Hölder estimates and Extension technique: Difference between pages

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Hölder continuity of the solutions can sometimes be proved only from ellipticity
The Dirichlet-to-Neumann operator for the upper half-plane maps the boundary value $U(x, 0)$ of a harmonic function $U(x, y)$ in the upper half-space $\R^{n+1}_+ = \R^n \times [0, \infty)$ to its outer normal derivative $-\partial_y U(x, 0)$. This operator coincides with the square root of the Laplace operator, $(-\Delta)^{1/2}$. Extension technique is a similar identification of non-local operators (most notably the [[fractional Laplacian]] $(-\Delta)^s$) as Dirichlet-to-Neumann operators for (possibly degenerate) elliptic equations. This construction is frequently used to turn nonlocal problems involving the fractional Laplacian into local problems in one more space dimension.
assumptions on the equation, without depending on smoothness of the
coefficients. This allows great flexibility in terms of applications of the
result. The corresponding result for elliptic equations of second order is the
[[Krylov-Safonov]] theorem in the non-divergence form, or the [[De Giorgi-Nash-Moser theorem]] in the divergence form.


The Hölder estimates are closely related to the [[Harnack inequality]]. In most cases, one can deduce the Hölder estimates from the Harnack inequality. However, there are simple example of integro-differential equations for which the Hölder estimates hold and the Harnack inequality does not <ref name="rang2013h" /> <ref name="bogdan2005harnack" />.
==Fractional Laplacian==


There are integro-differential versions of both [[De Giorgi-Nash-Moser theorem]]
The extension problem for the [[fractional Laplacian]] $(-\Delta)^s$, $s \in (0, 1)$ takes the following form.<ref name="CS"/> Let
and [[Krylov-Safonov theorem]]. The former uses variational techniques and is
$$U:\mathbb{R}^n \times \mathbb{R}_+ \longrightarrow \mathbb{R}$$
stated in terms of Dirichlet forms. The latter is based on comparison
be a function satisfying
principles.
\begin{equation}
\label{eqn:Main}
\nabla \cdot (y^{1-2s} \nabla U(x,y)) = 0
\end{equation}
on the upper half-space, lying inside the appropriately weighted Sobolev space $\dot{H}(1-2s,\mathbb{R}^{n+1}_+)$. Then if we let $u(x) = U(x,0)$, we have
\begin{equation}
\label{eqn:Neumann}
(-\Delta)^s u(x) = -C_{n,s} \lim_{y\rightarrow 0} y^{1-2s} \partial_y U(x,y).
\end{equation}
The energy associated with the operator in \eqref{eqn:Main} is  
\begin{equation}
\label{eqn:Energy}
\int y^{1-2s} |\nabla U|^2 dx dy
\end{equation}


A Hölder estimate says that a solution to an integro-differential equation with rough coefficients
The weight $y^{1-2s}$, for $0<s<1$, lies inside the Muckenhoupt $A_2$ class of weights. It is known that degenerate 2nd order elliptic PDEs with these weights satisfy many of the usual properties of uniformly elliptic PDEs, such as the maximum principle, the [[De Giorgi-Nash-Moser]] regularity theory, the [[boundary Harnack inequality]], and the Wiener criterion for regularity of a boundary point.<ref name="FKS"/><ref name="FKJ1"/><ref name="FKJ2"/>
$L_x u(x) = f(x)$ in $B_1$, is $C^\alpha$ in $B_{1/2}$ for some $\alpha>0$
(small). It is very important when an estimate allows for a very rough dependence of
$L_x$ with respect to $x$, since the result then applies to the linearization of
(fully) nonlinear equations without any extra a priori estimate. On the other
hand, the linearization of a [[fully nonlinear integro-differential equation]] (for example the [[Isaacs equation]] or the [[Bellman equation]]) would inherit the initial assumptions regarding for the kernels with
respect to $y$. Therefore, smoothness (or even structural) assumptions for the
kernels with respect to $y$ can be made keeping such result applicable.


In the non variational setting the integro-differential operators $L_x$ are
The translation invariance of the operator in the $x$-directions can be applied to obtain higher regularity results and Liouville type properties.<ref name="CSS"/>
assumed to belong to some family, but no continuity is assumed for its
dependence with respect to $x$. Typically, $L_x u(x)$ has the form
$$ L_x u(x) = a_{ij}(x) \partial_{ij} u + b(x) \cdot \nabla u + \int_{\R^n} (u(x+y) - u(x) - y \cdot \nabla u(x) \, \chi_{B_1}(y))
K(x,y) \, dy$$
Within the context of nonlocal equations, we would be interested on a regularization effect caused by the integral term and not the second order part of the equation. Because of that, the coefficients $a_{ij}(x)$ are usually assumed to be zero.


