Boundary problems for domains with conical exits at infinity and limit operators

Abstract

We consider a general boundary value problem1 in a smooth unbounded domain with conical exits at infinity, where the coefficients belong to the space of infinitely differentiable functions on bounded together with all derivatives. We associate with boundary value problem (1) a bounded linear operatorand we define for the family of limit operators. We prove that is a Fredholm operator if and only if the boundary value problem (1) is elliptic at every point and all limit operators are invertible.We also consider a realization of the differential operator as unbounded operator in the Hilbert space with domain We prove that if the boundary value problem (1) is uniformly elliptic in then the essential spectrum of is the union of the spectra of all limit operators.

Keywords:

• unbounded domain
• boundary value problem
• Fredholm theory
• essential spectrum
• limit operators

AMS Subject Classifications:

• 35J40
• 35J10
• 58C40

Acknowledgements

I thank the National System of Investigators of Mexico for their support.

Additional information

Funding

The work is partially supported by the CONACYT project 000000000179872.