Differentiability estimates and Conformally invariant operators: Difference between pages

From nonlocal pde
(Difference between pages)
Jump to navigation Jump to search
(Created page with "Given a fully nonlinear integro-differential equation $Iu=0$, uniformly elliptic with respect to certain class of operators, sometimes an interior $C^{1,\alpha}$ esti...")
 
imported>Tianling
No edit summary
 
Line 1: Line 1:
Given a [[fully nonlinear integro-differential equation]] $Iu=0$, [[uniformly elliptic]] with respect to certain [[class of operators]], sometimes an interior $C^{1,\alpha}$ estimate holds. Assume $I0=0$. The $C^{1,\alpha}$ estimate is a result like the following.
On a general compact Riemannian manifold $M$ with metric $g$, a metrically defined operator $A$ is said to be conformally invariant if under the conformal change in the metric $g_w=e^{2w}g$, the pair of the corresponding operators $A_w$ and $A$ are related by
\[
A_w(\varphi)=e^{-bw} A(e^{aw}\varphi)\quad\mbox{for all }\varphi \in C^{\infty}(M),
\]
where $a, b$ are constant.


'''Theorem'''. Let $u \in L^\infty(R^n) \cap C(\overline B_1)$ solve the equation \[Iu = 0 \ \ \text{in } B_1.\]
Examples of conformally invariant operators include:
Then $u \in C^{1,\alpha}(B_{1/2})$ and the following estimate holds
\[ ||u||_{C^{1,\alpha}(B_{1/2})} \leq C ||u||_{L^\infty}. \]


A theorem as above is known to hold under some assumptions on the [[nonlocal operator]] $I$. A list of valid assumptions is provided below.
* The conformal Laplacian:
\[
L_g=-\Delta_g + \frac{n-2}{4(n-1)}R_g,
\]
where $n$ is the dimension of the manifold, $-\Delta_g$ is the Laplace–Beltrami operator of $g$, and $R_g$ is the scalar curvature of $g$. This is a second order differential operator. One can check that in this case, $a=\frac{n-2}{2}$ and $b=\frac{n+2}{2}$.


Note that the result is stated for general [[fully nonlinear integro-differential equations]], but the most important cases to apply it are the [[Isaacs equation]] and [[Bellman equation]].
* The Paneitz operator <ref name="paneitz1983quartic"/> <ref name="paneitz2008quartic"/>:
\[
P=(-\Delta_g)^2-\mbox{div}_g (a_n R_g g+b_n Ric_g)d+\frac{n-4}{2}Q,
\]
where $\mbox{div}_g$ is the divergence operator, $d$ is the differential operator, $Ric_g$ is the Ricci tensor,
\[
Q=c_n|Ric_g|^2+d_nR_g^2-\frac{1}{2(n-2)}\Delta_gR
\]
and
\[
a_n=\frac{(n-2)^2+4}{2(n-1)(n-2)}, b_n=-\frac{4}{n-2}, c_n=-\frac{2}{(n-2)^2}, d_n=\frac{n^3-4n^2+16n-16}{8(n-1)^2(n-2)^2}.
\]
This is a fourth order operator with leading term $(-\Delta_g)^2$.


== Idea of the proof ==
* GJMS operators <ref name="GJMS"/>: this is a family of conformally invariant differential operators with leading term $(-\Delta_g)^k$ for all integers $k$ if $n$ is odd, and for $k\in \{1,2,\cdots,\frac{n}{2}\}$ if $n$ is even. A nonexistence result can be found in <ref name="gover2004conformally"/> for $k>\frac n2$ and $n\ge 4$ even. An explicit formula and a recursive formula each for GJMS operators and Q-curvatures have been found by Juhl <ref name="Juhl1"/><ref name="Juhl2"/> (see also Fefferman-Graham<ref name="FG13"/> ). The formula are more explicit when they are on the standard spheres.  
The idea to prove a $C^{1,\alpha}$ estimate is to apply [[Holder estimates]] to the derivatives of the solutions $u$. The directional derivatives $u_e$ satisfy the two inequalities
\[ M^+_{\mathcal L} u_e \geq 0 \text{ and } M^-_{\mathcal L} u_e \geq 0 \]
where $M^\pm_{\mathcal L}$ are the [[extremal operatos]] with respect to the corresponding class of operators $\mathcal L$. If the [[Holder estimates]] apply to this class of operators, one would expect that $u_e \in C^\alpha$ for any vector $e$, and therefore $u \in C^{1,\alpha}$.