Since [[linear integro-differential operators]] allow for a great flexibility of
The above extension technique is closely related to the concept of trace of a diffusion on a hyperplane.<ref name="MO"/><ref name="D"/>
equations, there are several variations on the result: different assumptions on
the kernels, mixed local terms, evolution equations, etc. The linear equation
with rough coefficients is equivalent to the function $u$ satisfying two
inequalities for the [[extremal operators]] corresponding to the family of
operators $L$, which stresses the nonlinear character of the estimates.


As with other estimates in this field too, some Hölder estimates blow up as the
==Fractional powers of more general operators==
order of the equation converges to two, and others pass to the limit. The
In a similar way, extension problem for fractional powers $L^s$ of a general self-adjoint nonnegative linear differential operators $L$ in a domain $\Omega \subset \mathbb{R}^n$ (or more generally, a Hilbert space) can be constructed. In this case, the extension into a "cylinder" $\Omega \times [0, \infty)$ is considered. Let $U$ be a solution of
blow-up is a matter of the techniques used in the proof. Only estimates which
$$\begin{cases}
are robust are a true generalization of either the [[De Giorgi-Nash-Moser theorem]] or
\partial_y (y^{1-2s} \partial_y U(x, y)) - L_x U(x, y) = 0,&\hbox{in}~\Omega\times(0,\infty),\\
[[Krylov-Safonov theorem]].  
U(x,0)=u(x),&\hbox{on}~\Omega,
\end{cases}$$
with boundary conditions along $\partial \Omega \times [0,\infty)$ equal to the boundary conditions for $L$. Then
\begin{equation}
L^su(x) = C_{s} \lim_{y\rightarrow 0} y^{1-2s} U_y(x,y).
\end{equation}
If $L$ has a purely discrete spectrum on $\Omega$, the operator $(-L)^s$ has the same eigenfunctions as $L$, and its eigenvalues are $\{\lambda_i^s\}$, where the $\{\lambda_i\}$ are the eigenvalues of $(-L)$.


== The general statement ==
For example, $L$ can be the Dirichlet Laplacian in $\Omega$.<ref name="CT"/> Note that $(-L)^s$ is not the same as the fractional Laplacian, except when $\Omega = \mathbb{R}^n$.


=== Elliptic form ===
==More general non-local operators==
The general form of the Hölder estimates for an elliptic problem say that if we have an equation which holds in a domain, and the solution is globally bounded, then the solution is Hölder continuous in the interior of the domain. Typically this is stated in the following form: if $u : \R^d \to \R$ solves
Let $L$ be as above, and consider the Dirichlet-to-Neumann operator $A$ related to the elliptic equation
\[
\[ \partial_y (w(y) \partial_y U(x, y)) + L_x U(x, y) = 0 \]
L(u,x) = 0 \ \ \text{in } B_1,
in the upper half-plane. Then $A = f(-L)$ for some [[operator monotone function]] $f$. Conversely, for any operator monotone $f$, there is an appropriate extension problem for $f(-L)$. (This identification requires some conditions on $w(y)$ which ensure the extension problem is well-posed.)
\]
and $u \in L^\infty(\R^d)$, then for some small $\alpha > 0$,
\[ \|u\|_{C^\alpha(B_{1/2})} \leq C \|u\|_{L^\infty(\R^d)}.\]


There is no lack of generality in assuming that $L$ is a '''linear''' integro-differential operator, provided that there is no regularity assumption on its $x$ dependence.
The relation between $w$ and $f$ is equivalent to the Krein correspondence, and can be described as follows.<ref name="KW"/><ref name="SSV"/> For $\lambda \ge 0$, let $g_\lambda$ be the nonincreasing positive solution of the ODE
\[ \partial_y (w(y) \partial_y g_\lambda(y)) = \lambda g_\lambda(y) \]
for $y \ge 0$, satisfying $g_\lambda(0) = 1$. Furthermore, let $h$ be the nondecreasing solution of
\[ \partial_y (w(y) \partial_y h(y)) = 0 \]
satisfying $h(0) = 0$ and $h(1) = 1$. Then
\[ f(\lambda) = \lim_{y \to 0^+} \frac{1 - g_\lambda(y)}{h(y)} . \]
One can prove that $f$ defined above is operator monotone, and conversely, for any operator monotone $f$ one can find $w$ for which the above identity holds. Noteworthy, there are relatively few explicit pairs of corresponding $w$ and $f$.


