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Well-posedness of general 1D initial boundary value problems for scalar balance laws
Limited regularity of solutions to fractional heat and Schrödinger equations
Department of Mathematical Sciences, Copenhagen University, Universitetsparken 5, DK-2100 Copenhagen, Denmark |
When $ P$ is the fractional Laplacian $ (-\Delta )^a$, $ 0<a<1$, or a pseudodifferential generalization thereof, the Dirichlet problem for the associated heat equation over a smooth set $ \Omega \subset{\Bbb R}^n$: $ r^+Pu(x, t)+\partial_tu(x, t) = f(x, t)$ on $ \Omega \times \, ]0, T[\, $, $ u(x, t) = 0$ for $ x\notin\Omega$, $ u(x, 0) = 0$, is known to be solvable in relatively low-order Sobolev or Hölder spaces. We now show that in contrast with differential operator cases, the regularity of $ u$ in $ x$ at $ \partial\Omega$ when $ f$ is very smooth cannot in general be improved beyond a certain estimate. An improvement requires the vanishing of a Neumann boundary value. --- There is a similar result for the Schrödinger Dirichlet problem $ r^+Pv(x)+Vv(x) = g(x)$ on $ \Omega$, $ \text{supp } v\subset\overline\Omega$, with $ V(x)\in C^\infty$. The proofs involve a precise description, of interest in itself, of the Dirichlet domains in terms of regular functions and functions with a $ \text{dist}(x, \partial\Omega )^a$ singularity.
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N. Abatangelo,
Large s-harmonic functions and boundary blow-up solutions for the fractional Laplacian, Discrete Contin. Dyn. Syst., 35 (2015), 5555-5607.
doi: 10.3934/dcds.2015.35.5555. |
[2] |
N. Abatangelo, S. Dipierro, M. M. Fall, S. Jarohs and A. Saldana, Positive powers of the Laplacian in the half-space under Dirichlet boundary conditions, arXiv: 1806.05128. |
[3] |
N. Abatangelo, S. Jarohs and A. Saldana, Integral representation of solutions to higher-order fractional Dirichlet problems on balls, Comm. Contemp. Math., 20 (2018), 1850002, 36 pp, arXiv: 1707.03603.
doi: 10.1142/S0219199718500025. |
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H. Amann,
Operator-valued Fourier multipliers, vector-valued Besov spaces, and applications, Math. Nachr., 186 (1997), 5-56.
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U. Biccari, M. Warma and E. Zuazua, Local regularity for fractional heat equations, arXiv: 1704.07562. |
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B. M. Blumenthal and R. K. Getoor,
The asymptotic distribution of the eigenvalues for a class of Markov operators, Pacific J. Math., 9 (1959), 399-408.
|
[7] |
K. Bogdan, K. Burdzy and Z.-Q. Chen,
Censored stable processes, Prob. Theory Related Fields, 127 (2003), 89-152.
doi: 10.1007/s00440-003-0275-1. |
[8] |
M. Bonforte, Y. Sire and J. L. Vazquez,
Existence, uniqueness and asymptotic behaviour for fractional porous medium equations on bounded domains, Discrete Contin. Dyn. Syst., 35 (2015), 5725-5767.
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L. Boutet de Monvel,
Boundary problems for pseudo-differential operators, Acta Math., 126 (1971), 11-51.
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L. Caffarelli and L. Silvestre,
An extension problem related to the fractional Laplacian, Comm. Part. Diff. Eq., 32 (2007), 1245-1260.
doi: 10.1080/03605300600987306. |
[11] |
H. Chang-Lara and G. Davila,
Regularity for solutions of non local parabolic equations, Calc. Var. Part. Diff. Equations, 49 (2014), 139-172.
doi: 10.1007/s00526-012-0576-2. |
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Z.-Q. Chen and R. Song,
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M. M. Fall, Regularity estimates for nonlocal Schrödinger equations, arXiv: 1711.02206. |
[17] |
M. Felsinger and M. Kassmann,
Local regularity for parabolic nonlocal operators, Comm. Part. Diff. Equations, 38 (2013), 1539-1573.
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[18] |
M. Felsinger, M. Kassmann and P. Voigt,
The Dirichlet problem for nonlocal operators, Math. Z., 279 (2015), 779-809.
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X. Fernandez-Real and X. Ros-Oton,
Regularity theory for general stable operators: Parabolic equations, J. Funct. Anal., 272 (2017), 4165-4221.
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Refined semiclassical asymptotics for fractional powers of the Laplace operator, J. Reine Angew. Math., 712 (2016), 1-37.
