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The Poisson problem for the fractional Hardy operator: Distributional identities and singular solutions
- Source :
- Transactions of the American Mathematical Society.
- Publication Year :
- 2021
- Publisher :
- American Mathematical Society (AMS), 2021.
-
Abstract
- The purpose of this paper is to study and classify singular solutions of the Poisson problem { L μ s u = f in Ω ∖ { 0 } , u = 0 in R N ∖ Ω \begin{equation*} \left \{ \begin {aligned} \mathcal {L}^s_\mu u = f \quad \ \text {in}\ \, \Omega \setminus \{0\},\\ u =0 \quad \ \text {in}\ \, \mathbb {R}^N \setminus \Omega \ \end{aligned} \right . \end{equation*} for the fractional Hardy operator L μ s u = ( − Δ ) s u + μ | x | 2 s u \mathcal {L}_\mu ^s u= (-\Delta )^s u +\frac {\mu }{|x|^{2s}}u in a bounded domain Ω ⊂ R N \Omega \subset \mathbb {R}^N ( N ≥ 2 N \ge 2 ) containing the origin. Here ( − Δ ) s (-\Delta )^s , s ∈ ( 0 , 1 ) s\in (0,1) , is the fractional Laplacian of order 2 s 2s , and μ ≥ μ 0 \mu \ge \mu _0 , where μ 0 = − 2 2 s Γ 2 ( N + 2 s 4 ) Γ 2 ( N − 2 s 4 ) > 0 \mu _0 = -2^{2s}\frac {\Gamma ^2(\frac {N+2s}4)}{\Gamma ^2(\frac {N-2s}{4})}>0 is the best constant in the fractional Hardy inequality. The analysis requires a thorough study of fundamental solutions and associated distributional identities. Special attention will be given to the critical case μ = μ 0 \mu = \mu _0 which requires more subtle estimates than the case μ > μ 0 \mu >\mu _0 .
- Subjects :
- Applied Mathematics
General Mathematics
Operator (physics)
010102 general mathematics
Order (ring theory)
35R11, 35J75, 35B40
01 natural sciences
Omega
010101 applied mathematics
Combinatorics
Mathematics - Analysis of PDEs
Domain (ring theory)
FOS: Mathematics
0101 mathematics
Fractional Laplacian
Poisson problem
Analysis of PDEs (math.AP)
Mathematics
Subjects
Details
- ISSN :
- 10886850 and 00029947
- Database :
- OpenAIRE
- Journal :
- Transactions of the American Mathematical Society
- Accession number :
- edsair.doi.dedup.....1d15a8fc582741dcca16019cc78b1321
- Full Text :
- https://doi.org/10.1090/tran/8443