For non variational problems, in order to adapt the situation to the [[viscosity solution]] framework, the equation may be replaced by two inequalities.
Suppose now that $U(x, y)$ is a sufficiently regular solution of the extension problem
\begin{align*}
\[ \partial_y (w(y) \partial_y U(x, y)) + L_x U(x, y) = 0 . \]
M^+u \geq 0 \ \ \text{in } B_1, \\
For simplicity, suppose that the spectrum of $L$ is discrete. Let $\varphi$ be an eigenfunction of $-L$ with eigenvalue $\lambda$, and denote $U_\varphi(y) = \langle U(\cdot, y), \varphi \rangle$. Then $U_\varphi$ is a solution of the ODE
M^-u \leq 0 \ \ \text{in } B_1.
\[ \partial_y (w(y) \partial_y U_\varphi(y)) + \lambda U_\varphi(y) = 0 . \]
\end{align*}
Hence, $U_\varphi(y) = U_\varphi(0) g_\lambda(y)$, and
where $M^+$ and $M^-$ are [[extremal operators]] with respect to some class.
\[ -\lim_{y \to 0^+} \frac{U_\varphi(y) - U_\varphi(0)}{h(y)} = f(\lambda) U_\varphi(0) . \]
It follows that
\[ -\lim_{y \to 0^+} \frac{U(\cdot, y) - U_\varphi(\cdot, 0)}{h(y)} = f(-L) U(\cdot, 0) , \]
or equivalently
\[ -\lim_{y \to 0^+} \frac{\partial_y U(\cdot, y)}{h'(y)} = f(-L) U(\cdot, 0) . \]
This proves that the Dirichlet-to-Neumann operator is indeed equal to $f(-L)$. The proof in the continuous spectrum case is similar.<ref name="K"/><ref name="KSV"/>


=== Parabolic form ===
[[Operator monotone function]]s, often called complete Bernstein functions, form a subclass of [[Bernstein function]]s. Hence, existence of the extension problem is closely related to the concept of [[subordination]]. Operators of the form $f(-\Delta)$ for an operator monotone $f$ admit an explicit [[Operator monotone function#Operator monotone functions of the Laplacian|description]]. This gives a fairly explicit condition for the existence of the extension problem for a given translation-invariant non-local operator.
The general form of the Hölder estimates for a parabolic problem is also an interior regularity statement for solutions of a parabolic equation. Typically this is stated in the following form: if $u : \R^d \times (-1,0] \to \R$ solves
\[
u_t - L(u,x) = 0 \ \ \text{in } (-1,0] \times B_1,
\]
and $u \in L^\infty(\R^d)$, then for some small $\alpha > 0$,
\[ \|u\|_{C^\alpha((-1/2,0] \times B_{1/2})} \leq C \|u\|_{L^\infty((-1,0] \times \R^d)}.\]


== List of results ==
==Relationship with Scattering operators==
There is an identification between the fractional Laplacian defined by the extension and the fractional Paneitz operator from Scattering Theory when the order of the operator is less than 1.<ref name="CG"/>
<!--This is a stub, to be expanded on later.-->


There are several Hölder estimates for elliptic and parabolic integro-differential equations which have been obtained. Here we list some of these results with a brief description of their main assumptions.
==References==
 
{{reflist|refs=
=== Variational equations ===
<ref name="CG">{{Citation | last1=González | first1=Maria del Mar | last2=Chang | first2=Sun-Yung Alice | title=Fractional Laplacian in conformal geometry | url=http://dx.doi.org/10.1016/j.aim.2010.07.016 | doi=10.1016/j.aim.2010.07.016 | year=2011 | journal=Advances in Mathematics | issn=0001-8708 | volume=226 | issue=2 | pages=1410–1432}}</ref>
A typical example of a symmetric nonlocal [[Dirichlet form]] is a bilinear form
<ref name="CS">{{Citation | last1=Caffarelli | first1=Luis | last2=Silvestre | first2=Luis | title=An extension problem related to the fractional Laplacian | url=http://dx.doi.org.ezproxy.lib.utexas.edu/10.1080/03605300600987306 | doi=10.1080/03605300600987306 | year=2007 | journal=Communications in Partial Differential Equations | issn=0360-5302 | volume=32 | issue=7 | pages=1245–1260}}</ref>
$E(u,v)$ satisfying
<ref name="CSS">{{Citation | last1=Caffarelli | first1=Luis | last2=Salsa | first2=Sandro | last3=Silvestre | first3=Luis | title=Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian | url=http://dx.doi.org/10.1007/s00222-007-0086-6 | doi=10.1007/s00222-007-0086-6 | year=2008 | journal=[[Inventiones Mathematicae]] | issn=0020-9910 | volume=171 | issue=2 | pages=425–461}}</ref>
$$ E(u,v) = \iint_{\R^n \times \R^n} (v(y)-u(x))(v(y)-v(x)) K(x,y) \, dx
<ref name="CT">{{Citation | last1=Tan | first1=Jinggang | last2=Cabré | first2=Xavier | title=Positive solutions of nonlinear problems involving the square root of the Laplacian | url=http://dx.doi.org/10.1016/j.aim.2010.01.025 | doi=10.1016/j.aim.2010.01.025 | year=2010 | journal=Advances in Mathematics | issn=0001-8708 | volume=224 | issue=5 | pages=2052–2093}}</ref>
\, dy $$
<ref name="FKS">{{Citation | last1=Fabes | first1=Eugene B. | last2=Kenig | first2=Carlos E. | last3=Serapioni | first3=Raul P. | title=The local regularity of solutions of degenerate elliptic equations | url=http://dx.doi.org/10.1080/03605308208820218 | doi=10.1080/03605308208820218 | year=1982 | journal=Communications in Partial Differential Equations | issn=0360-5302 | volume=7 | issue=1 | pages=77–116}}</ref>
on the closure of all $L^2$-functions with respect to $J(u)=E(u,u)$. Note
<ref name="FKJ1">{{Citation | last1=Fabes | first1=Eugene B. | last2=Kenig | first2=Carlos E. | last3=Jerison | first3=David | title=Conference on harmonic analysis in honor of Antoni Zygmund, Vol. I, II (Chicago, Ill., 1981) | publisher=Wadsworth | series=Wadsworth Math. Ser. | year=1983 | chapter=Boundary behavior of solutions to degenerate elliptic equations | pages=577–589}}</ref>
that $K$ can be assumed to be symmetric because the skew-symmetric part
<ref name="FKJ2">{{Citation | last1=Fabes | first1=Eugene B. | last2=Jerison | first2=David | last3=Kenig | first3=Carlos E. | title=The Wiener test for degenerate elliptic equations | url=http://www.numdam.org/item?id=AIF_1982__32_3_151_0 | year=1982 | journal=[[Annales de l'Institut Fourier|Université de Grenoble. Annales de l'Institut Fourier]] | issn=0373-0956 | volume=32 | issue=3 | pages=151–182}}</ref>
of $K$ would be ignored by the bilinear form.  
<ref name="D">{{Citation | last1=DeBlassie | first1=R. D. | title=The first exit time of a two-dimensional symmetric stable process from a wedge | url=http://dx.doi.org/10.1214/aop/1176990735 | doi=10.1214/aop/1176990735 | year=1990 | journal=Ann. Probab. | issn=0091-1798 | volume=18 | pages=1034–1070}}</ref>
 
