# American Institute of Mathematical Sciences

October  2015, 35(10): 4905-4929. doi: 10.3934/dcds.2015.35.4905

## Wave extension problem for the fractional Laplacian

 1 Department of Mathematics and Statistics, University of Helsinki, FI-00014 Helsinki, Finland 2 Department of Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, SE-412 96 Göteborg, Sweden 3 Departamento de Matemáticas, Universidad Autónoma de Madrid and ICMAT, 28049 Madrid, Spain

Received  November 2014 Revised  February 2015 Published  April 2015

We show that the fractional Laplacian can be viewed as a Dirichlet-to-Neumann map for a degenerate hyperbolic problem, namely, the wave equation with an additional diffusion term that blows up at time zero. A solution to this wave extension problem is obtained from the Schrödinger group by means of an oscillatory subordination formula, which also allows us to find kernel representations for such solutions. Asymptotics of related oscillatory integrals are analysed in order to determine the correct domains for initial data in the general extension problem involving non-negative self-adjoint operators. An alternative approach using Bessel functions is also described.
Citation: Mikko Kemppainen, Peter Sjögren, José Luis Torrea. Wave extension problem for the fractional Laplacian. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 4905-4929. doi: 10.3934/dcds.2015.35.4905
##### References:
 [1] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian,, Comm. Partial Differential Equations, 32 (2007), 1245.  doi: 10.1080/03605300600987306.  Google Scholar [2] L. C. Evans, Partial Differential Equations,, American Mathematical Society, (1998).   Google Scholar [3] J. E. Galé, P. J. Miana and P. R. Stinga, Extension problem and fractional operators: Semigroups and wave equations,, J. Evol. Equ., 13 (2013), 343.  doi: 10.1007/s00028-013-0182-6.  Google Scholar [4] N. N. Lebedev, Special Functions and Their Applications,, Revised English edition, (1965).   Google Scholar [5] P. R. Stinga and J. L. Torrea, Extension problem and Harnack's inequality for some fractional operators,, Comm. Partial Differential Equations, 35 (2010), 2092.  doi: 10.1080/03605301003735680.  Google Scholar [6] R. S. Strichartz, Convolutions with kernels having singularities on a sphere,, Trans. Amer. Math. Soc., 148 (1970), 461.  doi: 10.1090/S0002-9947-1970-0256219-1.  Google Scholar

show all references

##### References:
 [1] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian,, Comm. Partial Differential Equations, 32 (2007), 1245.  doi: 10.1080/03605300600987306.  Google Scholar [2] L. C. Evans, Partial Differential Equations,, American Mathematical Society, (1998).   Google Scholar [3] J. E. Galé, P. J. Miana and P. R. Stinga, Extension problem and fractional operators: Semigroups and wave equations,, J. Evol. Equ., 13 (2013), 343.  doi: 10.1007/s00028-013-0182-6.  Google Scholar [4] N. N. Lebedev, Special Functions and Their Applications,, Revised English edition, (1965).   Google Scholar [5] P. R. Stinga and J. L. Torrea, Extension problem and Harnack's inequality for some fractional operators,, Comm. Partial Differential Equations, 35 (2010), 2092.  doi: 10.1080/03605301003735680.  Google Scholar [6] R. S. Strichartz, Convolutions with kernels having singularities on a sphere,, Trans. Amer. Math. Soc., 148 (1970), 461.  doi: 10.1090/S0002-9947-1970-0256219-1.  Google Scholar
 [1] Gregory Beylkin, Lucas Monzón. Efficient representation and accurate evaluation of oscillatory integrals and functions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (8) : 4077-4100. doi: 10.3934/dcds.2016.36.4077 [2] Miaomiao Niu, Zhongwei Tang. Least energy solutions for nonlinear Schrödinger equation involving the fractional Laplacian and critical growth. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3963-3987. doi: 10.3934/dcds.2017168 [3] Ran Zhuo, Yan Li. Nonexistence and symmetry of solutions for Schrödinger systems involving fractional Laplacian. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1595-1611. doi: 10.3934/dcds.2019071 [4] Shaoming Guo. Oscillatory integrals related to Carleson's theorem: fractional monomials. Communications on Pure & Applied Analysis, 2016, 15 (3) : 929-946. doi: 10.3934/cpaa.2016.15.929 [5] Hans Zwart, Yann Le Gorrec, Bernhard Maschke. Relating systems properties of the wave and the Schrödinger equation. Evolution Equations & Control Theory, 2015, 4 (2) : 233-240. doi: 10.3934/eect.2015.4.233 [6] Takahisa Inui. Global dynamics of solutions with group invariance for the nonlinear schrödinger equation. Communications on Pure & Applied Analysis, 2017, 16 (2) : 557-590. doi: 10.3934/cpaa.2017028 [7] M.T. Boudjelkha. Extended Riemann Bessel functions. Conference Publications, 2005, 2005 (Special) : 121-130. doi: 10.3934/proc.2005.2005.121 [8] Vincenzo Ambrosio, Teresa Isernia. Multiplicity and concentration results for some nonlinear Schrödinger equations with the fractional p-Laplacian. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5835-5881. doi: 10.3934/dcds.2018254 [9] Wulong Liu, Guowei Dai. Multiple solutions for a fractional nonlinear Schrödinger equation with local potential. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2105-2123. doi: 10.3934/cpaa.2017104 [10] Xudong Shang, Jihui Zhang. Multiplicity and concentration of positive solutions for fractional nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2239-2259. doi: 10.3934/cpaa.2018107 [11] Patricio Felmer, César Torres. Radial symmetry of ground states for a regional fractional Nonlinear Schrödinger Equation. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2395-2406. doi: 10.3934/cpaa.2014.13.2395 [12] Zhengping Wang, Huan-Song Zhou. Radial sign-changing solution for fractional Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 499-508. doi: 10.3934/dcds.2016.36.499 [13] Van Duong Dinh, Binhua Feng. On fractional nonlinear Schrödinger equation with combined power-type nonlinearities. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4565-4612. doi: 10.3934/dcds.2019188 [14] David Gómez-Castro, Juan Luis Vázquez. The fractional Schrödinger equation with singular potential and measure data. Discrete & Continuous Dynamical Systems - A, 2019, 39 (12) : 7113-7139. doi: 10.3934/dcds.2019298 [15] Hassan Emamirad, Arnaud Rougirel. Feynman path formula for the time fractional Schrödinger equation. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020246 [16] Weiming Liu, Lu Gan. Multi-bump positive solutions of a fractional nonlinear Schrödinger equation in $\mathbb{R}^N$. Communications on Pure & Applied Analysis, 2016, 15 (2) : 413-428. doi: 10.3934/cpaa.2016.15.413 [17] Binhua Feng. On the blow-up solutions for the fractional nonlinear Schrödinger equation with combined power-type nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1785-1804. doi: 10.3934/cpaa.2018085 [18] César E. Torres Ledesma. Existence and concentration of solutions for a non-linear fractional Schrödinger equation with steep potential well. Communications on Pure & Applied Analysis, 2016, 15 (2) : 535-547. doi: 10.3934/cpaa.2016.15.535 [19] Chenmin Sun, Hua Wang, Xiaohua Yao, Jiqiang Zheng. Scattering below ground state of focusing fractional nonlinear Schrödinger equation with radial data. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 2207-2228. doi: 10.3934/dcds.2018091 [20] Van Duong Dinh. On blow-up solutions to the focusing mass-critical nonlinear fractional Schrödinger equation. Communications on Pure & Applied Analysis, 2019, 18 (2) : 689-708. doi: 10.3934/cpaa.2019034

2018 Impact Factor: 1.143