On the bounds of the sum of eigenvalues for a Dirichlet problem involving mixed fractional Laplacians

Journal of Differential Equations - Tập 317 - Trang 1-31 - 2022
Huyuan Chen1, Mousomi Bhakta2, Hichem Hajaiej3
1School of Mathematics and Statistics, Jiangxi Normal University, Nanchang, Jiangxi 330022, PR China
2Department of Mathematics, Indian Institute of Science Education and Research (IISER-Pune), Pune 411008, India
3California State University, Los Angeles, 5151, USA

Tài liệu tham khảo

Autuori, 2017, Mathematical models for nonlocal elastic composite material, Adv. Nonlinear Anal., 6, 355, 10.1515/anona-2016-0186 Bhakta Bhakta, 2019, Multiplicity results for (p,q) fractional elliptic equations involving critical nonlinearities, Adv. Differ. Equ., 24, 185 Bhakta, 2017, Multiplicity results and sign changing solutions of non-local equations with concave-convex nonlinearities, Differ. Integral Equ., 30, 387 Biagi Caffarelli, 2008, Regularity estimates for the solution and the free boundary to the obstacle problem for the fractional Laplacian, Invent. Math., 171, 425, 10.1007/s00222-007-0086-6 Chen, 2018, Liouville theorem for the fractional Lane-Emden equation in an unbounded domain, J. Math. Pures Appl., 111, 21, 10.1016/j.matpur.2017.07.010 Chen, 2015, Large solution to elliptic equations involving fractional Laplacian, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 32, 1199, 10.1016/j.anihpc.2014.08.001 Chen, 2014, Lower and upper bounds of Dirichlet eigenvalues for totally characteristic degenerate elliptic operators, Sci. China Math., 57, 2235, 10.1007/s11425-014-4895-y Chen, 2018, Classification of isolated singularities of nonnegative solutions to fractional semilinear elliptic equations and the existence results, J. Lond. Math. Soc., 97, 196, 10.1112/jlms.12104 Chen, 2019, Initial trace of positive solutions to fractional diffusion equations with absorption, J. Funct. Anal., 276, 1145, 10.1016/j.jfa.2018.10.013 Chen, 2019, The Dirichlet problem for the logarithmic Laplacian, Commun. Partial Differ. Equ., 44, 1100, 10.1080/03605302.2019.1611851 Chen, 2017, Universal inequality and upper bounds of eigenvalues for non-integer poly-Laplacian on a bounded domain, Calc. Var. Partial Differ. Equ., 56, 10.1007/s00526-017-1220-y Cheng, 2007, Bounds on eigenvalues of Dirichlet Laplacian, Math. Ann., 337, 159, 10.1007/s00208-006-0030-x Cheng, 2011, A lower bound for eigenvalues of a clamped plate problem, Calc. Var. Partial Differ. Equ., 42, 579, 10.1007/s00526-011-0399-6 Di Nezza, 2012, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136, 521, 10.1016/j.bulsci.2011.12.004 Elshahed, 2003, A fractional calculus model in semilunar heart valve vibrations Frank, 2018, Eigenvalue bounds for Schrödinger operators with complex potentials. III, Trans. Am. Math. Soc., 370, 219, 10.1090/tran/6936 Frank, 2018, Eigenvalue bounds for the fractional Laplacian: a review, 210 Frank, 2016, Uniqueness of radial solutions for the fractional Laplacian, Commun. Pure Appl. Math., 69, 1671, 10.1002/cpa.21591 Geisinger, 2014, A short proof of Weyl's law for fractional differential operators, J. Math. Phys., 10.1063/1.4861935 Harrell, 2009, Eigenvalue inequalities for Klein-Gordon operators, J. Funct. Anal., 256, 3977, 10.1016/j.jfa.2008.12.008 Hajaiej, 2013, Existence of minimizers of functional involving the fractional gradient in the absence of compactness, symmetry and monotonicity, J. Math. Anal. Appl., 399, 17, 10.1016/j.jmaa.2012.09.023 Hajaiej, 2013, On the optimality of the conditions used to prove the symmetry of the minimizers of some fractional constrained variational problems, Ann. Inst. Henri Poincaré, 14, 1425, 10.1007/s00023-012-0212-x Hajaiej, 2014, Symmetry of minimizers of some fractional problems, Appl. Anal., 94, 1 Jin, 2014, On a fractional Nirenberg problem, part I: blow up analysis and compactness of solutions, J. Eur. Math. Soc., 16, 1111, 10.4171/JEMS/456 Kröger, 1994, Estimates for sums of eigenvalues of the Laplacian, J. Funct. Anal., 126, 217, 10.1006/jfan.1994.1146 Li, 1983, On the Schrödinger equation and the eigenvalue problem, Commun. Math. Phys., 88, 309, 10.1007/BF01213210 Lieb, 1980, The number of bound states of one-body Schrödinger operators and the Weyl problem, Proc. Symp. Pure Math., 36, 241, 10.1090/pspum/036/573436 Magin, 2004, Fractional calculus in bioengineering 1, 2, 3, Crit. Rev. Biomed. Eng., 1, 32 Magin, 2007, Modelling of pulsating peripheral bioheat transfusing fractional calculus and constructal theory, J. Design Nature, 1, 18 Magin, 2009, Modeling the cardiac tissue electrode interface using fractional calculus, J. Vib. Control, 19, 1431 Melas, 2003, A lower bound for sums of eigenvalues of the Laplacian, Proc. Am. Math. Soc., 131, 631, 10.1090/S0002-9939-02-06834-X Musina, 2014, On fractional Laplacians, Commun. Partial Differ. Equ., 39, 1780, 10.1080/03605302.2013.864304 Pólya, 1961, On the eigenvalues of vibrating membranes (in memoriam Hermann Weyl), Proc. Lond. Math. Soc., 3, 419, 10.1112/plms/s3-11.1.419 Pucci, 2017, Asymptotic stability for nonlinear damped Kirchhoff systems involving the fractional p-Laplacian operator, J. Differ. Equ., 263, 2375, 10.1016/j.jde.2017.02.039 Ros-Oton, 2014, The Dirichlet problem for the fractional Laplacian: regularity up to the boundary, J. Math. Pures Appl., 101, 275, 10.1016/j.matpur.2013.06.003 Ros-Oton, 2014, The Pohozaev identity for the fractional Laplacian, Arch. Ration. Mech. Anal., 213, 587, 10.1007/s00205-014-0740-2 Servadei, 2013, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33, 2105, 10.3934/dcds.2013.33.2105 Triebel, 1983, Theory of Function Space, vol. 78 Weyl, 1912, Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung), Math. Ann., 71, 441, 10.1007/BF01456804 Y. Wang, H. Hichem, Kröger's type upper bounds for Dirichlet eigenvalues of the fractional Laplacian, preprint, 2020. Yolcu, 2013, Estimates for the sums of eigenvalues of the fractional Laplacian on a bounded domain, Commun. Contemp. Math., 15 Yolcu, 2014, Sharper estimates on the eigenvalues of Dirichlet fractional Laplacian, Discrete Contin. Dyn. Syst., 35, 2209, 10.3934/dcds.2015.35.2209