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T. Ghosh, M. Salo and G. Uhlmann, The Calderón problem for the fractional Schrödinger equation, arXiv: 1609.09248. |
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M. Gonzalez, R. Mazzeo and Y. Sire,
Singular solutions of fractional order conformal Laplacians, J. Geom. Anal., 22 (2012), 845-863.
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G. Grubb,
Pseudo-differential boundary problems in Lp spaces, Comm. Part. Diff. Eq., 15 (1990), 289-340.
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G. Grubb, Functional Calculus of Pseudodifferential Boundary Problems, Progress in Math. vol. 65, Second Edition, Birkhäuser, Boston, 1996.
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[25] |
G. Grubb, Distributions and Operators. Graduate Texts in Mathematics, 252, Springer, New York, 2009. |
[26] |
G. Grubb,
Local and nonlocal boundary conditions for µ-transmission and fractional elliptic pseudodifferential operators, Analysis and P.D.E., 7 (2014), 1649-1682.
doi: 10.2140/apde.2014.7.1649. |
[27] |
G. Grubb,
Fractional Laplacians on domains, a development of Hörmander's theory of µ-transmission pseudodifferential operators, Adv. Math., 268 (2015), 478-528.
doi: 10.1016/j.aim.2014.09.018. |
[28] |
G. Grubb,
Spectral results for mixed problems and fractional elliptic operators, J. Math. Anal. Appl., 421 (2015), 1616-1634.
doi: 10.1016/j.jmaa.2014.07.081. |
[29] |
G. Grubb,
Regularity of spectral fractional Dirichlet and Neumann problems, Math. Nachr., 289 (2016), 831-844.
doi: 10.1002/mana.201500041. |
[30] |
G. Grubb,
Integration by parts and Pohozaev identities for space-dependent fractional-order operators, J. Diff. Eq., 261 (2016), 1835-1879.
doi: 10.1016/j.jde.2016.04.017. |
[31] |
G. Grubb,
Regularity in Lp Sobolev spaces of solutions to fractional heat equations, J. Funct. Anal., 274 (2018), 2634-2660.
doi: 10.1016/j.jfa.2017.12.011. |
[32] |
G. Grubb, Green's formula and a Dirichlet-to-Neumann operator for fractional-order pseudodifferential operators, Comm. Part. Diff. Equ., 43 (2018), 750–789, arXiv: 1611.03024, https://www.tandfonline.com/doi/full/10.1080/03605302.2018.1475487.
doi: 10.1080/03605302.2018.1475487. |
[33] |
G. Grubb, Fractional-order operators: Boundary problems, heat equations, in Mathematical Analysis and Applications –- Plenary Lectures, ISAAC 2017, Vaxjo Sweden, Springer Proceedings in Mathematics and Statistics, (L. G. Rodino and J. Toft, eds.), Springer, Switzerland, 2018, pp. 51–81. |
[34] |
W. Hoh and N. Jacob,
On the Dirichlet problem for pseudodifferential operators generating Feller semigroups, J. Functional Anal., 137 (1996), 19-48.
doi: 10.1006/jfan.1996.0039. |
[35] |
L. Hörmander, Seminar notes on pseudo-differential operators and boundary problems, Lectures at IAS Princeton 1965-66, available from Lund University, https://lup.lub.lu.se/search/. |
[36] |
L. Hörmander, The Analysis of Linear Partial Differential Operators, III, Springer Verlag, Berlin, 1985. |
[37] |
N. Jacob, Pseudo Differential Operators and Markov Processes, Vol. I–3, Imperial College Press, London, 2001.
doi: 10.1142/9781860949746. |
[38] |
T. Jakubowski,
The estimates for the Green function in Lipschitz domains for the symmetric stable processes, Probab. Math. Statist., 22 (2002), 419-441.
|
[39] |
T. Jin and J. Xiong,
Schauder estimates for solutions of linear parabolic integro-differential equations, Discrete Contin. Dyn. Syst., 35 (2015), 5977-5998.
doi: 10.3934/dcds.2015.35.5977. |
[40] |
J. Johnsen,
Elliptic boundary problems and the Boutet de Monvel calculus in Besov and Triebel-Lizorkin spaces, Math. Scand., 79 (1996), 25-85.
doi: 10.7146/math.scand.a-12593. |
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T. Kulczycki,
Properties of Green function of symmetric stable processes, Probab. Math. Statist., 17 (1997), 339-364.
