_+$ if $2s\not =1$ and $\tau _{1/2}:=\alpha + \varepsilon$ if $2s=1$. We also assume that

$$\begin{aligned} \Vert u\Vert _{C^{k+2s+\alpha +\varepsilon _s}({\mathbb {R}}^N )}\le 1, \end{aligned}$$

where $\varepsilon _s:=0$ if $2s\not =1$ and $\varepsilon _s:=\varepsilon$ if $2s=1$.

We consider the case $m=0$. Since $\Vert u\Vert _{L^\infty ({\mathbb {R}}^N)}\le 1$, we have

$$\begin{aligned} |\delta ^e u(x,r,\theta )| \le C \min (1,r^{2s+\alpha }). \end{aligned}$$

(7.1)

Here, for $2s\ge 1$, we use the fact that $2 \delta ^e u(x,r)=r \int _0^1(\nabla u(x+t r \theta )-\nabla u(x-t r \theta ))\cdot \theta \, dt$. Moreover for $x_1,x_2\in {\mathbb {R}}^N$ and $r>0$, then for $2s+\alpha <1$, we have

$$\begin{aligned} |\delta ^e u(x_1,r,\theta )-\delta ^e u(x_2,r,\theta )| \le C \min (r^{2s+\alpha }, |x_1-x_2|^{2s+\alpha }) \end{aligned}$$

(7.2)

and if $2s\ge 1$, we have

$$\begin{aligned} |\delta ^e u(x_1,r,\theta )-\delta ^e u(x_2,r,\theta )| \le C \min (r^{2s+\alpha }, r |x_1-x_2|^{\tau _s}). \end{aligned}$$

(7.3)

On the other hand, for all $s\in (0,1)$,

$$\begin{aligned} |\delta ^o u(x,r,\theta ) | \le C \min (1,r )^{\min (2s+\alpha ,1)}, \end{aligned}$$

(7.4)

and

$$\begin{aligned} |\delta ^o u(x_1,r,\theta )-\delta ^o u(x_2,r,\theta )| \le C \min (r, |x_1-x_2| )^{\min (2s+\alpha ,1)}. \end{aligned}$$

(7.5)

Using (7.4), for $s\in (0,1)$, we estimate

$$\begin{aligned} |{{\mathcal {O}}}^s_{B,u}(x)|&\le C \int _0^\infty \min (r,1)^{\min (2s+\alpha ,1)} \min (r, 1)^{(2s-1)_++\alpha } r^{-1-2s}\, dr\\&\le C\int _0^1r^{\min (2s+\alpha ,1)} r ^{(2s-1)_++\alpha } r^{-1-2s}\, dr+ C \int _1^\infty r^{-1-2s}\, dr, \end{aligned}$$

so that,

$$\begin{aligned} \Vert {{\mathcal {O}}}^s_{B,u}\Vert _{L^\infty ({\mathbb {R}}^N)}\le C. \end{aligned}$$

(7.6)

We consider next ${{\mathcal {E}}}^s_{A,u}$. For all $x\in {\mathbb {R}}^N$ and for all $s\in (0,1)$, by (7.1), we have

$$\begin{aligned} |{{\mathcal {E}}}^s_{A,u}(x)|&\le C \int _0^\infty \min ( r^{2s+\alpha } ,1) r^{-1-2s}\, dr \le C \int _0^1r^{\alpha -1} \, dr+ C \int _1^\infty r^{-1-2s}\, dr, \end{aligned}$$

yielding

$$\begin{aligned} \Vert {{\mathcal {E}}}^s_{A,u}\Vert _{L^\infty ({\mathbb {R}}^N)}\le C. \end{aligned}$$

(7.7)

Let $x_1, x_2\in {\mathbb {R}}^N$ with $|x_1-x_2|\le 1$. Using (7.5), for $s\in (0,1)$ we have

$$\begin{aligned} |{{\mathcal {O}}}^s_{B,u}(x_1)&-{{\mathcal {O}}}^s_{B,u}(x_2)|\le C \int _0^\infty \min (r,|x_1-x_2|)^{\min (2s+\alpha ,1)} \min (r, 1)^{\tau _s} r^{-1-2s}\, dr\\&+C \int _0^\infty \min (r,1)^{\min (2s+\alpha ,1)} \min (r, |x_1-x_2|)^{\tau _s} r^{-1-2s}\, dr\\&\le C\int _0^{|x_1-x_2|}r^{\min (2s+\alpha ,1)+\tau _s} r^{-1-2s}\, dr +C|x_1-x_2|^{\min (2s+\alpha ,1) } \int _{|x_1-x_2|}^1 r^{\tau _s -1-2s } \, dr\\&+C|x_1-x_2|^{\tau _s } \int _{|x_1-x_2|}^1 r^{\min (2s+\alpha ,1)-1-2s } \, dr\\&+C |x_1-x_2|^{\min (2s+\alpha ,1) } \int _{1}^\infty r^{-1-2s}\, dr+C |x_1-x_2|^{ \tau _s } \int _{1}^\infty r^{-1-2s}\, dr\\&\le C |x_1-x_2|^{\alpha } . \end{aligned}$$