There is a technical problem with the idea above. The Holder estimates indicate that $u_e$ is $C^\alpha$ in some $B_{1/2}$ provided that $u_e$ is already known to be bounded in $L^\infty(\R^n)$. In order to obtain the estimate starting from $u \in L^\infty(\R^n)$, one applies the Holder estimates successively to gain regularity at every step and then prove iteratively that $u \in C^\alpha \Rightarrow u \in C^{2\alpha} \Rightarrow u \in C^{3\alpha} \Rightarrow \dots \Rightarrow u \in C^{1,\alpha}$. The last step in the iteration illustrates the difficulty. Imagine that we have already proved that $u$ is Lipschitz in $B_{3/4}$, so we know that $u_e \in L^\infty(B_{3/4})$ for any vector $e$. This is not enough to apply the Holder estimates to $u_e$ since we would need $u_e \in L^\infty(\R^n)$.
*Scattering operators <ref name="graham2003scattering"/>, or the conformally invariant fractional powers of the Laplacian <ref name="chang2011fractional"/>: This is a family of conformally invariant pseudo-differential operators $P_\sigma$ defined on the conformal infinity of asymptotically hyperbolic manifolds with leading term $(-\Delta_g)^\sigma$ for all real numbers $\sigma\in (0,\frac n2)$ except at most finite values. The authors <ref name="chang2011fractional"/> reconciled the way of defining $P_\sigma$ in <ref name="graham2003scattering"/> and the localization method of Caffarelli-Silvestre  <ref name="CSextension"/> for the fractional Laplacian $(-\Delta)^\sigma$ in the Euclidean space $\mathbb{R}^n$.


The only known solution to this difficulty is to add an extra smoothness assumption to the family of kernels that allows to integrate by parts the tails in the integral representation of each linear operator $L u_e$ in the class $\mathcal L$ and write its tail as an integral in terms of $u$. It is an interesting [[open problem]] whether a better solution exist.


==Classes of kernels for which the estimate holds ==
Special cases:


=== Translation invariant, uniformly elliptic of order $s$, and some smoothness in the tails of the kernels ===
* On the Euclidean space $\mathbb{R}^n$: the operators mentioned above are just the fractional Laplacians.


The first situation in which the interior $C^{1,\alpha}$ estimate was proved for a nonlocal equation was if $I$ is translation invariant and [[uniformly elliptic]] with respect to the class of kernels satisfying the following hypothesis for some $\rho_0$ small enough<ref name="CS"/>.
* On the standard sphere $(\mathbb{S}^n, g_{\mathbb{S}^n})$ (which is the conformal infinity of the standar Poincare disk): they are the following intertwining operator <ref name="branson1987group"/> of explicit formula:
\begin{align*}
\[
\frac{(2-s)\lambda}{|y|^{n+s}} \leq K(y) &\leq \frac{(2-s)\Lambda}{|y|^{n+s}} && \text{(standard unif. ellipticity of order $s$)}\\
P_\sigma=\frac{\Gamma(B+\frac{1}{2}+\sigma)}{\Gamma(B+\frac{1}{2}-\sigma)},\quad B=\sqrt{-\Delta_{g_{\mathbb{S}^n}}+\left(\frac{n-1}{2}\right)^2},
\int_{\R^n \setminus B_{\rho_0}} \frac{|K(y)-K(y-h)|}{|h|} \mathrm d y &\leq C \qquad \text{every time $|h|<\frac {\rho_0} 2$} && \text{(kernel tails in $W^{1,1}$)}
\]
\end{align*}
where $\Gamma$ is the Gamma function and $\Delta_{g_{\mathbb{S}^n}}$ is the Laplace-Beltrami operator on $(\mathbb{S}^n, g_{\mathbb{S}^n})$. Moreover, the operator $P_{\sigma}$  
* is the pull back of $(-\Delta)^{\sigma}$ under stereographic projections,


