Note on abstract elliptic equations with nonlocal boundary in time condition

Abstract

Our main purpose of this paper is to study the linear elliptic equation with nonlocal in time condition. The problem is taken in abstract Hilbert space $H$. In concrete form, the elliptic equation has been extensively investigated in many practical areas, such as geophysics, plasma physics, bioelectric field problems. Under some assumptions of the input data, we obtain the well-posed result for the solution. In the first part, we study the regularity of the solution. In the second part, we investigate the asymptotic behaviour when some paramteres tend to zero.

Keywords

• Cauchy problem
• elliptic equations
• well-posedness
• regularity

Kaynakça

1. [1] H.N. Dinh, D.V. Nguyen, H. Sahli, A non-local boundary value problem method for the Cauchy problem for elliptic equations, Inverse Problems, 25:055002, 2009.
2. [2] M.M. Lavrentev, V.G. Romanov, and S.P. Shishatskii, Ill-posed problems of mathematical physics and analysis, Transla- tions of Mathematical Monographs, Vol. 64, American Mathematical Society, Providence, RI, 1986.
3. [3] C.R. Johnson, Computational and numerical methods for bioelectric field problems, Crit. Rev. Biomed. Eng. 25 (1997), pp. 1–81.
4. [4] R. Gorenflo, Funktionentheoretische bestimmung des aussenfeldes zu einer zweidimensionalen magnetohydrostatischen konfiguration, Z. Angew. Math. Phys. 16 (1965), pp. 279–290.
5. [5] J. Hadamard, Lectures on the Cauchy Problem in Linear Differential Equations, Yale University Press, New Haven, CT, 1923.
6. [6] T. Hohage. Regularization of exponentially ill-posed problems, Numer. Funct. Anal. Optim., 21(3-4):439–464, 2000.
7. [7] V. Isakov, Inverse Problems for Partial Differential Equations, Volume 127 of Applied Mathematical Sciences, Springer, New York, second edition, 2006.
8. [8] N.H. Tuan, D.D. Trong, P.H. Quan, A note on a Cauchy problem for the Laplace equation: regularization and error estimates, Appl. Math. Comput. 217 (2010), no. 7, 2913–2922.

9. [9] N.A. Tuan, D. O’regan, D. Baleanu, N.H. Tuan, On time fractional pseudo-parabolic equations with nonlocal integral conditions, Evolution Equation and Control Theory, 10.3934/eect.2020109.
10. [10] N.H. Can, N.H. Tuan, V.V. Au, L.D. Thang, Regularization of Cauchy abstract problem for a coupled system for nonlinear elliptic equations, J. Math. Anal. Appl. 462 (2018), no. 2, 1148–1177.
11. [11] M.M. Lavrentev, V.G. Romanov, and S.P. Shishatskii, Ill-posed problems of mathematical physics and analysis, Transla- tions of Mathematical Monographs, Vol. 64, American Mathematical Society, Providence, RI, 1986.
12. [12] C.R. Johnson, Computational and numerical methods for bioelectric field problems, Crit. Rev. Biomed. Eng. 25 (1997), pp. 1–81.
13. [13] V.A. Khoa, M.T.N. Truong, N.H.M. Duy, N.H. Tuan, The Cauchy problem of coupled elliptic sine-Gordon equations with noise: Analysis of a general kernel-based regularization and reliable tools of computing, Comput. Math. Appl. 73 (2017), no. 1, 141-162.
14. [14] N.H. Tuan, L.D. Thang, D. Lesnic, A new general filter regularization method for Cauchy problems for elliptic equations with a locally Lipschitz nonlinear source, J. Math. Anal. Appl. 434 (2016), no. 2, 1376-1393.
15. [15] B. Kaltenbacher, A. Kirchner, B. Vexler, Adaptive discretizations for the choice of a Tikhonov regularization parameter in nonlinear inverse problems, Inverse Problems 27 (2011) 125008.
16. [16] B. Kaltenbacher, W. Polifke, Some regularization methods for a thermoacoustic inverse problem, in: Special Issue M.V. Klibanov, J. Inverse Ill-Posed Probl. 18 (2011) 997-1011.
17. [17] B. Kaltenbacher, F. Schoepfer, Th. Schuster, Convergence of some iterative methods for the regularization of nonlinear ill-posed problems in Banach spaces, Inverse Problems 25 (2009) 065003, 19 pp.
18. [18] M. Asaduzzaman, Existence Results for a Nonlinear Fourth Order Ordinary Differential Equation with Four-Point Bound- ary Value Conditions . Advances in the Theory of Nonlinear Analysis and its Application , 4 (4) (2020), 233–242.
19. [19] M. Massar, On a fourth-order elliptic Kirchhoff type problem with critical Sobolev exponent, Advances in the Theory of Nonlinear Analysis and its Application , 4 (4) , 394–401.
20. [20] I. Kim, Semilinear problems involving nonlinear operators of monotone type, Results in Nonlinear Analysis , 2 (1)(2009), 25–35.
21. [21] S. Weng, X. Liu, Z. Chao, Some Fixed Point Theorems for Cyclic Mapping in a Complete b-metric-like Space, Results in Nonlinear Analysis , 2020 V3 I4 , 207-213.