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 and Applied Analysis, 2020, 19 (3) : 1463-1483. doi: 10.3934/cpaa.2020074
References:
[1]

R. A. Adams, Sobolev Spaces, Academic, New York, 1975.

[2]

S. Agmon, The Lp approach to the Dirichlet problem. 1. regularity theorems, Ann. Scuola Norm. Sup. Pisa

[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

[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.

[5]

A. Anop and T. Kasirenko, Elliptic boundary-value problems in Hörmander spaces, meth. Funct. Anal. Topol, 22 (2016), 295–310.

[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.

[7]

H. O. Cordes, Elliptic Pseudodifferential Operators- an Abstact Theory, Lect. Notes in Maths., 756, Springer, Berlin, 1979.

[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.

[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.

[10]

G. Grubb, Functional Calculus of Pseudodifferential Boundary Problems, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-0769-6.

[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.

[12]

L. Hörmander, Linear Partial Differential Operators, Springer, Berlin, 1963.

[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.

[14]

T. Kato, Perturbation Theory for Linear Operators, 2nd edition, Springer, Berlin, 1995.

[15]

J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems, I, Springer, Berlin, 1972.

[16]

A. S. Markus, Introduction to the Spectral Theory of Polynomial Operator Pencils, Amer. Math. Soc., Providence, R.I, 1988.

[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.

[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.

[19]

V. A. Mikhailets and A. A. Murach, Hörmander Spaces, Interpolation, and Elliptic Problems, De Gruyter, Berlin, 2014. doi: 10.1515/9783110296891.

[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.

[21]

C. E. Rickart, General Theory of Banach Algebras, Van Nostrand, New York, 1960.

[22]

M. A. Shubin, Pseudodifferential Operators and Spectral Theory, Berlin, Springer, 2001. doi: 10.1007/978-3-642-56579-3.

[23]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amsterdam, 1978.

[24]

L. R. Volevich and B. P. Paneyakh, Certain spaces of generalized functions and embedding theorems, Russ. Math. Surv., 20 (1965), 1-73. 

[25]

V. Volpert, Elliptic Partial Differential Equations, Vol.1, Birkhäuser, Basel, 2011. doi: 10.1007/978-3-0348-0813-2.

show all references

References:
[1]

R. A. Adams, Sobolev Spaces, Academic, New York, 1975.

[2]

S. Agmon, The Lp approach to the Dirichlet problem. 1. regularity theorems, Ann. Scuola Norm. Sup. Pisa

[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

[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.

[5]

A. Anop and T. Kasirenko, Elliptic boundary-value problems in Hörmander spaces, meth. Funct. Anal. Topol, 22 (2016), 295–310.

[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.

[7]

H. O. Cordes, Elliptic Pseudodifferential Operators- an Abstact Theory, Lect. Notes in Maths., 756, Springer, Berlin, 1979.

[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.

[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.

[10]

G. Grubb, Functional Calculus of Pseudodifferential Boundary Problems, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-0769-6.

[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.

[12]

L. Hörmander, Linear Partial Differential Operators, Springer, Berlin, 1963.

[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.

[14]

T. Kato, Perturbation Theory for Linear Operators, 2nd edition, Springer, Berlin, 1995.

[15]

J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems, I, Springer, Berlin, 1972.

[16]

A. S. Markus, Introduction to the Spectral Theory of Polynomial Operator Pencils, Amer. Math. Soc., Providence, R.I, 1988.

[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.

[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.

[19]

V. A. Mikhailets and A. A. Murach, Hörmander Spaces, Interpolation, and Elliptic Problems, De Gruyter, Berlin, 2014. doi: 10.1515/9783110296891.

[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.

[21]

C. E. Rickart, General Theory of Banach Algebras, Van Nostrand, New York, 1960.

[22]

M. A. Shubin, Pseudodifferential Operators and Spectral Theory, Berlin, Springer, 2001. doi: 10.1007/978-3-642-56579-3.

[23]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amsterdam, 1978.

[24]

L. R. Volevich and B. P. Paneyakh, Certain spaces of generalized functions and embedding theorems, Russ. Math. Surv., 20 (1965), 1-73. 

[25]

V. Volpert, Elliptic Partial Differential Equations, Vol.1, Birkhäuser, Basel, 2011. doi: 10.1007/978-3-0348-0813-2.

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