=== Variant if the kernel tails are $C^1$ ===
* has the eigenfunctions of spherical harmonics, and


A small variation of the previous result is to assume the class of kernels satisfying the slightly stronger assumptions. A scale invariant class for which interior $C^{1,\alpha}$ regularity holds is <ref name="CS2"/>
* is the inverse of a spherical Riesz potential.
\begin{align*}
\frac{(2-s)\lambda}{|y|^{n+s}} \leq K(y) &\leq \frac{(2-s)\Lambda}{|y|^{n+s}} && \text{(standard unif. ellipticity of order $s$)}\\
\nabla K(y) &\leq \frac{\Lambda}{|y|^{n+s+1}} && \text{appropriate decay of the kernel in $C^1$.}
\end{align*}


Then, any solution of $Iu=0$ in $B_r$ satisfies the estimate
\[ [u]_{C^{1,\alpha}(B_{r/2})} \leq C \left(\frac 1 {r^{1+\alpha}} ||u||_{L^\infty(B_r)} + \frac 1 {r^{1+\alpha-s}} \int_{\R^n \setminus B_r} \frac{|u(y)|}{|y|^{n+s}} \mathrm d y \right). \]
Other $C^{1,\alpha}$ estimates are obtained from this one using [[perturbation methods]] <ref name="CS2"/>.


== References ==
== References ==
{{reflist|refs=
{{reflist|refs=
<ref name="CS">{{Citation | last1=Caffarelli | first1=Luis | last2=Silvestre | first2=Luis | title=Regularity theory for fully nonlinear integro-differential equations | url=http://dx.doi.org/10.1002/cpa.20274 | doi=10.1002/cpa.20274 | year=2009 | journal=[[Communications on Pure and Applied Mathematics]] | issn=0010-3640 | volume=62 | issue=5 | pages=597–638}}</ref>
 
<ref name="CS2">{{Citation | last1=Caffarelli | first1=Luis | last2=Silvestre | first2=Luis | title=Regularity results for nonlocal equations by approximation | publisher=[[Springer-Verlag]] | location=Berlin, New York | year=2009 | journal=Archive for Rational Mechanics and Analysis | issn=0003-9527 | pages=1–30}}</ref>
<ref name="branson1987group">{{Citation | last1=Branson | first1= Thomas P | title=Group representations arising from Lorentz conformal geometry | journal=Journal of functional analysis | year=1987 | volume=74 | pages=199--291}}</ref>
 
<ref name="CSextension">{{Citation | last1=Caffarelli | first1= Luis | last2=Silvestre | first2= Luis | title=An extension problem related to the fractional Laplacian | journal=Communications in Partial Differential Equations | year=2007 | volume=32 | pages=1245--1260}}</ref>
 
<ref name="chang2011fractional">{{Citation | last1=Chang | first1= Sun-Yung Alice | last2=González | first2= Maria del Mar | title=Fractional Laplacian in conformal geometry | journal=Advances in Mathematics | year=2011 | volume=226 | pages=1410--1432}}</ref>
 
<ref name="FG13">{{Citation | last1=Fefferman | first1= Charles | last2=Graham | first2= C | title=Juhl’s formulae for GJMS operators and 𝑄-curvatures | journal=Journal of the American Mathematical Society|year=2013 | volume=26 | pages=1191--1207}}</ref>
 
