# American Institute of Mathematical Sciences

March  2020, 19(3): 1463-1483. doi: 10.3934/cpaa.2020074

## Fredholm theory for an elliptic differential operator defined on $\mathbb{R}^n$ and acting on generalized Sobolev spaces

 School of Mathematics and Statistics, The University of New South Wales, UNSW SYDNEY, Sydney, NSW 2052, Australia

Received  May 2019 Revised  July 2019 Published  November 2019

We consider a spectral problem for an elliptic differential operator debined on $\mathbb{R}^n$ and acting on the generalized Sobolev space $W^{0, \chi}_p(\mathbb{R}^n)$ for $1 < p < \infty$. We note that similar problems, but with $\mathbb{R}^n$ replaced by either a bounded region in $\mathbb{R}^n$ or by a closed manifold have been the subject of investigation by various authors. Then in this paper we establish, under the assumption of parameter-ellipticity, results pertaining to the existence and uniqueness of solutions of the spectral problem. Furthrermore, by utilizing the aforementioned results, we obain results pertaining to the spectral properties of the Banach space operator induced by the spectral problem.

Citation: Melvin Faierman. Fredholm theory for an elliptic differential operator defined on $\mathbb{R}^n$ and acting on generalized Sobolev spaces. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1463-1483. doi: 10.3934/cpaa.2020074
##### References:
 [1] R. A. Adams, Sobolev Spaces, Academic, New York, 1975.  Google Scholar [2] S. Agmon, The Lp approach to the Dirichlet problem. 1. regularity theorems, Ann. Scuola Norm. Sup. Pisa  Google Scholar [3] M. S. Agranovich, R. Denk and M. Faierman, Weakly smooth nonselfadjoint elliptic boundary problems, in Advances in Partial Differential Equations: Specral Theory, Microlocal Analysis Singular Manifolds  Google Scholar [4] M. S. Agranovich and M. I. Vishik, Elliptic problems with a parameter and parabolic problems of general type, Russ. Math. Surv., 19 (1964), 53–157.  Google Scholar [5] A. Anop and T. Kasirenko, Elliptic boundary-value problems in Hörmander spaces, meth. Funct. Anal. Topol, 22 (2016), 295–310.  Google Scholar [6] H. O. Cordes, On compactness of commutators of multiplication and convolutions, and boundedness of pseudo differential operators, J. Funct. Anal., 18 (1975), 115-131.  doi: 10.1016/0022-1236(75)90020-8.  Google Scholar [7] H. O. Cordes, Elliptic Pseudodifferential Operators- an Abstact Theory, Lect. Notes in Maths., 756, Springer, Berlin, 1979.  Google Scholar [8] H. O. Cordes, Spectral Theory of Linear Differential Operators and Comparison Algebras, London Math. Soc., Lecture Notes Series, 76, Camebridge Univ. Press, Cambridge, 1987. doi: 10.1017/CBO9780511662836.  Google Scholar [9] M. Faierman, Fredholm theory connected with a Douglis-Nirenberg system of differential equations over $\mathbf{R}^n$, Meth. Funct. Anal. Topol., 22 (2016), 330–345.  Google Scholar [10] G. Grubb, Functional Calculus of Pseudodifferential Boundary Problems, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-0769-6.  Google Scholar [11] G. Grubb and N. J. Kokholm, A global calculus of paramater dependant pseududifferential boundary problems, in Lp Sobolev spaces, Acta. Math., 171 (1993), 1–100. doi: 10.1007/BF02392532.  Google Scholar [12] L. Hörmander, Linear Partial Differential Operators, Springer, Berlin, 1963.  Google Scholar [13] R. Illner, On algebras of pseudo differential operators in $L_p(\mathbb{R}^n)$, Comm. Partial Differ. Equ., 2 (1977), 359–393. doi: 10.1080/03605307708820034.  Google Scholar [14] T. Kato, Perturbation Theory for Linear Operators, 2nd edition, Springer, Berlin, 1995.  Google Scholar [15] J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems, I, Springer, Berlin, 1972.  Google Scholar [16] A. S. Markus, Introduction to the Spectral Theory of Polynomial Operator Pencils, Amer. Math. Soc., Providence, R.I, 1988.  Google Scholar [17] V. A. Mikhailets and A. A. Murach, Elliptic operators in a refined scale of function spaces, Ukr. Math. J., 57, (2005), 817–825. doi: 10.1007/s11253-005-0231-6.  Google Scholar [18] V. A. Mikhailets and A. A. Murach, Elliptic systems of pseudodifferential equations in a refined scale on a closed manifold, Bull. Pol. Acad. Sci. Math., 56 (2008), 213-224.  doi: 10.4064/ba56-3-4.  Google Scholar [19] V. A. Mikhailets and A. A. Murach, Hörmander Spaces, Interpolation, and Elliptic Problems, De Gruyter, Berlin, 2014. doi: 10.1515/9783110296891.  Google Scholar [20] P. Rabier, Fredholm and regularity theory of Douglis-Nirenberg elliptic systems on $\mathbb{R}^n$, Math. Z., 270 (2012), 369–313. doi: 10.1007/s00209-010-0802-6.  Google Scholar [21] C. E. Rickart, General Theory of Banach Algebras, Van Nostrand, New York, 1960.  Google Scholar [22] M. A. Shubin, Pseudodifferential Operators and Spectral Theory, Berlin, Springer, 2001. doi: 10.1007/978-3-642-56579-3.  Google Scholar [23] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amsterdam, 1978.  Google Scholar [24] L. R. Volevich and B. P. Paneyakh, Certain spaces of generalized functions and embedding theorems, Russ. Math. Surv., 20 (1965), 1-73.   Google Scholar [25] V. Volpert, Elliptic Partial Differential Equations, Vol.1, Birkhäuser, Basel, 2011. doi: 10.1007/978-3-0348-0813-2.  Google Scholar