|
[42] |
N. S. Landkof, Foundations of Modern Potential Theory, Die Grundlehren der mathematischen Wissenschaften, Band 180, (Translated from the Russian by A. P. Doohovskoy.), Springer-Verlag, New York-Heidelberg, 1972. |
[43] |
T. Leonori, I. Peral, A. Primo and F. Soria,
Basic estimates for solutions of a class of nonlocal elliptic and parabolic equations, Discrete Contin. Dyn. Syst., 35 (2015), 6031-6068.
doi: 10.3934/dcds.2015.35.6031. |
[44] |
R. Musina and A. I. Nazarov,
On fractional Laplacians, Comm. Part. Diff. Eq., 39 (2014), 1780-1790.
doi: 10.1080/03605302.2013.864304. |
[45] |
X. Ros-Oton,
Nonlocal elliptic equations in bounded domains: A survey, Publ. Mat., 60 (2016), 3-26.
|
[46] |
X. Ros-Oton, Boundary regularity, Pohozaev identities and nonexistence results, Recent Developments in Nonlocal Theory, De Gruyter, Berlin, 2018, pp. 335–358. |
[47] |
X. Ros-Oton and J. Serra,
The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, J. Math. Pures Appl., 101 (2014), 275-302.
doi: 10.1016/j.matpur.2013.06.003. |
[48] |
X. Ros-Oton and J. Serra,
The Pohozaev identity for the fractional Laplacian, Arch. Rat. Mech. Anal., 213 (2014), 587-628.
doi: 10.1007/s00205-014-0740-2. |
[49] |
X. Ros-Oton and J. Serra,
Regularity theory for general stable operators, J. Differential Equations, 260 (2016), 8675-8715.
doi: 10.1016/j.jde.2016.02.033. |
[50] |
X. Ros-Oton and H. Vivas, Higher-order boundary regularity estimates for nonlocal parabolic equations, Calc. Var. Partial Differential Equations, 57 (2018), Art. 111, 20 pp.
doi: 10.1007/s00526-018-1399-6. |
[51] |
R. Servadei and E. Valdinoci,
On the spectrum of two different fractional operators, Proc. Roy. Soc. Edinburgh, 144 (2014), 831-855.
doi: 10.1017/S0308210512001783. |
[52] |
L. Silvestre,
Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112.
doi: 10.1002/cpa.20153. |
[53] |
M. E. Taylor, Pseudodifferential Operators, Princeton University Press, Princeton, NJ, 1981.
![]() ![]() |
show all references
References:
[1] |
N. Abatangelo,
Large s-harmonic functions and boundary blow-up solutions for the fractional Laplacian, Discrete Contin. Dyn. Syst., 35 (2015), 5555-5607.
doi: 10.3934/dcds.2015.35.5555. |
[2] |
N. Abatangelo, S. Dipierro, M. M. Fall, S. Jarohs and A. Saldana, Positive powers of the Laplacian in the half-space under Dirichlet boundary conditions, arXiv: 1806.05128. |
[3] |
N. Abatangelo, S. Jarohs and A. Saldana, Integral representation of solutions to higher-order fractional Dirichlet problems on balls, Comm. Contemp. Math., 20 (2018), 1850002, 36 pp, arXiv: 1707.03603.
doi: 10.1142/S0219199718500025. |
[4] |
H. Amann,
Operator-valued Fourier multipliers, vector-valued Besov spaces, and applications, Math. Nachr., 186 (1997), 5-56.
doi: 10.1002/mana.3211860102. |
[5] |
U. Biccari, M. Warma and E. Zuazua, Local regularity for fractional heat equations, arXiv: 1704.07562. |
[6] |
B. M. Blumenthal and R. K. Getoor,
The asymptotic distribution of the eigenvalues for a class of Markov operators, Pacific J. Math., 9 (1959), 399-408.
|
[7] |
K. Bogdan, K. Burdzy and Z.-Q. Chen,
Censored stable processes, Prob. Theory Related Fields, 127 (2003), 89-152.
doi: 10.1007/s00440-003-0275-1. |
[8] |
M. Bonforte, Y. Sire and J. L. Vazquez,
Existence, uniqueness and asymptotic behaviour for fractional porous medium equations on bounded domains, Discrete Contin. Dyn. Syst., 35 (2015), 5725-5767.
doi: 10.3934/dcds.2015.35.5725. |
[9] |
L. Boutet de Monvel,
Boundary problems for pseudo-differential operators, Acta Math., 126 (1971), 11-51.
doi: 10.1007/BF02392024. |
[10] |
L. Caffarelli and L. Silvestre,
An extension problem related to the fractional Laplacian, Comm. Part. Diff. Eq., 32 (2007), 1245-1260.