In the above estimate, it is used that $\tau _{s}=\alpha +\varepsilon$, for $s=1/2$.

This together with (7.6) imply that $\Vert {{\mathcal {O}}}^s_{B,u}\Vert _{C^{0,\alpha }({\mathbb {R}}^N)}\le C$, for all $s\in (0,1)$.

Now for $2s\ge 1$, let $x_1 \not = x_2\in {\mathbb {R}}^N$ with $|x_1-x_2|\le 1$. Using (7.3) and (7.1) we have

$$\begin{aligned}&|{{\mathcal {E}}}^s_{A,u}(x_1)-{{\mathcal {E}}}^s_{A,u}(x_2)|\\&\le C \int _0^\infty \min (r^{ 2s+\alpha } , r|x_1-x_2|^{ \tau _s} ) r^{-1-2s }\, dr +C |x_1-x_2|^{\alpha } \int _0^\infty \min (r ^{ 2s+\alpha },1 ) r^{-1-2s}\, dr\\&\le C \int _0^{|x_1-x_2|} r^{\alpha -1}\, dr+C|x_1-x_2|^{\tau _s}\int _{|x_1-x_2|}^\infty r^{-2s}\, dr +C |x_1-x_2|^{\alpha } \le C |x_1-x_2|^\alpha . \end{aligned}$$

Hence using (7.7), for $2s\ge 1$, we get $\Vert {{\mathcal {E}}}^s_{A,u}\Vert _{C^{0,\alpha }({\mathbb {R}}^N)}\le C$.

We now consider the case $2s+\alpha < 1$. For $x_1, x_2\in {\mathbb {R}}^N$, $|x_1-x_2|\le 1$, by (7.2), we estimate

$$\begin{aligned}&|{{\mathcal {E}}}^s_{A,u}(x_1)-{{\mathcal {E}}}^s_{A,u}(x_2)|\\&\le C\int _0^\infty \min (r,|x_1-x_2|)^{ 2s+\alpha } r^{-1-2s}\, dr +C|x_1-x_2|^{\alpha } \int _0^\infty \min (r ^{ 2s+\alpha },1 ) r^{-1-2s}\, dr\\&\le C \int _0^{|x_1-x_2|}r^{-1+\alpha } \, dr+C|x_1-x_2|^{2s+\alpha }\int _{|x_1-x_2|}^\infty r^{-1-2s}\, dr+C |x_1-x_2|^{\alpha } \le C |x_1-x_2|^\alpha . \end{aligned}$$

We then conclude from this and (7.7) that $\Vert {{\mathcal {E}}}^s_{A,u}\Vert _{C^{0,\alpha }({\mathbb {R}}^N)}\le C$, provided $2s+\alpha <1$.

If $m>1$, we can use the Leibniz formula for the derivatives of the product of two functions. Note that for all $\gamma \in {\mathbb {N}}^N$ with $|\gamma |\le m$, we have that $\delta ^e \partial ^\gamma u$ (resp. $\delta ^o \partial ^\gamma u$) satisfies (7.1) and (7.2) (resp. (7.4) and (7.5)).

Case $2s+\alpha >2$. We first observe from the arguments in the previous case that

$$\begin{aligned} \Vert {{\mathcal {E}}}_{A,u}^s\Vert _{L^{\infty }({\mathbb {R}}^N)}\le C \Vert A\Vert _{C^{0}(Q_\infty ) \times L^\infty (S^{N-1} )} \Vert u\Vert _{C^{2s+\alpha }({\mathbb {R}}^N)},\nonumber \\ \Vert {{\mathcal {O}}}_{B,u}^s\Vert _{L^{\infty }({\mathbb {R}}^N)} \le C \Vert A\Vert _{{{\mathcal {C}}}^{0}_{1}(Q_\infty )\times L^\infty (S^{N-1} )} \Vert u\Vert _{C^{2s+\alpha }({\mathbb {R}}^N)} . \end{aligned}$$