<ref name="gover2004conformally">{{Citation | last1=Gover | first1= A | last2=Hirachi | first2= Kengo | title=Conformally invariant powers of the Laplacian—a complete nonexistence theorem | journal=Journal of the American Mathematical Society |year=2004 |volume=17 | pages=389--405}}</ref>
 
<ref name="GJMS">{{Citation | last1=Graham | first1= C Robin | last2=Jenne | first2= Ralph | last3=Mason | first3= Lionel J | last4=Sparling | first4= George AJ | title=Conformally invariant powers of the Laplacian, I: Existence | journal=Journal of the London Mathematical Society | year=1992 | volume=2 | pages=557--565}}</ref>
 
<ref name="graham2003scattering">{{Citation | last1=Graham | first1= C Robin | last2=Zworski | first2= Maciej | title=Scattering matrix in conformal geometry | journal=Inventiones mathematicae | year=2003 | volume=152 | pages=89--118}}</ref>
 
<ref name="Juhl1">{{Citation | last1=Juhl | first1= Andreas | title=On the recursive structure of Branson’s Q-curvature | journal=arXiv preprint arXiv:1004.1784}}</ref>
 
<ref name="Juhl2">{{Citation | last1=Juhl | first1= Andreas | title=Explicit formulas for GJMS-operators and Q-curvatures | journal=Geometric and Functional Analysis | year=2013|volume=23 | pages=1278--1370}}</ref>
 
<ref name="paneitz1983quartic">{{Citation | last1=Paneitz | first1= S | title=A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds |year=1983 | journal=preprint}}</ref>
 
<ref name="paneitz2008quartic">{{Citation | last1=Paneitz | first1= S | title=A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds (summary) | url=http://dx.doi.org/10.3842/SIGMA.2008.036 | doi:10.3842/SIGMA.2008.036 | year=2008 | journal=SIGMA Symmetry Integrability Geom. Methods Appl. | issue=4 | Paper=036}}</ref>
 
}}
}}

Revision as of 22:05, 23 September 2013

On a general compact Riemannian manifold $M$ with metric $g$, a metrically defined operator $A$ is said to be conformally invariant if under the conformal change in the metric $g_w=e^{2w}g$, the pair of the corresponding operators $A_w$ and $A$ are related by \[ A_w(\varphi)=e^{-bw} A(e^{aw}\varphi)\quad\mbox{for all }\varphi \in C^{\infty}(M), \] where $a, b$ are constant.

Examples of conformally invariant operators include:

  • The conformal Laplacian:

\[ L_g=-\Delta_g + \frac{n-2}{4(n-1)}R_g, \] where $n$ is the dimension of the manifold, $-\Delta_g$ is the Laplace–Beltrami operator of $g$, and $R_g$ is the scalar curvature of $g$. This is a second order differential operator. One can check that in this case, $a=\frac{n-2}{2}$ and $b=\frac{n+2}{2}$.

\[ P=(-\Delta_g)^2-\mbox{div}_g (a_n R_g g+b_n Ric_g)d+\frac{n-4}{2}Q, \] where $\mbox{div}_g$ is the divergence operator, $d$ is the differential operator, $Ric_g$ is the Ricci tensor, \[ Q=c_n|Ric_g|^2+d_nR_g^2-\frac{1}{2(n-2)}\Delta_gR \] and \[ a_n=\frac{(n-2)^2+4}{2(n-1)(n-2)}, b_n=-\frac{4}{n-2}, c_n=-\frac{2}{(n-2)^2}, d_n=\frac{n^3-4n^2+16n-16}{8(n-1)^2(n-2)^2}. \] This is a fourth order operator with leading term $(-\Delta_g)^2$.

