Schauder estimates for Bessel operators

Giorgio Metafune; Luigi Negro; Chiara Spina

doi:10.1515/forum-2023-0334

Article

Schauder estimates for Bessel operators

Giorgio Metafune , Luigi Negro and Chiara Spina

Published/Copyright: November 20, 2023

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From the journal Forum Mathematicum Volume 36 Issue 5

Abstract

We prove Schauder estimates for elliptic and parabolic problems governed by the degenerate operator

ℒ = Δ x + D y ⁢ y + c y ⁢ D y ,

in the half-space Ω = { ( x , y ) : x ∈ ℝ N , y > 0 } , under Neumann boundary conditions at y = 0 .

Keywords: Degenerate elliptic operators; boundary degeneracy; Schauder estimates

MSC 2020: 35K67; 35B45; 47D07; 35J70; 35J75

Communicated by Matthias Hieber

A Boundedness of a family of integral operators

We consider a two-parameters family of integral operators ( S α , β ⁢ ( t ) ) t > 0 on L ∞ ⁢ ( ℝ M ) , defined for α , β ∈ ℝ and t > 0 by

S α , β ⁢ ( t ) ⁢ f ⁢ ( y ) = t - M 2 ⁢ ( | y | t ∧ 1 ) - α ⁢ ∫ ℝ M ( | z | t ∧ 1 ) - β ⁢ exp ⁡ ( - | y - z | 2 κ ⁢ t ) ⁢ f ⁢ ( z ) ⁢ 𝑑 z ,

where κ is a positive constant. We omit the dependence on κ even though in some proofs we need to vary it.

We start by observing that the scale homogeneity of S α , β is 2 since a simple change of variable in the integral yields

S α , β ⁢ ( t ) ⁢ ( I s ⁢ f ) = I s ⁢ ( S α , β ⁢ ( s 2 ⁢ t ) ⁢ f ) , I s ⁢ f ⁢ ( y ) = f ⁢ ( s ⁢ y ) , t , s > 0 ,

which gives

S α , β ⁢ ( t ) = I 1 / t ⁢ S α , β ⁢ ( 1 ) ⁢ I t .

The boundedness of S α , β ⁢ ( t ) ⁢ f in L ∞ ⁢ ( ℝ M ) is then equivalent to that for t = 1 and ∥ S α , β ⁢ ( t ) ∥ ∞ = ∥ S α , β ⁢ ( 1 ) ∥ ∞ , since I s is an isometry.

Proposition A.1.

The following properties are equivalent:

For every t > 0 , ( S α , β ⁢ ( t ) ) t ≥ 0 is bounded from L ∞ to L ∞ and

∥ S α , β ⁢ ( t ) ∥ L ∞ → L ∞ ≤ C .
S α , β ⁢ ( 1 ) is bounded from L ∞ to L ∞ .
α ≤ 0 < M - β .

Proof.

The equivalence of (i) and (ii) has been already proved above. Condition (iii) means that | y | - α ∈ L ∞ ⁢ ( B ) , | y | - β ∈ L 1 ⁢ ( B ) , where B is the unit ball of ℝ M , and these are necessary for the boundedness of S α , β ⁢ ( 1 ) .

To prove that (iii) implies (ii), write S α , β ⁢ ( 1 ) ⁢ f = S 1 ⁢ ( f ) + S 2 ⁢ ( f ) , where for f ≥ 0

S 1 ⁢ ( f ) ⁢ ( y ) = ( | y | ∧ 1 ) - α ⁢ ∫ B | z | - β ⁢ e - | y - z | 2 κ ⁢ f ⁢ ( z ) ⁢ 𝑑 z ≤ ∫ B | z | - β ⁢ f ⁢ ( z ) ⁢ 𝑑 z ≤ C ⁢ ∥ f ∥ ∞

and

S 2 ⁢ ( f ) ⁢ ( y ) = ( | y | ∧ 1 ) - α ⁢ ∫ ℝ M ∖ B e - | y - z | 2 κ ⁢ f ⁢ ( z ) ⁢ 𝑑 z ≤ ∥ f ∥ ∞ ⁢ ∫ ℝ M e - | z | 2 k ⁢ 𝑑 z ≤ C ⁢ ∥ f ∥ ∞ . ∎

