Extension technique

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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.

Contents

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} \tag{1} \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} \tag{2} (-\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 (1) is \begin{equation} \tag{3} \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 $$\begin{cases} - L_x U(x,y)+\frac{1-2s}{y}U_y(x,y)+U_{yy}(x,y) = 0,&\hbox{in}~\Omega\times(0,\infty),\\ U(x,0)=u(x),&\hbox{on}~\Omega, \end{cases}$$ 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} Here $C_s$ is a constant depending only on $s$. Moreover, the solution can be explicitly written as $$U(x,y)=\frac{y^{2s}}{4^s\Gamma(s)}\int_0^\infty e^{-tL}u(x)e^{-y^2/(4t)}\,\frac{dt}{t^{1+s}}.$$ For details see [8] and [9].

See for example [10] where this technique is applied to obtain global regularity estimates for fractional powers of divergence form elliptic operators with Dirichlet or Neumann boundary conditions.

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 $\Omega$.[11] 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.[12][13] 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.[14][15]

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.[16]

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. Stinga, P. R. (2010), "Fractional powers of second order partial differential operators: extension problem and regularity theory", PhD. Thesis - Universidad Aut\'onoma de Madrid, https://www.ma.utexas.edu/users/stinga/Stinga%20-%20PhD%20Thesis.pdf 
  9. Stinga, P. R.; Torrea, J. L. (2010), "Extension problem and Harnack's inequality for some fractional operators", Comm. Partial Differential Equations 35 (11): 2092-2122, http://www.tandfonline.com/doi/abs/10.1080/03605301003735680#.VcJb9UWmR1Q 
  10. Caffarelli, L. A.; Stinga, P. R. (2015), "Fractional elliptic equations, Caccioppoli estimates and regularity", Ann. Inst. H. Poincar\'e (C) Anal. Non Lin\'eaire: to appear, http://www.sciencedirect.com/science/article/pii/S0294144915000153 
  11. 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 
  12. 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 
  13. 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 
  14. 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 
  15. 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 
  16. 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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