  • GJMS operators [3]: this is a family of conformally invariant differential operators with leading term $(-\Delta_g)^k$ for all integers $k$ if $n$ is odd, and for $k\in \{1,2,\cdots,\frac{n}{2}\}$ if $n$ is even. A nonexistence result can be found in [4] for $k>\frac n2$ and $n\ge 4$ even. An explicit formula and a recursive formula each for GJMS operators and Q-curvatures have been found by Juhl [5][6] (see also Fefferman-Graham[7] ). The formula are more explicit when they are on the standard spheres.
  • Scattering operators [8], or the conformally invariant fractional powers of the Laplacian [9]: This is a family of conformally invariant pseudo-differential operators $P_\sigma$ defined on the conformal infinity of asymptotically hyperbolic manifolds with leading term $(-\Delta_g)^\sigma$ for all real numbers $\sigma\in (0,\frac n2)$ except at most finite values. The authors [9] reconciled the way of defining $P_\sigma$ in [8] and the localization method of Caffarelli-Silvestre [10] for the fractional Laplacian $(-\Delta)^\sigma$ in the Euclidean space $\mathbb{R}^n$.


Special cases:

  • On the Euclidean space $\mathbb{R}^n$: the operators mentioned above are just the fractional Laplacians.
  • On the standard sphere $(\mathbb{S}^n, g_{\mathbb{S}^n})$ (which is the conformal infinity of the standar Poincare disk): they are the following intertwining operator [11] of explicit formula:

\[ P_\sigma=\frac{\Gamma(B+\frac{1}{2}+\sigma)}{\Gamma(B+\frac{1}{2}-\sigma)},\quad B=\sqrt{-\Delta_{g_{\mathbb{S}^n}}+\left(\frac{n-1}{2}\right)^2}, \] where $\Gamma$ is the Gamma function and $\Delta_{g_{\mathbb{S}^n}}$ is the Laplace-Beltrami operator on $(\mathbb{S}^n, g_{\mathbb{S}^n})$. Moreover, the operator $P_{\sigma}$

  • is the pull back of $(-\Delta)^{\sigma}$ under stereographic projections,
  • has the eigenfunctions of spherical harmonics, and
  • is the inverse of a spherical Riesz potential.


References

  1. ↑ Paneitz, S (1983), "A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds", preprint 
  2. ↑ Paneitz, S (2008), "A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds (summary)", SIGMA Symmetry Integrability Geom. Methods Appl. (4), http://dx.doi.org/10.3842/SIGMA.2008.036 
  3. ↑ Graham, C Robin; Jenne, Ralph; Mason, Lionel J; Sparling, George AJ (1992), "Conformally invariant powers of the Laplacian, I: Existence", Journal of the London Mathematical Society 2: 557--565 
  4. ↑ Gover, A; Hirachi, Kengo (2004), "Conformally invariant powers of the Laplacian—a complete nonexistence theorem", Journal of the American Mathematical Society 17: 389--405 
  5. ↑ Juhl, Andreas, "On the recursive structure of Branson’s Q-curvature", arXiv preprint arXiv:1004.1784 
  6. ↑ Juhl, Andreas (2013), "Explicit formulas for GJMS-operators and Q-curvatures", Geometric and Functional Analysis 23: 1278--1370 
  7. ↑ Fefferman, Charles; Graham, C (2013), "Juhl’s formulae for GJMS operators and 𝑄-curvatures", Journal of the American Mathematical Society 26: 1191--1207 
  8. ↑ 8.0 8.1 Graham, C Robin; Zworski, Maciej (2003), "Scattering matrix in conformal geometry", Inventiones mathematicae 152: 89--118 
  9. ↑ 9.0 9.1 Chang, Sun-Yung Alice; González, Maria del Mar (2011), "Fractional Laplacian in conformal geometry", Advances in Mathematics 226: 1410--1432 
  10. ↑ Caffarelli, Luis; Silvestre, Luis (2007), "An extension problem related to the fractional Laplacian", Communications in Partial Differential Equations 32: 1245--1260 
  11. ↑ Branson, Thomas P (1987), "Group representations arising from Lorentz conformal geometry", Journal of functional analysis 74: 199--291 
Retrieved from "https://web.ma.utexas.edu/mediawiki/index.php?title=Differentiability_estimates&oldid=1325"