Limited regularity of solutions to fractional heat and Schrödinger equations

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Limited regularity of solutions to fractional heat and Schrödinger equations. / Grubb, Gerd.

I: Discrete and Continuous Dynamical Systems- Series A, Bind 39, Nr. 6, 2019, s. 3609-3634.

Publikation: Bidrag til tidsskriftTidsskriftartikelForskningfagfællebedømt

Harvard

Grubb, G 2019, 'Limited regularity of solutions to fractional heat and Schrödinger equations', Discrete and Continuous Dynamical Systems- Series A, bind 39, nr. 6, s. 3609-3634. https://doi.org/10.3934/dcds.2019148

APA

Grubb, G. (2019). Limited regularity of solutions to fractional heat and Schrödinger equations. Discrete and Continuous Dynamical Systems- Series A, 39(6), 3609-3634. https://doi.org/10.3934/dcds.2019148

Vancouver

Grubb G. Limited regularity of solutions to fractional heat and Schrödinger equations. Discrete and Continuous Dynamical Systems- Series A. 2019;39(6):3609-3634. https://doi.org/10.3934/dcds.2019148

Author

Grubb, Gerd. / Limited regularity of solutions to fractional heat and Schrödinger equations. I: Discrete and Continuous Dynamical Systems- Series A. 2019 ; Bind 39, Nr. 6. s. 3609-3634.

Bibtex

@article{6a5be4c9721446bb9b6041fd9bcb456f,
title = "Limited regularity of solutions to fractional heat and Schr{\"o}dinger equations",
abstract = "When P is the fractional Laplacian (-Δ) a , 0 < a < 1, or a pseudodifferential generalization thereof, the Dirichlet problem for the associated heat equation over a smooth set (Equation presented), is known to be solvable in relatively low-order Sobolev or Holder spaces. We now show that in contrast with differential operator cases, the regularity of u in x at δΩ 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 Schr{\"o}dinger Dirichlet problem r+Pv(x) + Vv(x) = g(x) on Ω, supp u ⊂ Ω, with V(x) ϵ C ∞ . The proofs involve a precise description, of interest in itself, of the Dirichlet domains in terms of regular functions and functions with a dist(x, δΩ) a singularity.",
keywords = "Fractional heat equation, Fractional Laplacian, Fractional schrodinger dirichlet problem, Limited spatial regularity, Lp and holder estimates, Pseudodifferential operator, Stable process",
author = "Gerd Grubb",
year = "2019",
doi = "10.3934/dcds.2019148",
language = "English",
volume = "39",
pages = "3609--3634",
journal = "Discrete and Continuous Dynamical Systems. Series A",
issn = "1078-0947",
publisher = "American Institute of Mathematical Sciences",
number = "6",

}

RIS

TY - JOUR

T1 - Limited regularity of solutions to fractional heat and Schrödinger equations

AU - Grubb, Gerd

PY - 2019

Y1 - 2019

N2 - When P is the fractional Laplacian (-Δ) a , 0 < a < 1, or a pseudodifferential generalization thereof, the Dirichlet problem for the associated heat equation over a smooth set (Equation presented), is known to be solvable in relatively low-order Sobolev or Holder spaces. We now show that in contrast with differential operator cases, the regularity of u in x at δΩ 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 Ω, supp u ⊂ Ω, with V(x) ϵ C ∞ . The proofs involve a precise description, of interest in itself, of the Dirichlet domains in terms of regular functions and functions with a dist(x, δΩ) a singularity.

AB - When P is the fractional Laplacian (-Δ) a , 0 < a < 1, or a pseudodifferential generalization thereof, the Dirichlet problem for the associated heat equation over a smooth set (Equation presented), is known to be solvable in relatively low-order Sobolev or Holder spaces. We now show that in contrast with differential operator cases, the regularity of u in x at δΩ 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 Ω, supp u ⊂ Ω, with V(x) ϵ C ∞ . The proofs involve a precise description, of interest in itself, of the Dirichlet domains in terms of regular functions and functions with a dist(x, δΩ) a singularity.

KW - Fractional heat equation

KW - Fractional Laplacian

KW - Fractional schrodinger dirichlet problem

KW - Limited spatial regularity

KW - Lp and holder estimates

KW - Pseudodifferential operator

KW - Stable process

UR - http://www.scopus.com/inward/record.url?scp=85063774528&partnerID=8YFLogxK

U2 - 10.3934/dcds.2019148

DO - 10.3934/dcds.2019148

M3 - Journal article

AN - SCOPUS:85063774528

VL - 39

SP - 3609

EP - 3634

JO - Discrete and Continuous Dynamical Systems. Series A

JF - Discrete and Continuous Dynamical Systems. Series A

SN - 1078-0947

IS - 6

ER -

ID: 236554799