References

[1] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau Stand. Appl. Math. 55, U. S. Government Printing Office, Washington, 1964. 10.1115/1.3625776Search in Google Scholar

[2] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations 32 (2007), no. 7–9, 1245–1260. 10.1080/03605300600987306Search in Google Scholar

[3] H. Dong and T. Phan, On parabolic and elliptic equations with singular or degenerate coefficients, preprint (2020), https://arxiv.org/abs/2007.04385. Search in Google Scholar

[4] H. Dong and T. Phan, Parabolic and elliptic equations with singular or degenerate coefficients: the Dirichlet problem, Trans. Amer. Math. Soc. 374 (2021), no. 9, 6611–6647. 10.1090/tran/8397Search in Google Scholar

[5] H. Dong and T. Phan, Weighted mixed-norm L p -estimates for elliptic and parabolic equations in non-divergence form with singular coefficients, Rev. Mat. Iberoam. 37 (2021), no. 4, 1413–1440. 10.4171/rmi/1233Search in Google Scholar

[6] H. Dong and T. Phan, Weighted mixed-norm L p estimates for equations in non-divergence form with singular coefficients: The Dirichlet problem, J. Funct. Anal. 285 (2023), no. 2, Paper No. 109964. 10.1016/j.jfa.2023.109964Search in Google Scholar

[7] P. M. N. Feehan and C. A. Pop, A Schauder approach to degenerate-parabolic partial differential equations with unbounded coefficients, J. Differential Equations 254 (2013), no. 12, 4401–4445. 10.1016/j.jde.2013.03.006Search in Google Scholar

[8] A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Progr. Nonlinear Differential Equations Appl. 16, Birkhäuser, Basel, 1995. 10.1007/978-3-0348-9234-6Search in Google Scholar

[9] A. Lunardi, Interpolation Theory, Appunti. Sc. Norm. Super. Pisa (N. S.) 16, Edizioni della Normale, Pisa, 1999. Search in Google Scholar

[10] G. Metafune, L. Negro and C. Spina, Degenerate operators on the half-line, J. Evol. Equ. 22 (2022), no. 3, Paper No. 60. 10.1007/s00028-022-00814-6Search in Google Scholar

[11] G. Metafune, L. Negro and C. Spina, L p estimates for the Caffarelli–Silvestre extension operators, J. Differential Equations 316 (2022), 290–345. 10.1016/j.jde.2022.01.049Search in Google Scholar

[12] G. Metafune, L. Negro and C. Spina, A unified approach to degenerate problems in the half-space, J. Differential Equations 351 (2023), 63–99. 10.1016/j.jde.2022.12.018Search in Google Scholar

[13] G. Metafune, L. Negro and C. Spina, Elliptic and parabolic problems for a Bessel-type operator, Recent Advances in Mathematical Analysis—Celebrating the 70th Anniversary of Francesco Altomare, Trends Math., Birkhäuser/Springer, Cham (2023), 397–424. 10.1007/978-3-031-20021-2_20Search in Google Scholar

[14] E. Sinestrari, On the abstract Cauchy problem of parabolic type in spaces of continuous functions, J. Math. Anal. Appl. 107 (1985), no. 1, 16–66. 10.1016/0022-247X(85)90353-1Search in Google Scholar

Received: 2023-09-15

Published Online: 2023-11-20

Published in Print: 2024-09-01

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Articles in the same Issue

https://doi.org/10.1515/forum-2023-0334

Keywords for this article

Degenerate elliptic operators; boundary degeneracy; Schauder estimates