doi: 10.1080/03605300600987306. |
[11] |
H. Chang-Lara and G. Davila,
Regularity for solutions of non local parabolic equations, Calc. Var. Part. Diff. Equations, 49 (2014), 139-172.
doi: 10.1007/s00526-012-0576-2. |
[12] |
Z.-Q. Chen and R. Song,
Estimates on Green functions and Poisson kernels for symmetric stable processes, Math. Ann., 312 (1998), 465-501.
doi: 10.1007/s002080050232. |
[13] |
R. Cont and P. Tankov, Financial Modelling with Jump Processes, Financial Mathematics Series, Chapman & Hall/CRC, Boca Raton, FL, 2004. |
[14] |
R. Courant and D. Hilbert, Methods of Mathematical Physics II, Interscience Publishers, New York, 1962. |
[15] |
J. I. Diaz, D. Gomez-Castro and J. L. Vazquez, The fractional Schrödinger equation with general nonnegative potentials, The weighted space approach, arXiv: 1804.08398. |
[16] |
M. M. Fall, Regularity estimates for nonlocal Schrödinger equations, arXiv: 1711.02206. |
[17] |
M. Felsinger and M. Kassmann,
Local regularity for parabolic nonlocal operators, Comm. Part. Diff. Equations, 38 (2013), 1539-1573.
doi: 10.1080/03605302.2013.808211. |
[18] |
M. Felsinger, M. Kassmann and P. Voigt,
The Dirichlet problem for nonlocal operators, Math. Z., 279 (2015), 779-809.
doi: 10.1007/s00209-014-1394-3. |
[19] |
X. Fernandez-Real and X. Ros-Oton,
Regularity theory for general stable operators: Parabolic equations, J. Funct. Anal., 272 (2017), 4165-4221.
doi: 10.1016/j.jfa.2017.02.015. |
[20] |
R. Frank and L. Geisinger,
Refined semiclassical asymptotics for fractional powers of the Laplace operator, J. Reine Angew. Math., 712 (2016), 1-37.
doi: 10.1515/crelle-2013-0120. |
[21] |
T. Ghosh, M. Salo and G. Uhlmann, The Calderón problem for the fractional Schrödinger equation, arXiv: 1609.09248. |
[22] |
M. Gonzalez, R. Mazzeo and Y. Sire,
Singular solutions of fractional order conformal Laplacians, J. Geom. Anal., 22 (2012), 845-863.
doi: 10.1007/s12220-011-9217-9. |
[23] |
G. Grubb,
Pseudo-differential boundary problems in Lp spaces, Comm. Part. Diff. Eq., 15 (1990), 289-340.
doi: 10.1080/03605309908820688. |
[24] |
G. Grubb, Functional Calculus of Pseudodifferential Boundary Problems, Progress in Math. vol. 65, Second Edition, Birkhäuser, Boston, 1996.
doi: 10.1007/978-1-4612-0769-6. |
[25] |
G. Grubb, Distributions and Operators. Graduate Texts in Mathematics, 252, Springer, New York, 2009. |
[26] |
G. Grubb,
Local and nonlocal boundary conditions for µ-transmission and fractional elliptic pseudodifferential operators, Analysis and P.D.E., 7 (2014), 1649-1682.
doi: 10.2140/apde.2014.7.1649. |
[27] |
G. Grubb,
Fractional Laplacians on domains, a development of Hörmander's theory of µ-transmission pseudodifferential operators, Adv. Math., 268 (2015), 478-528.
doi: 10.1016/j.aim.2014.09.018. |
[28] |
G. Grubb,
Spectral results for mixed problems and fractional elliptic operators, J. Math. Anal. Appl., 421 (2015), 1616-1634.
doi: 10.1016/j.jmaa.2014.07.081. |
[29] |
G. Grubb,
Regularity of spectral fractional Dirichlet and Neumann problems, Math. Nachr., 289 (2016), 831-844.
doi: 10.1002/mana.201500041. |
[30] |
G. Grubb,
Integration by parts and Pohozaev identities for space-dependent fractional-order operators, J. Diff. Eq., 261 (2016), 1835-1879.
doi: 10.1016/j.jde.2016.04.017. |
[31] |
G. Grubb,
Regularity in Lp Sobolev spaces of solutions to fractional heat equations, J. Funct. Anal., 274 (2018), 2634-2660.
doi: 10.1016/j.jfa.2017.12.011. |
[32] |
G. Grubb, Green's formula and a Dirichlet-to-Neumann operator for fractional-order pseudodifferential operators, Comm. Part. Diff. Equ., 43 (2018), 750–789, arXiv: 1611.03024, https://www.tandfonline.com/doi/full/10.1080/03605302.2018.1475487.
