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Existence of a unique positive entropy solution to a singular fractional Laplacian
, Article Complex Variables and Elliptic Equations ; 2020 ; Hesaaraki, M ; Sharif University of Technology
Taylor and Francis Ltd
2020
Abstract
In this paper, we study the existence of a positive solution to the elliptic problem: (Formula presented.) Here (Formula presented.) (N>2s) is an open bounded domain with smooth boundary, (Formula presented.) and (Formula presented.). For (Formula presented.), we take advantage of the convexity of Ω. The operator (Formula presented.) indicates the restricted fractional Laplacian, and μ is a non-negative Radon measure as a source term. The assumptions on f and h will be precised later. Besides, we will discuss the notion of entropy solution and its uniqueness for some specific measures. © 2020, © 2020 Informa UK Limited, trading as Taylor & Francis Group
A fractional Laplacian problem with mixed singular nonlinearities and nonregular data
, Article Journal of Elliptic and Parabolic Equations ; Volume 7, Issue 2 , 2021 , Pages 787-814 ; 22969020 (ISSN) ; Hesaaraki, M ; Sharif University of Technology
Birkhauser
2021
Abstract
In this note, we study on the existence and uniqueness of a positive solution to the following doubly singular fractional problem: {(-Δ)su=K(x)uq+f(x)uγ+μinΩ,u>0inΩ,u=0in(RN\u03a9).Here Ω ⊂ RN (N> 2 s) is an open bounded domain with smooth boundary, s∈ (0 , 1) , q> 0 , γ> 0 , and K(x) is a positive Hölder continuous function in which behaves as dist (x, ∂Ω) -β near the boundary with 0 ≤ β< 2 s. Also, 0 ≤ f, μ∈ L1(Ω) , or non-negative bounded Radon measures in Ω. Moreover, we assume that 0<βs+q<1, or βs+q>1 with 2 β+ q(2 s- 1) < (2 s+ 1). For s∈(0,12), we take advantage of the convexity of Ω. For any γ> 0 , we will prove the existence of a positive weak (distributional) solution to the above...
Nonlocal Lazer–McKenna-type problem perturbed by the Hardy’s potential and its parabolic equivalence
, Article Boundary Value Problems ; Volume 2021, Issue 1 , 2021 ; 16872762 (ISSN) ; Hesaaraki, M ; Karim Hamdani, M ; Thanh Chung, N ; Sharif University of Technology
Springer Science and Business Media Deutschland GmbH
2021
Abstract
In this paper, we study the effect of Hardy potential on the existence or nonexistence of solutions to the following fractional problem involving a singular nonlinearity: {(−Δ)su=λu|x|2s+μuγ+fin Ω,u>0in Ω,u=0in (RN∖Ω). Here 0 < s< 1 , λ> 0 , γ> 0 , and Ω ⊂ RN (N> 2 s) is a bounded smooth domain such that 0 ∈ Ω. Moreover, 0 ≤ μ, f∈ L1(Ω). For 0 < λ≤ Λ N,s, Λ N,s being the best constant in the fractional Hardy inequality, we find a necessary and sufficient condition for the existence of a positive weak solution to the problem with respect to the data μ and f. Also, for a regular datum of f, under suitable assumptions, we obtain some existence and uniqueness results and calculate the rate of...
Existence of a unique positive entropy solution to a singular fractional Laplacian
, Article Complex Variables and Elliptic Equations ; Volume 66, Issue 5 , 2021 , Pages 783-800 ; 17476933 (ISSN) ; Hesaaraki, M ; Sharif University of Technology
Taylor and Francis Ltd
2021
Abstract
In this paper, we study the existence of a positive solution to the elliptic problem: (Formula presented.) Here (Formula presented.) (N>2s) is an open bounded domain with smooth boundary, (Formula presented.) and (Formula presented.). For (Formula presented.), we take advantage of the convexity of Ω. The operator (Formula presented.) indicates the restricted fractional Laplacian, and μ is a non-negative Radon measure as a source term. The assumptions on f and h will be precised later. Besides, we will discuss the notion of entropy solution and its uniqueness for some specific measures. © 2020 Informa UK Limited, trading as Taylor & Francis Group
Singular PDEs with Irregular Data
, Ph.D. Dissertation Sharif University of Technology ; Hesaaraki, Mahmoud (Supervisor) ; Fotouhi Firoozabad, Morteza (Co-Supervisor)
Abstract
Singular differential equations have a wide range of applications. Hardy singularities, which are connected to inequalities of the same name and have various extensions, are the most well-known singularities. The application of Hardy inequalities in quantum physics and also in the linearization of reaction-diffusion equations in thermodynamics and combustion theory motivates researchers to examine them. Singularities on a domain's boundary are another well-known type of singularity. In the study of fluid mechanics and pseudoplastic flows, these singularities emerge.Differential equations with coefficients or functions that are simply functions belonging to $ L^1 $, or bounded Radon measures,...