<ref name="MO">{{Citation | last1=Molchanov | first1=S. A. | last2=Ostrovski | first2=E. | title=Symmetric stable processes as traces of degenerate diffusion processes | url=http://dx.doi.org/10.1137/1114012 | doi=10.1137/1114012 | year=1969 | journal=Theor Probab. Appl. | volume=14 | pages=128–131}}</ref>
Minimizers of the corresponding quadratic forms satisfy the nonlocal Euler
<ref name="KW">{{Citation | last1=Kotani | first1=S. | last2=Watanabe | first1=S. | title=Krein’s spectral theory of strings and
equation
generalized diffusion processes | url=http://dx.doi.org/10.1007/BFb0093046 | doi=10.1007/BFb009304 | year=1982 | pages=235–259 | booktitle=Functional Analysis in Markov Processes | series=Lecture Notes in Mathematics | volume=923 | editor1-last=Fukushima | editor1-first=M. | publisher=Springer, Berlin / Heidelberg | isbn=978-3-540-11484-0}}</ref>
$$ \lim_{\varepsilon \to 0} \int_{|x-y|>\varepsilon} (u(y) - u(x) ) K(x,y) \, dy = 0,$$
<ref name="SSV">{{Citation | last1=Schilling | first1=R. | last2=Song | first2=R. | last3=Vondraček | first3=Z. | title=Bernstein functions. Theory and Applications | year=2010 | publisher=de Gruyter, Berlin | series=Studies in Mathematics | volume=37 | url=http://dx.doi.org/10.1515/9783110215311 | doi=10.1515/9783110215311}}</ref>
which should be understood in the sense of distributions.
<ref name="K">{{Citation | last1=Kwaśnicki | first1=M. | title=Spectral analysis of subordinate Brownian motions on the half-line | year=2011 | journal=Studia Math. | volume=206 | pages=211–271 | url=http://dx.doi.org/10.4064/sm206-3-2 | doi=10.4064/sm206-3-2}}</ref>
 
<ref name="KSV">{{Citation | last1=Kim | first1=P. | last2=Song | first2=R. | last3=Vondraček | first3=Z. | title=On harmonic functions for trace processes | url=http://dx.doi.org/10.1002/mana.200910008 | doi=10.1002/mana.200910008 | year=2011 | journal=Math. Nachr. | volume=284 | pages=1889–1902}}</ref>
It is known that the gradient flow of a Dirichlet form (parabolic version of the
}}
result) becomes instantaneously Hölder continuous <ref name="CCV"/>. The method
of the proof builds an integro-differential version of the parabolic De Giorgi
technique that was developed for the study of critical [[Surface quasi-geostrophic equation]].  
 
At some point in the original proof of De Giorgi, it is used that the
characteristic functions of a set of positive measure do not belong to $H^1$.
Moreover, a quantitative estimate is required about the measure of
''intermediate'' level sets for $H^1$ functions. In the integro-differential
context, the required statement to carry out the proof would be the same with
the $H^{s/2}$ norm. This required statement is not true for $s$ small, and would
even require a non trivial proof for $s$ close to $2$. The difficulty is
bypassed though an argument that takes advantage of the nonlocal character of
the equation, and hence the estimate blows up as the order approaches two.
 