doi: 10.1080/03605302.2018.1475487. |
[33] |
G. Grubb, Fractional-order operators: Boundary problems, heat equations, in Mathematical Analysis and Applications –- Plenary Lectures, ISAAC 2017, Vaxjo Sweden, Springer Proceedings in Mathematics and Statistics, (L. G. Rodino and J. Toft, eds.), Springer, Switzerland, 2018, pp. 51–81. |
[34] |
W. Hoh and N. Jacob,
On the Dirichlet problem for pseudodifferential operators generating Feller semigroups, J. Functional Anal., 137 (1996), 19-48.
doi: 10.1006/jfan.1996.0039. |
[35] |
L. Hörmander, Seminar notes on pseudo-differential operators and boundary problems, Lectures at IAS Princeton 1965-66, available from Lund University, https://lup.lub.lu.se/search/. |
[36] |
L. Hörmander, The Analysis of Linear Partial Differential Operators, III, Springer Verlag, Berlin, 1985. |
[37] |
N. Jacob, Pseudo Differential Operators and Markov Processes, Vol. I–3, Imperial College Press, London, 2001.
doi: 10.1142/9781860949746. |
[38] |
T. Jakubowski,
The estimates for the Green function in Lipschitz domains for the symmetric stable processes, Probab. Math. Statist., 22 (2002), 419-441.
|
[39] |
T. Jin and J. Xiong,
Schauder estimates for solutions of linear parabolic integro-differential equations, Discrete Contin. Dyn. Syst., 35 (2015), 5977-5998.
doi: 10.3934/dcds.2015.35.5977. |
[40] |
J. Johnsen,
Elliptic boundary problems and the Boutet de Monvel calculus in Besov and Triebel-Lizorkin spaces, Math. Scand., 79 (1996), 25-85.
doi: 10.7146/math.scand.a-12593. |
[41] |
T. Kulczycki,
Properties of Green function of symmetric stable processes, Probab. Math. Statist., 17 (1997), 339-364.
|
[42] |
N. S. Landkof, Foundations of Modern Potential Theory, Die Grundlehren der mathematischen Wissenschaften, Band 180, (Translated from the Russian by A. P. Doohovskoy.), Springer-Verlag, New York-Heidelberg, 1972. |
[43] |
T. Leonori, I. Peral, A. Primo and F. Soria,
Basic estimates for solutions of a class of nonlocal elliptic and parabolic equations, Discrete Contin. Dyn. Syst., 35 (2015), 6031-6068.
doi: 10.3934/dcds.2015.35.6031. |
[44] |
R. Musina and A. I. Nazarov,
On fractional Laplacians, Comm. Part. Diff. Eq., 39 (2014), 1780-1790.
doi: 10.1080/03605302.2013.864304. |
[45] |
X. Ros-Oton,
Nonlocal elliptic equations in bounded domains: A survey, Publ. Mat., 60 (2016), 3-26.
|
[46] |
X. Ros-Oton, Boundary regularity, Pohozaev identities and nonexistence results, Recent Developments in Nonlocal Theory, De Gruyter, Berlin, 2018, pp. 335–358. |
[47] |
X. Ros-Oton and J. Serra,
The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, J. Math. Pures Appl., 101 (2014), 275-302.
doi: 10.1016/j.matpur.2013.06.003. |
[48] |
X. Ros-Oton and J. Serra,
The Pohozaev identity for the fractional Laplacian, Arch. Rat. Mech. Anal., 213 (2014), 587-628.
doi: 10.1007/s00205-014-0740-2. |
[49] |
X. Ros-Oton and J. Serra,
Regularity theory for general stable operators, J. Differential Equations, 260 (2016), 8675-8715.
doi: 10.1016/j.jde.2016.02.033. |
[50] |
X. Ros-Oton and H. Vivas, Higher-order boundary regularity estimates for nonlocal parabolic equations, Calc. Var. Partial Differential Equations, 57 (2018), Art. 111, 20 pp.
doi: 10.1007/s00526-018-1399-6. |
[51] |
R. Servadei and E. Valdinoci,
On the spectrum of two different fractional operators, Proc. Roy. Soc. Edinburgh, 144 (2014), 831-855.
doi: 10.1017/S0308210512001783. |
[52] |
L. Silvestre,
Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112.
doi: 10.1002/cpa.20153. |
[53] |
M. E. Taylor, Pseudodifferential Operators, Princeton University Press, Princeton, NJ, 1981.
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