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| 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. | | The nonlocal porous medium equation of order $\sigma$ is the name currently given to two very different equations, namely |
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| ==Fractional Laplacian== | | \[ u_t = \nabla \cdot \left ( u \nabla \mathcal{K_\alpha} (u) \right )\] |
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| The extension problem for the [[fractional Laplacian]] $(-\Delta)^s$, $s \in (0, 1)$ takes the following form.<ref name="CS"/> Let
| | \[\mbox{ where } \mathcal{K}_\alpha(u) := C_{n,\alpha}\; u * |x|^{-n+\alpha},\;\; \alpha+2=\sigma \] |
| $$U:\mathbb{R}^n \times \mathbb{R}_+ \longrightarrow \mathbb{R}$$
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| be a function satisfying
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| \begin{equation}
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| \label{eqn:Main}
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| \nabla \cdot (y^{1-2s} \nabla U(x,y)) = 0
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| \end{equation}
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| 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
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| \begin{equation}
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| \label{eqn:Neumann}
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| (-\Delta)^s u(x) = -C_{n,s} \lim_{y\rightarrow 0} y^{1-2s} \partial_y U(x,y).
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| \end{equation} | |
| The energy associated with the operator in \eqref{eqn:Main} is
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| \begin{equation} | |
| \label{eqn:Energy} | |
| \int y^{1-2s} |\nabla U|^2 dx dy | |
| \end{equation}
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| 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"/>
| | and |
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| 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"/>
| | \[ u_t +(-\Delta)^{s}(u^m) = 0 \] |
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| The above extension technique is closely related to the concept of trace of a diffusion on a hyperplane.<ref name="MO"/><ref name="D"/>
| | These equations agree when $s=1$ and $m=2$, otherwise they are not only different superficially, they also exhibit extremely different behaviors. They are both fractional order [[Quasilinear equations]]. |
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| ==Fractional powers of more general operators==
| | The first of the two has the remarkable property (for nonlocal equations at least) that any initial data with compact support remains with compact support for all later times, the opposite is true of the second equation, for which [[instantaneous speed of propagation]] holds. |
| 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
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| $$\begin{cases}
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| \partial_y (y^{1-2s} \partial_y U(x, y)) - L_x U(x, y) = 0,&\hbox{in}~\Omega\times(0,\infty),\\
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| U(x,0)=u(x),&\hbox{on}~\Omega,
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| \end{cases}$$
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| with boundary conditions along $\partial \Omega \times [0,\infty)$ equal to the boundary conditions for $L$. Then | |
| \begin{equation}
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| L^su(x) = C_{s} \lim_{y\rightarrow 0} y^{1-2s} U_y(x,y).
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| \end{equation}
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| Here $C_s$ is a constant depending only on $s$. Moreover, the solution can be explicitly written as
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| $$U(x,y)=\frac{y^{2s}}{4^s\Gamma(s)}\int_0^\infty e^{-tL}u(x)e^{-y^2/(4t)}\,\frac{dt}{t^{1+s}}.$$
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| For details see
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| 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 $(-L)$.
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| 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 $\Omega = \mathbb{R}^n$.
| | This means that the first model presents us with a [[free boundary problem]]. For this model global existence and Hölder continuity of weak solutions have been recently obtained <ref name="CV1"/>, there is almost nothing known about the properties of its free boundary, making it a rich source of open questions. |
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| ==More general non-local operators==
| | For the second equation, both the Cauchy problem and long time behavior have been extensively studied in recent years <ref name="PQRV" /ref>. |
| Let $L$ be as above, and consider the Dirichlet-to-Neumann operator $A$ related to the elliptic equation
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| \[ \partial_y (w(y) \partial_y U(x, y)) + L_x U(x, y) = 0 \]
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| 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.) | |
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| 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
| | == References == |
| \[ \partial_y (w(y) \partial_y g_\lambda(y)) = \lambda g_\lambda(y) \]
| | {{reflist|refs= |
| for $y \ge 0$, satisfying $g_\lambda(0) = 1$. Furthermore, let $h$ be the nondecreasing solution of
| | <ref name="CV1"> {{Citation | last1=Caffarelli | first1=Luis | last2=Vazquez | first2=Juan | title=Nonlinear Porous Medium Flow with Fractional Potential Pressure | url=http://dx.doi.org/10.1007/s00205-011-0420-4 | publisher=Springer Berlin / Heidelberg | year=2011 | journal=Archive for Rational Mechanics and Analysis | issn=0003-9527 | pages=1–29}} </ref> |
| \[ \partial_y (w(y) \partial_y h(y)) = 0 \]
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| satisfying $h(0) = 0$ and $h(1) = 1$. Then
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| \[ f(\lambda) = \lim_{y \to 0^+} \frac{1 - g_\lambda(y)}{h(y)} . \]
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| 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 $f$.
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| Suppose now that $U(x, y)$ is a sufficiently regular solution of the extension problem
| | <ref name="PQRV">{{Citation | last1=Pablo | first1=Arturo de | last2=Quirós | first2=Fernando | last3=Rodríguez | first3=Ana | last4=Vazquez | first4=Juan Luis | title=A fractional porous medium equation | url=http://www.sciencedirect.com/science/article/pii/S0001870810003130 | doi=DOI: 10.1016/j.aim.2010.07.017 | year=2011 | journal=Advances in Mathematics | issn=0001-8708 | volume=226 | issue=2 | pages=1378–1409}} |
| \[ \partial_y (w(y) \partial_y U(x, y)) + L_x U(x, y) = 0 . \]
| | </ref> |
| 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
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| \[ \partial_y (w(y) \partial_y U_\varphi(y)) + \lambda U_\varphi(y) = 0 . \]
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| Hence, $U_\varphi(y) = U_\varphi(0) g_\lambda(y)$, and
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| \[ -\lim_{y \to 0^+} \frac{U_\varphi(y) - U_\varphi(0)}{h(y)} = f(\lambda) U_\varphi(0) . \]
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| It follows that
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| \[ -\lim_{y \to 0^+} \frac{U(\cdot, y) - U_\varphi(\cdot, 0)}{h(y)} = f(-L) U(\cdot, 0) , \]
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| or equivalently
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| \[ -\lim_{y \to 0^+} \frac{\partial_y U(\cdot, y)}{h'(y)} = f(-L) U(\cdot, 0) . \]
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| 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"/>
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| [[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.
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| ==Relationship with Scattering operators==
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| 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"/>
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| <!--This is a stub, to be expanded on later.-->
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| ==References==
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| {{reflist|refs=
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| <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> | |
| <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>
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| <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>
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| <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>
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| <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>
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| <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>
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| <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>
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| <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>
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| <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>
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| <ref name="KW">{{Citation | last1=Kotani | first1=S. | last2=Watanabe | first1=S. | title=Krein’s spectral theory of strings and
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| 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>
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| <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>
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| <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>
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| <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>
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| | [[Category:Quasilinear equations]] [[Category:Evolution equations]] [[Category:Free boundary problems]] |