In the stationary case, it is known that minimizers of Dirichlet forms are
Hölder continuous by adapting Moser's proof of [[De Giorgi-Nash-Moser theorem]] to the
nonlocal setting <ref name="K"/>. In this result, the constants do not blow up as the order of the equation goes to two.


{{stub}}
{{stub}}
=== Non variational equations ===
The results below are nonlocal versions of [Krylov-Safonov theorem].  No regularity needs to be
assumed for $K$ with respect to $x$.
* The first result was obtained by Bass and Levin using probabilistic techniques.<ref
name="BL"/> It applies to elliptic integro-differential equations with symmetric and uniformly elliptic kernels (bounded pointwise). The constants obtained in the estimates are not uniform as the order of the equation goes to two. An extension of this result was obtained by Song and Vondracev <ref name="song2004" />.
The estimate says the following. Assume a bounded function $u: \R^n \to \R$ solves
$$ \int_{\R^n} (u(x+y) - u(x)) K(x,y) \mathrm{d}y = 0 \qquad \text{for all $x$ in } B_1, $$
where $K$ satisfies the symmetry assumption $K(x,y) = K(x,-y)$ and
\begin{equation} \label{pointwisebound}
\frac{\lambda}{|y|^{n+s}} \leq K(x,y) \leq \frac{\Lambda}{|y|^{n+s}} \qquad \text{for all } x,y \in \R^n,
\end{equation}
where $s \in (0,2)$ and $\Lambda \geq \lambda > 0$ are given parameters. Then
\[ \|u\|_{C^\alpha(B_{1/2})} \leq C \|u\|_{L^\infty(\R^n)}.\]
* A result by Bass and Kassmann also uses probabilistic techniques.<ref
name="BK"/> It applies to elliptic integro-differential equations with a rather general set of assumptions in the kernels. The main novelty is that the order of the equation may vary (continuously) from point to point. The constants obtained in the estimates are not uniform as the order of the equation goes to two.
* The first purely analytic proof was obtained by Luis Silvestre <ref
name="S"/>. The assumptions on the kernel are similar to those of Bass and Kassmann except that the order of the equation may change abruptly from point to point. The constants obtained in the estimates are not uniform as the order of the equation goes to two.
* The first estimate which remains uniform as the order of the equation goes to two was obtained by Caffarelli and Silvestre <ref name="CS"/>. The equations here are elliptic, with symmetric and uniformly elliptic kernels (pointwise bounded as in \ref{pointwisebound}).
* An estimate for parabolic equations of order one with a bounded drift was obtained by Silvestre <ref name="silvestre2011differentiability"/>. The equation here is parabolic, with symmetric and uniformly elliptic kernels (pointwise bounded as in \ref{pointwisebound}). The order of the equation is set to be one, but the proof also gives the estimate for any order greater than one, or also less than one if there is no drift. The constants in the estimates blow up as the order of the equation converges to two.
* Davila and Chang-Lara studied equations with nonsymmetric kernels, elliptic and parabolic. Their first result is for elliptic equations so that the odd part of the kernels is of lower order compared to their symmetric part <ref name="lara2012regularity" />. A later work provides estimates for parabolic equations which stay uniform as the order of the equation goes to two.<ref name="lara2014regularity" />. In a newer paper they improved both of their previous results by providing estimates for parabolic equations, with nonsymmetric kernels <ref name="chang2014h" />. For these three results,the kernels are required to be uniformly elliptic (pointwise bounded as in \ref{pointwisebound}).
* The first relaxation of the pointwise bound \eqref{pointwisebound} on the kernels was given by Bjorland, Caffarelli and Figalli <ref name="bjorland2012" /> for elliptic equations. They consider symmetric kernels which are bounded above everywhere, but are bounded below only in a cone of directions. A similar result is obtained by Kassmann, Rang and Schwab <ref name="rang2013h"/>.