show all references

##### References:
 [1] R. A. Adams, Sobolev Spaces, Academic, New York, 1975.  Google Scholar [2] S. Agmon, The Lp approach to the Dirichlet problem. 1. regularity theorems, Ann. Scuola Norm. Sup. Pisa  Google Scholar [3] M. S. Agranovich, R. Denk and M. Faierman, Weakly smooth nonselfadjoint elliptic boundary problems, in Advances in Partial Differential Equations: Specral Theory, Microlocal Analysis Singular Manifolds  Google Scholar [4] M. S. Agranovich and M. I. Vishik, Elliptic problems with a parameter and parabolic problems of general type, Russ. Math. Surv., 19 (1964), 53–157.  Google Scholar [5] A. Anop and T. Kasirenko, Elliptic boundary-value problems in Hörmander spaces, meth. Funct. Anal. Topol, 22 (2016), 295–310.  Google Scholar [6] H. O. Cordes, On compactness of commutators of multiplication and convolutions, and boundedness of pseudo differential operators, J. Funct. Anal., 18 (1975), 115-131.  doi: 10.1016/0022-1236(75)90020-8.  Google Scholar [7] H. O. Cordes, Elliptic Pseudodifferential Operators- an Abstact Theory, Lect. Notes in Maths., 756, Springer, Berlin, 1979.  Google Scholar [8] H. O. Cordes, Spectral Theory of Linear Differential Operators and Comparison Algebras, London Math. Soc., Lecture Notes Series, 76, Camebridge Univ. Press, Cambridge, 1987. doi: 10.1017/CBO9780511662836.  Google Scholar [9] M. Faierman, Fredholm theory connected with a Douglis-Nirenberg system of differential equations over $\mathbf{R}^n$, Meth. Funct. Anal. Topol., 22 (2016), 330–345.  Google Scholar [10] G. Grubb, Functional Calculus of Pseudodifferential Boundary Problems, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-0769-6.  Google Scholar [11] G. Grubb and N. J. Kokholm, A global calculus of paramater dependant pseududifferential boundary problems, in Lp Sobolev spaces, Acta. Math., 171 (1993), 1–100. doi: 10.1007/BF02392532.  Google Scholar [12] L. Hörmander, Linear Partial Differential Operators, Springer, Berlin, 1963.  Google Scholar [13] R. Illner, On algebras of pseudo differential operators in $L_p(\mathbb{R}^n)$, Comm. Partial Differ. Equ., 2 (1977), 359–393. doi: 10.1080/03605307708820034.  Google Scholar [14] T. Kato, Perturbation Theory for Linear Operators, 2nd edition, Springer, Berlin, 1995.  Google Scholar [15] J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems, I, Springer, Berlin, 1972.  Google Scholar [16] A. S. Markus, Introduction to the Spectral Theory of Polynomial Operator Pencils, Amer. Math. Soc., Providence, R.I, 1988.  Google Scholar [17] V. A. Mikhailets and A. A. Murach, Elliptic operators in a refined scale of function spaces, Ukr. Math. J., 57, (2005), 817–825. doi: 10.1007/s11253-005-0231-6.  Google Scholar [18] V. A. Mikhailets and A. A. Murach, Elliptic systems of pseudodifferential equations in a refined scale on a closed manifold, Bull. Pol. Acad. Sci. Math., 56 (2008), 213-224.  doi: 10.4064/ba56-3-4.  Google Scholar [19] V. A. Mikhailets and A. A. Murach, Hörmander Spaces, Interpolation, and Elliptic Problems, De Gruyter, Berlin, 2014. doi: 10.1515/9783110296891.  Google Scholar [20] P. Rabier, Fredholm and regularity theory of Douglis-Nirenberg elliptic systems on $\mathbb{R}^n$, Math. Z., 270 (2012), 369–313. doi: 10.1007/s00209-010-0802-6.  Google Scholar [21] C. E. Rickart, General Theory of Banach Algebras, Van Nostrand, New York, 1960.  Google Scholar [22] M. A. Shubin, Pseudodifferential Operators and Spectral Theory, Berlin, Springer, 2001. doi: 10.1007/978-3-642-56579-3.  Google Scholar [23] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amsterdam, 1978.  Google Scholar [24] L. R. Volevich and B. P. Paneyakh, Certain spaces of generalized functions and embedding theorems, Russ. Math. Surv., 20 (1965), 1-73.   Google Scholar [25] V. Volpert, Elliptic Partial Differential Equations, Vol.1, Birkhäuser, Basel, 2011. doi: 10.1007/978-3-0348-0813-2.  Google Scholar
