Large $s$-harmonic functions and boundary blow-up solutions for the fractional Laplacian

By Nicola Abatangelo. 2013

127 citations recorded by the source corpus.

We present a notion of weak solution for the Dirichlet problem driven by the fractional Laplacian, following the Stampacchia theory. Then, we study semilinear problems on bounded domains $\Omega$ with two different boundary conditions at the same time: the shape of the solution outside $\Omega$ and a weighted limit to the boundary. We allow the nonlinearity to be positive or negative and we look for solutions blowing up at the boundary. Our starting observation is the existence of s-harmonic functions which explode at the boundary: these will be used both as supersolutions in the case of negative nonlinearity and as subsolutions in the positive case.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/2e0ca99163adc6f57cd265d9b806be265d55efe1

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Citations in the corpus

Nonlocal elliptic equations in bounded domains: a survey (2015, methodology): “…Serra [55]; • Boundary regularity of solutions: Bogdan [8], the author and Serra [45, 46, 47], Grubb [29, 30], Chen-Song [16], Barles-Chasseigne-Imbert [4], BogdanGrzywny-Ryznar [9], and Bogdan-Kumagai-Kwaśnicki [10]; • Other qualitative properties of solutions: Birkner-López-Wakolbinger [7],…”
Fractional Adams–Moser–Trudinger type inequalities (2015, unknown): “Proposition 14 For an open set Ω ⊂ Rn, let σ ∈ (0, 1) and u ∈ Lσ(R n) ∩ C0,α(Ω) for some α ∈ (σ, 1], or σ ∈ [1, 2) and u ∈ Lσ(R n)∩C1,α(Ω) for some α ∈ (σ− 1, 1] .”
A new look at the fractional Poisson problem via the logarithmic Laplacian (2019, methodology): “Starting with the work of Bogdan [9], these formulas have been derived under rather general assumptions on Ω, see [1, 12] and the references therein.”
Regularity of spectral fractional Dirichlet and Neumann problems (2014, background): “When f ∈ H s−2a p (Ω), the solutions are in spaces H (a−1)(s) p (Ω), which allow a blowup of u (of the form da−1) at ∂Ω, see also Abatangelo [A14].”
Entire $s$-harmonic functions are affine (2014, background): “ville theorem holds as well and was proved by K. Bogdan et al. in [4]. Applications of Liouville theorems for nonlocal operators in the study of nonlocal elliptic systems of equations can be found in [1,8,13,14]. The aim of this paper is to classify all s-harmonic functions in RN, thereby obtaining ”
Local and nonlocal boundary conditions for μ-transmission and fractional elliptic pseudodifferential operators (2014, methodology): “d in Ω with problems prescribing a value of uon the exterior Rn \ Ω, obtaining new results for the latter, that were treated recently by e.g. Felsinger, Kassman and Voigt in [FKV14] and Abatangelo in [A14]. For nonhomogeneous problems the solutions can be “large” at the boundary, cf. [A14] and its r”
Getting Acquainted with the Fractional Laplacian (2017, background): “See [1] also for a justification of the boundary behavior, as well as the study of Dirichlet problems prescribing a singular boundary trace.”