* A generalization of all previous results was obtained by Schwab and Silvestre <ref name="schwab2014regularity" />. They consider parabolic equations. The estimates stay uniform as the order of the equation goes to two. The kernels are assumed to be bounded above only in average and bounded below only in sets of positive density. There is no symmetry assumption. It corresponds to the class of operators described [[Linear_integro-differential_operator#More singular/irregular kernels|here]].
* A non scale invariant family of kernels was studied by Kassmann and Mimica <ref name="kassmann2013intrinsic"/>. They study elliptic equations with symmetric kernels that satisfy pointwise bounds. However, the assumption is different from \eqref{pointwisebound} in the sense that $K(x,y)$ is required to be comparable to a function of $|y|$ which is not necessarily power-like.
== Other variants ==
* There are Holder estimates for equations in divergence form that are non local in time <ref name="zacher2013" />
* If we allow for continuous dependence on the coefficients with respect to $x$, there are Hölder estimates for a rather singular class of integral equations <ref name="barles2011" />.
== References ==
{{reflist|refs=
<ref name="S">{{Citation | last1=Silvestre | first1=Luis | title=Hölder
estimates for solutions of integro-differential equations like the fractional
Laplace | url=http://dx.doi.org/10.1512/iumj.2006.55.2706 |
doi=10.1512/iumj.2006.55.2706 | year=2006 | journal=Indiana University
Mathematics Journal | issn=0022-2518 | volume=55 | issue=3 |
pages=1155–1174}}</ref>
<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="CCV">{{Citation | last1=Caffarelli | first1=Luis | last2=Chan |
first2=Chi Hin | last3=Vasseur | first3=Alexis | title= |
doi=10.1090/S0894-0347-2011-00698-X | year=2011 | journal=[[Journal of the
American Mathematical Society]] | issn=0894-0347 | issue=24 |
pages=849–869}}</ref>
<ref name="K">{{Citation | last1=Kassmann | first1=Moritz | title=A priori
estimates for integro-differential operators with measurable kernels |
url=http://dx.doi.org/10.1007/s00526-008-0173-6 | doi=10.1007/s00526-008-0173-6
| year=2009 | journal=Calculus of Variations and Partial Differential Equations
| issn=0944-2669 | volume=34 | issue=1 | pages=1–21}}</ref>
<ref name="BK">{{Citation | last1=Bass | first1=Richard F. | last2=Kassmann |
first2=Moritz | title=Hölder continuity of harmonic functions with respect to
operators of variable order | url=http://dx.doi.org/10.1080/03605300500257677 |
doi=10.1080/03605300500257677 | year=2005 | journal=Communications in Partial
Differential Equations | issn=0360-5302 | volume=30 | issue=7 |
pages=1249–1259}}</ref>
<ref name="BL">{{Citation | last1=Bass | first1=Richard F. | last2=Levin |
first2=David A. | title=Harnack inequalities for jump processes |
url=http://dx.doi.org/10.1023/A:1016378210944 | doi=10.1023/A:1016378210944 |
year=2002 | journal=Potential Analysis. An International Journal Devoted to the
Interactions between Potential Theory, Probability Theory, Geometry and
Functional Analysis | issn=0926-2601 | volume=17 | issue=4 |
pages=375–388}}</ref>
<ref name="zacher2013">{{Citation | last1=Zacher | first1= Rico | title=A De Giorgi--Nash type theorem for time fractional diffusion equations | url=http://dx.doi.org/10.1007/s00208-012-0834-9 | journal=Math. Ann. | issn=0025-5831 | year=2013 | volume=356 | pages=99--146 | doi=10.1007/s00208-012-0834-9}}</ref>
<ref name="barles2011">{{Citation | last1=Barles | first1= Guy | last2=Chasseigne | first2= Emmanuel | last3=Imbert | first3= Cyril | title=H\"older continuity of solutions of second-order non-linear elliptic integro-differential equations | url=http://dx.doi.org/10.4171/JEMS/242 | journal=J. Eur. Math. Soc. (JEMS) | issn=1435-9855 | year=2011 | volume=13 | pages=1--26 | doi=10.4171/JEMS/242}}</ref>
<ref name="rang2013h">{{Citation | last1=Rang | first1= Marcus | last2=Kassmann | first2= Moritz | last3=Schwab | first3= Russell W | title=H$\backslash$" older Regularity For Integro-Differential Equations With Nonlinear Directional Dependence | journal=arXiv preprint arXiv:1306.0082}}</ref>
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Revision as of 14:09, 5 August 2015

