Homogeneous eigenvalue problems in Orlicz-Sobolev spaces
DOI:
https://doi.org/10.12775/TMNA.2023.008Słowa kluczowe
Orlicz spaces, nonlinear eigenvalues, asymptotic behaviorAbstrakt
In this article we consider a homogeneous eigenvalue problem ruled by the fractional $g$-Laplacian operator whose Euler-Lagrange equation is obtained by minimization of a quotient involving Luxemburg norms. We prove existence of an infinite sequence of variational eigenvalues and study its behavior as the fractional parameter $s\uparrow 1$ among other stability results.Bibliografia
A. Alberico, A. Cianchi, L. Pick and L. Slaviková, On fractional Orlicz–Sobolev spaces, Anal. Math. Phys. 11 (2021), no. 2, 1–21.
A. Alberico, A. Cianchi, L. Pick and L. Slaviková, On the limit as s → 1− of possibly non-separable fractional Orlicz–Sobolev spaces, Rend. Lincei Mat. Appl. 31 (2021), no. 4, 879–899.
D. Applebaum, Lévy processes – from probability to finance and quantum groups, Notices Amer. Math. Soc. 51 (2004), 1336–1347.
S. Bahrouni and A. Salort, Neumann and Robin type boundary conditions in fractional Orlicz–Sobolev spaces, ESAIM Control Optim. Calc. Var. 27, article no. S15.
J. Bourgain, H. Brezis and P. Mironescu, Another look at Sobolev spaces, Optimal Control and Partial Differential Equations (2001), 439–455.
J. Bourgain, H. Brezis and P. Mironescu, Limiting embedding theorems for W s,p when s ↑ 1 and applications, J. Anal. Math. 87 (2002), no. 1, 77–101.
H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer Science & Business Media, 2010.
C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, vol. 1, Springer International Publishing, Switzerland, 2016.
N. Cantizano, A. Salort and J. Spedaletti, Continuity of solutions for the G-Laplacian operator, Proc. Roy. Soc. Edinburgh Sect. A, 1–28.
T. Champion and L. De Pascale, Asymptotic behaviour of nonlinear eigenvalue problems involving p-Laplacian-type operators, Proc. Roy. Soc. Edinburgh Sect. , 137 (2007), 1179–1195.
R. Cont and P. Tankov, Financial Modelling With Jump Processes, Financial Mathematics Series, Chapman & Hall/CRC, Boca Raton, FL, 2004.
J. Fernández Bonder, M. Pérez-Llanos and A. Salort, A Hölder infinity Laplacian obtained as limit of Orlicz fractional Laplacians, Rev. Mat. Complut., DOI:10.1007/s13163-021-00390-2.
J. Fernández Bonder, J.P. Pinasco and A. Salort, Quasilinear eigenvalues, Rev. Un. Mat. Argentina 56 (2015), no. 1, 1–25.
J. Fernández Bonder and A. Salort, Fractional order Orlicz–Sobolev spaces, J. Funct. Anal. 277 (2019), no. 2, 333–367.
J. Fernández Bonder and A. Salort, Magnetic fractional order Orlicz-Sobolev spaces, Studia Math. 259 (2021), no. 1, 1–24.
J. Fernández Bonder, A. Salort and H. Vivas, Interior and up to the boundary regularity for the fractional g-Laplacian: the convex case Nonlinear Anal. 223 (2022), 113060.
J. Fernández Bonder, A. Salort and H. Vivas, Global Hölder reglarity for eigenfunctions of the fractional g-Laplacian J. Math. Anal. Appl. 526 (2023), no. 1, 127332.
J. Fernández Bonder, A. Silva and J. Spedaletti, Gamma convergence and asymptotic behavior for eigenvalues of nonlocal problems, Discrete Contin. Dynam. Systems 41 (2021), no. 5, 2125–2140.
G. Franzina and P. Lindqvist, An eigenvalue problem with variable exponents, Nonlinear Anal. 85 (2013), 1–16.
M. Garcı́a-Huidobro, V.K. Le, R. Manásevich and K. Schmitt, On principal eigenvalues for quasilinear elliptic differential operators: an Orlicz–Sobolev space setting, NoDEA Nonlinear Differential Equations Appl. 6 (1999), no. 2, 207–225.
N. Garofalo, Fractional thoughts (2017), preprint, arXiv: 1712.03347.
J. Giacomoni, D. Kumar and K. Sreenadh, Boundary regularity results for strongly nonhomogeneous (p, q)-fractional problems (2021), preprint, arXiv-2102.
J.P. Gossez and R. Mansevich, On a nonlinear eigenvalue problem in Orlicz–Sobolev spaces, Proc. Roy. Soc. Edinburgh Sect. A 132 (2002), no. 4, 891–909.
M. Krasnosel’skiı̆ and J. Rutickiı̆, Convex Functions and Orlicz Spaces, P. Noordhoff Ltd., Groningen, 1961.
J. Lamperti, On the isometries of certain function-spaces, Pacific J. Math. 8 (1958), no. 3, 459–466.
P. Lindqvist, A nonlinear eigenvalue problem, Topics in Mathematical Analysis 3 (2008), 175–203.
P. Lindqvist, Notes on the p-Laplace Equation, no. 161, University of Jyväskylä, 2017.
S. Molina, A. Salort and H. Vivas, Maximum principles, Liouville theorem and symmetry results for the fractional g-Laplacian, Nonlinear Anal. 212 (2021), 112465.
D. Motreanu, V. Motreanu and N. Papageorgiou, Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems, Springer–Verlag, New York, 2014.
I. Peral, Multiplicity of Solutions for the p-Laplacian, International Center for Theoretical Physics Lecture Notes, Trieste, 1997.
A. Ponce, A new approach to Sobolev spaces and connections to Γ-convergence, Calc. Var. Partial Differential Equations 19 (2004), no. 3, 229–255.
P. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, CBMS Regional Conference Series in Mathematics, vol. 65, Conference Board of the Mathematical Sciences, Washington, DC, American Mathematical Society, Providence, RI, 1986, MR 845785.
A. Salort, Eigenvalues and minimizers for a non-standard growth non-local operator, J. Differential Equations 268 (2020), no. 9, 5413–5439.
A. Salort and H. Vivas, Fractional eigenvalues in Orlicz spaces with no ∆2 condition, J. Differential Equations 327 (2022), 166–188.
G. Samorodnitsky and M. Taqqu, Stable Non-Gaussian Random Processes: Stochastic Models With Infinite Variance, Chapman and Hall, New York, 1994.
Pobrania
Opublikowane
Jak cytować
Numer
Dział
Licencja
Prawa autorskie (c) 2023 Julián Fernández Bonder, Ariel Salort, Hernán Vivas
Utwór dostępny jest na licencji Creative Commons Uznanie autorstwa – Bez utworów zależnych 4.0 Międzynarodowe.
Statystyki
Liczba wyświetleń i pobrań: 0
Liczba cytowań: 0