 [1] Alessandro Carbotti, Giovanni E. Comi. A note on Riemann-Liouville fractional Sobolev spaces. Communications on Pure & Applied Analysis, 2021, 20 (1) : 17-54. doi: 10.3934/cpaa.2020255 [2] Noah Stevenson, Ian Tice. A truncated real interpolation method and characterizations of screened Sobolev spaces. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5509-5566. doi: 10.3934/cpaa.2020250 [3] Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020049 [4] Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020320 [5] Yichen Zhang, Meiqiang Feng. A coupled $p$-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior. Electronic Research Archive, 2020, 28 (4) : 1419-1438. doi: 10.3934/era.2020075 [6] Karoline Disser. Global existence and uniqueness for a volume-surface reaction-nonlinear-diffusion system. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 321-330. doi: 10.3934/dcdss.2020326 [7] Gongbao Li, Tao Yang. Improved Sobolev inequalities involving weighted Morrey norms and the existence of nontrivial solutions to doubly critical elliptic systems involving fractional Laplacian and Hardy terms. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020469 [8] Jiaquan Liu, Xiangqing Liu, Zhi-Qiang Wang. Sign-changing solutions for a parameter-dependent quasilinear equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020454 [9] Chongyang Liu, Meijia Han, Zhaohua Gong, Kok Lay Teo. Robust parameter estimation for constrained time-delay systems with inexact measurements. Journal of Industrial & Management Optimization, 2021, 17 (1) : 317-337. doi: 10.3934/jimo.2019113 [10] Felix Finster, Jürg Fröhlich, Marco Oppio, Claudio F. Paganini. Causal fermion systems and the ETH approach to quantum theory. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020451 [11] Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020384 [12] Giulia Cavagnari, Antonio Marigonda. Attainability property for a probabilistic target in wasserstein spaces. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 777-812. doi: 10.3934/dcds.2020300 [13] Pierre-Etienne Druet. A theory of generalised solutions for ideal gas mixtures with Maxwell-Stefan diffusion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020458 [14] Juan Pablo Pinasco, Mauro Rodriguez Cartabia, Nicolas Saintier. Evolutionary game theory in mixed strategies: From microscopic interactions to kinetic equations. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020051 [15] Sergey Rashkovskiy. Hamilton-Jacobi theory for Hamiltonian and non-Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (4) : 563-583. doi: 10.3934/jgm.2020024 [16] Anna Abbatiello, Eduard Feireisl, Antoní Novotný. Generalized solutions to models of compressible viscous fluids. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 1-28. doi: 10.3934/dcds.2020345 [17] Qianqian Han, Xiao-Song Yang. Qualitative analysis of a generalized Nosé-Hoover oscillator. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020346 [18] Mark F. Demers. Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 217-256. doi: 10.3934/dcds.2020217 [19] Federico Rodriguez Hertz, Zhiren Wang. On $\epsilon$-escaping trajectories in homogeneous spaces. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 329-357. doi: 10.3934/dcds.2020365 [20] Giulia Luise, Giuseppe Savaré. Contraction and regularizing properties of heat flows in metric measure spaces. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 273-297. doi: 10.3934/dcdss.2020327

2019 Impact Factor: 1.105