The Dirichlet-to-Neumann operator for the upper half-plane maps the boundary value $U(x, 0)$ of a harmonic function $U(x, y)$ in the upper half-space $\R^{n+1}_+ = \R^n \times [0, \infty)$ to its outer normal derivative $-\partial_y U(x, 0)$. This operator coincides with the square root of the Laplace operator, $(-\Delta)^{1/2}$. Extension technique is a similar identification of non-local operators (most notably the fractional Laplacian $(-\Delta)^s$) as Dirichlet-to-Neumann operators for (possibly degenerate) elliptic equations. This construction is frequently used to turn nonlocal problems involving the fractional Laplacian into local problems in one more space dimension.

Fractional Laplacian

The extension problem for the fractional Laplacian $(-\Delta)^s$, $s \in (0, 1)$ takes the following form.[1] Let $$U:\mathbb{R}^n \times \mathbb{R}_+ \longrightarrow \mathbb{R}$$ be a function satisfying \begin{equation} \label{eqn:Main} \nabla \cdot (y^{1-2s} \nabla U(x,y)) = 0 \end{equation} on the upper half-space, lying inside the appropriately weighted Sobolev space $\dot{H}(1-2s,\mathbb{R}^{n+1}_+)$. Then if we let $u(x) = U(x,0)$, we have \begin{equation} \label{eqn:Neumann} (-\Delta)^s u(x) = -C_{n,s} \lim_{y\rightarrow 0} y^{1-2s} \partial_y U(x,y). \end{equation} The energy associated with the operator in \eqref{eqn:Main} is \begin{equation} \label{eqn:Energy} \int y^{1-2s} |\nabla U|^2 dx dy \end{equation}

The weight $y^{1-2s}$, for $0<s<1$, lies inside the Muckenhoupt $A_2$ class of weights. It is known that degenerate 2nd order elliptic PDEs with these weights satisfy many of the usual properties of uniformly elliptic PDEs, such as the maximum principle, the De Giorgi-Nash-Moser regularity theory, the boundary Harnack inequality, and the Wiener criterion for regularity of a boundary point.[2][3][4]

The translation invariance of the operator in the $x$-directions can be applied to obtain higher regularity results and Liouville type properties.[5]

The above extension technique is closely related to the concept of trace of a diffusion on a hyperplane.[6][7]

Fractional powers of more general operators

In a similar way, extension problem for fractional powers $L^s$ of a general self-adjoint nonnegative linear differential operators $L$ in a domain $\Omega \subset \mathbb{R}^n$ (or more generally, a Hilbert space) can be constructed. In this case, the extension into a "cylinder" $\Omega \times [0, \infty)$ is considered. Let $U$ be a solution of $$\begin{cases} \partial_y (y^{1-2s} \partial_y U(x, y)) - L_x U(x, y) = 0,&\hbox{in}~\Omega\times(0,\infty),\\ U(x,0)=u(x),&\hbox{on}~\Omega, \end{cases}$$ with boundary conditions along $\partial \Omega \times [0,\infty)$ equal to the boundary conditions for $L$. Then \begin{equation} L^su(x) = C_{s} \lim_{y\rightarrow 0} y^{1-2s} U_y(x,y). \end{equation} If $L$ has a purely discrete spectrum on $\Omega$, the operator $(-L)^s$ has the same eigenfunctions as $L$, and its eigenvalues are $\{\lambda_i^s\}$, where the $\{\lambda_i\}$ are the eigenvalues of $(-L)$.

For example, $L$ can be the Dirichlet Laplacian in $\Omega$.[8] Note that $(-L)^s$ is not the same as the fractional Laplacian, except when $\Omega = \mathbb{R}^n$.

More general non-local operators

Let $L$ be as above, and consider the Dirichlet-to-Neumann operator $A$ related to the elliptic equation \[ \partial_y (w(y) \partial_y U(x, y)) + L_x U(x, y) = 0 \] in the upper half-plane. Then $A = f(-L)$ for some operator monotone function $f$. Conversely, for any operator monotone $f$, there is an appropriate extension problem for $f(-L)$. (This identification requires some conditions on $w(y)$ which ensure the extension problem is well-posed.)

The relation between $w$ and $f$ is equivalent to the Krein correspondence, and can be described as follows.[9][10] For $\lambda \ge 0$, let $g_\lambda$ be the nonincreasing positive solution of the ODE \[ \partial_y (w(y) \partial_y g_\lambda(y)) = \lambda g_\lambda(y) \] for $y \ge 0$, satisfying $g_\lambda(0) = 1$. Furthermore, let $h$ be the nondecreasing solution of \[ \partial_y (w(y) \partial_y h(y)) = 0 \] satisfying $h(0) = 0$ and $h(1) = 1$. Then \[ f(\lambda) = \lim_{y \to 0^+} \frac{1 - g_\lambda(y)}{h(y)} . \] One can prove that $f$ defined above is operator monotone, and conversely, for any operator monotone $f$ one can find $w$ for which the above identity holds. Noteworthy, there are relatively few explicit pairs of corresponding $w$ and $f$.

Suppose now that $U(x, y)$ is a sufficiently regular solution of the extension problem \[ \partial_y (w(y) \partial_y U(x, y)) + L_x U(x, y) = 0 . \] For simplicity, suppose that the spectrum of $L$ is discrete. Let $\varphi$ be an eigenfunction of $-L$ with eigenvalue $\lambda$, and denote $U_\varphi(y) = \langle U(\cdot, y), \varphi \rangle$. Then $U_\varphi$ is a solution of the ODE \[ \partial_y (w(y) \partial_y U_\varphi(y)) + \lambda U_\varphi(y) = 0 . \] Hence, $U_\varphi(y) = U_\varphi(0) g_\lambda(y)$, and \[ -\lim_{y \to 0^+} \frac{U_\varphi(y) - U_\varphi(0)}{h(y)} = f(\lambda) U_\varphi(0) . \] It follows that \[ -\lim_{y \to 0^+} \frac{U(\cdot, y) - U_\varphi(\cdot, 0)}{h(y)} = f(-L) U(\cdot, 0) , \] or equivalently \[ -\lim_{y \to 0^+} \frac{\partial_y U(\cdot, y)}{h'(y)} = f(-L) U(\cdot, 0) . \] This proves that the Dirichlet-to-Neumann operator is indeed equal to $f(-L)$. The proof in the continuous spectrum case is similar.[11][12]

Operator monotone functions, often called complete Bernstein functions, form a subclass of Bernstein functions. Hence, existence of the extension problem is closely related to the concept of subordination. Operators of the form $f(-\Delta)$ for an operator monotone $f$ admit an explicit description. This gives a fairly explicit condition for the existence of the extension problem for a given translation-invariant non-local operator.

