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.
In: Discrete and Continuous Dynamical Systems- Series A, Vol. 39, No. 6, 2019, p. 3609-3634.Research output: Contribution to journal › Journal article › Research › peer-review
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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