Semilinear Dirichlet problem for the fractional Laplacian

Bogdan, Krzysztof; Jarohs, Sven; Kania, Edyta

doi:10.1016/j.na.2019.04.011

Cited by 18 publications

(19 citation statements)

References 27 publications

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“…Throughout the remainder of this section, let Ω ⊂ R N be a fixed open and bounded set with C 2boundary. We recall the following result from [1] (see also [8][9][10]).…”

Section: The Solution Map and Related Operatorsmentioning

confidence: 97%

A new look at the fractional Poisson problem via the Logarithmic Laplacian

Jarohs¹,

Saldaña²,

Weth³

2019

Preprint

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We analyze the s-dependence of solutions u s to the family of fractional Poisson problemsin an open bounded set Ω ⊂ R N , s ∈ (0, 1). In the case where Ω is of class C 2 and f ∈ C α (Ω) for some α > 0, we show that the map (0, 1) → L ∞ (Ω), s → u s is of class C 1 , and we characterize the derivative ∂ s u s in terms of the logarithmic Laplacian of f . As a corollary, we derive pointwise monotonicity properties of the solution map s → u s under suitable assumptions on f and Ω. Moreover, we derive explicit bounds for the corresponding Green operator on arbitrary bounded domains which are new even for the case s = 1, i.e., for the local Dirichlet problem −∆u = f in Ω, u ≡ 0 on ∂ Ω.

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“…Throughout the remainder of this section, let Ω ⊂ R N be a fixed open and bounded set with C 2boundary. We recall the following result from [1] (see also [8][9][10]).…”

Section: The Solution Map and Related Operatorsmentioning

confidence: 97%

A new look at the fractional Poisson problem via the Logarithmic Laplacian

Jarohs¹,

Saldaña²,

Weth³

2019

Preprint

Self Cite

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“…In the paper [2] on semilinear perturbation of fractional Laplacians −(−∆) α/2 on Ê d , d ∈ AE, 0 < α < 2 ∧ d, a Kato class J α (D) of measurable functions q on an open set D in Ê d is defined by uniform integrability of the functions G D (x, •)q, x ∈ D, with respect to Lebesgue measure on D, where G D denotes the corresponding Green function on D (see [2,Definition 1.23]). Let C 0 (D), B b (D), respectively, denote the space of all real functions on D which are continuous and vanish at infinity with respect to D, are Borel measurable and bounded, respectively.…”

Section: Introduction Notation and First Propertiesmentioning

confidence: 99%

“…Our main results are that F ui (G) ⊂ F co (G) if there exists a strictly positive function in F co (G) (Theorem 2.1), and F co (G) ⊂ F ui (G) if there exists a strictly positive function in F ui (G) (Theorem 2.2). In Section 3 we provide general examples (covering the situation in [2]), where the assumptions are satisfied. Before studying the relation between F ui (G) and F co (G), let us recall that a subset F of E + is called uniformly integrable if, for every ε > 0, there exists an integrable g ∈ E + such that {f ≥g} f dµ ≤ ε for every f ∈ F .…”

Section: Introduction Notation and First Propertiesmentioning

confidence: 99%

Compactness of integral operators and uniform integrability on measure spaces

Hansen¹

2022

Preprint

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Let (E, E, µ) be a measure space and G : E × E → [0, ∞] be measurable. Moreover, let F ui denote the set of all q ∈ E + (measurable numerical functions q ≥ 0 on E) such that {G(x, •)q : x ∈ E} is uniformly integrable, and let F co denote the set of all q ∈ E + such that the mapping f → G(f q) := G(•, y)f (y)q(y) dµ(y) is a compact operator on the space E b of bounded measurable functions on E (equipped with the sup-norm).It is shown that F ui = F co provided both F ui and F co contain strictly positive functions.

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“…We prove that for every η > 0 there is δ > 0 such that for every D ⊂ B(0, R) and measures λ 1 and λ 2 on B(0, R) In the article we also study the operator W D , see Definition 4.5. The operator W D is a boundary trace operator introduced in [4] building on results in [5]. In [4] it plays a significant role in the semilinear Dirichlet problem for the fractional Laplacian.…”

Section: Introductionmentioning

confidence: 99%

“…The operator W D is a boundary trace operator introduced in [4] building on results in [5]. In [4] it plays a significant role in the semilinear Dirichlet problem for the fractional Laplacian. We generalize the operator for the case of the subordinate Brownian motion and use it as a tool to obtain a finite measure for the Martin integral in the representation.…”

Section: Introductionmentioning

confidence: 99%