Relationship with Scattering operators

There is an identification between the fractional Laplacian defined by the extension and the fractional Paneitz operator from Scattering Theory when the order of the operator is less than 1.[13]

References

  1. ↑ Caffarelli, Luis; Silvestre, Luis (2007), "An extension problem related to the fractional Laplacian", Communications in Partial Differential Equations 32 (7): 1245–1260, doi:10.1080/03605300600987306, ISSN 0360-5302, http://dx.doi.org.ezproxy.lib.utexas.edu/10.1080/03605300600987306 
  2. ↑ Fabes, Eugene B.; Kenig, Carlos E.; Serapioni, Raul P. (1982), "The local regularity of solutions of degenerate elliptic equations", Communications in Partial Differential Equations 7 (1): 77–116, doi:10.1080/03605308208820218, ISSN 0360-5302, http://dx.doi.org/10.1080/03605308208820218 
  3. ↑ Fabes, Eugene B.; Kenig, Carlos E.; Jerison, David (1983), "Boundary behavior of solutions to degenerate elliptic equations", Conference on harmonic analysis in honor of Antoni Zygmund, Vol. I, II (Chicago, Ill., 1981), Wadsworth Math. Ser., Wadsworth, pp. 577–589 
  4. ↑ Fabes, Eugene B.; Jerison, David; Kenig, Carlos E. (1982), "The Wiener test for degenerate elliptic equations", Université de Grenoble. Annales de l'Institut Fourier 32 (3): 151–182, ISSN 0373-0956, http://www.numdam.org/item?id=AIF_1982__32_3_151_0 
  5. ↑ Caffarelli, Luis; Salsa, Sandro; Silvestre, Luis (2008), "Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian", Inventiones Mathematicae 171 (2): 425–461, doi:10.1007/s00222-007-0086-6, ISSN 0020-9910, http://dx.doi.org/10.1007/s00222-007-0086-6 
  6. ↑ Molchanov, S. A.; Ostrovski, E. (1969), "Symmetric stable processes as traces of degenerate diffusion processes", Theor Probab. Appl. 14: 128–131, doi:10.1137/1114012, http://dx.doi.org/10.1137/1114012 
  7. ↑ DeBlassie, R. D. (1990), "The first exit time of a two-dimensional symmetric stable process from a wedge", Ann. Probab. 18: 1034–1070, doi:10.1214/aop/1176990735, ISSN 0091-1798, http://dx.doi.org/10.1214/aop/1176990735 
  8. ↑ Tan, Jinggang; Cabré, Xavier (2010), "Positive solutions of nonlinear problems involving the square root of the Laplacian", Advances in Mathematics 224 (5): 2052–2093, doi:10.1016/j.aim.2010.01.025, ISSN 0001-8708, http://dx.doi.org/10.1016/j.aim.2010.01.025 
  9. ↑ Kotani, S.; Watanabe (1982), Fukushima, M., ed., [http://dx.doi.org/10.1007/BFb0093046 Krein’s spectral theory of strings and generalized diffusion processes], Lecture Notes in Mathematics, 923, Springer, Berlin / Heidelberg, pp. 235–259, doi:10.1007/BFb009304, ISBN 978-3-540-11484-0, http://dx.doi.org/10.1007/BFb0093046 
  10. ↑ Schilling, R.; Song, R.; Vondraček, Z. (2010), Bernstein functions. Theory and Applications, Studies in Mathematics, 37, de Gruyter, Berlin, doi:10.1515/9783110215311, http://dx.doi.org/10.1515/9783110215311 
  11. ↑ Kwaśnicki, M. (2011), "Spectral analysis of subordinate Brownian motions on the half-line", Studia Math. 206: 211–271, doi:10.4064/sm206-3-2, http://dx.doi.org/10.4064/sm206-3-2 
  12. ↑ Kim, P.; Song, R.; Vondraček, Z. (2011), "On harmonic functions for trace processes", Math. Nachr. 284: 1889–1902, doi:10.1002/mana.200910008, http://dx.doi.org/10.1002/mana.200910008 
  13. ↑ González, Maria del Mar; Chang, Sun-Yung Alice (2011), "Fractional Laplacian in conformal geometry", Advances in Mathematics 226 (2): 1410–1432, doi:10.1016/j.aim.2010.07.016, ISSN 0001-8708, http://dx.doi.org/10.1016/j.aim.2010.07.016 

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