(7.8)

Since $B(y,0,\theta ) =0$, we have

$$\begin{aligned} B(x_1,r,\theta ) - B(x_2,r,\theta )= r\int _0^1 (D_rB(x_1,\varrho r,\theta ) - D_r B(x_2,\varrho r,\theta )) \, d\varrho . \end{aligned}$$

On the other hand

$$\begin{aligned} B(x_1,r,\theta ) - B(x_2,r,\theta )= \int _0^1 D_x B(\varrho x_1+(1-\varrho ) x_2, r,\theta )\cdot (x_1-x_2) \, d\varrho . \end{aligned}$$

The above two estimates yield

$$\begin{aligned}&|B(x_1,r,\theta ) - B(x_2,r,\theta )|\le ( \Vert B\Vert _{C^{2s+\alpha -1}(Q_\infty )\times L^\infty (S^{N-1} )} \nonumber \\&\quad + \Vert B\Vert _{{{\mathcal {C}}}^1_{2s+\alpha -2} (Q_\infty ) \times L^\infty (S^{N-1} )} ) \min (r|x_1-x_2|^{2s+\alpha -2}, r^{2s+\alpha -2} |x_1-x_2| ). \end{aligned}$$

(7.9)

In addition, we have

$$\begin{aligned} \delta ^o u(x_1,r,\theta ) - \delta ^o u(x_2,r,\theta ) = \int _0^1 D_x \delta ^ou(\varrho x_1+(1-\varrho ) x_2, r,\theta )\cdot (x_1-x_2) d\varrho , \end{aligned}$$

so that

$$\begin{aligned} | \delta ^o u(x_1,r,\theta ) - \delta ^o u(x_2,r,\theta ) ) | \le C \Vert u\Vert _{C^{2s+\alpha }({\mathbb {R}}^N)} \min (r, r^{2s+\alpha -2} |x_1-x_2|). \end{aligned}$$

Using this and (7.9), we find that, for all $x_1,x_2\in {\mathbb {R}}^N$,

$$\begin{aligned}&|{{\mathcal {O}}}_{B,u}^s(x_1)- {{\mathcal {O}}}_{B,u}^s(x_2)| \le C ( \Vert B\Vert _{C^{2s+\alpha -1}(Q_\infty )\times L^\infty (S^{N-1} )}\nonumber \\&\quad + \Vert B\Vert _{{{\mathcal {C}}}^1_{2s+\alpha -2} (Q_\infty ) \times L^\infty (S^{N-1} )} ) |x_1-x_2|^\alpha . \end{aligned}$$

(7.10)

Next, we write $2 \delta ^e u(x,r)=r \int _0^1(\nabla u(x+t r \theta )-\nabla u(x-t r \theta ))\cdot \theta \, dt$ from which we deduce that

$$\begin{aligned} \delta ^e u(x,r)&= r^2 \int _0^1 t \int _0^1 D^2_x \delta ^o u(x, \varrho t r, \theta ) [\theta ,\theta ] \, d\varrho dt \end{aligned}$$

and

$$\begin{aligned} \delta ^e u(x_1,r)- \delta ^e u(x_2,r)&=r \int _0^1 \int _0^1D^2_x \delta ^o u( \varrho x_1+ (1-\varrho ) x_2,t r, \theta )[x_1-x_2,\theta ] \, dt d\varrho . \end{aligned}$$

By combining the above two estimates, we get

$$\begin{aligned} | \delta ^e u(x_1,r)- \delta ^e u(x_2,r)| \le C \Vert u\Vert _{C^{2s+\alpha }({\mathbb {R}}^N)} \min (r^{2s+\alpha }, r^{2s+\alpha -1} |x_1-x_2| ). \end{aligned}$$

Using now the above estimate and the fact that $A\in {C^{\alpha }(Q_\infty ) \times L^\infty (S^{N-1} )}$, we immediately deduce that $[{{\mathcal {E}}}_{A,u}^s]_{C^\alpha ({\mathbb {R}}^N)}\le C \Vert A\Vert _{C^{\alpha }(Q_\infty )\times L^\infty (S^{N-1} )} \Vert u\Vert _{C^{2s+\alpha }({\mathbb {R}}^N)}$. From this, (7.8) and (7.10), we get the statement in the lemma for $m=0$ and $2s+\alpha >2$. In the general case that $m\ge 1$, we can use the Leibniz formula for the derivatives of the product of two functions and argue as above to get the desired estimates. $\square$