# Singular perturbation for abstract elliptic equations and application

## TL;DR

This paper investigates the behavior of solutions to abstract elliptic equations with small parameters in Banach spaces, providing explicit formulas and uniform regularity results, and analyzing the limit as the parameter approaches zero.

## Contribution

It introduces a singular perturbation approach for abstract elliptic equations in Banach spaces, including explicit solutions and regularity properties.

## Key findings

- Explicit solution formulas derived
- Uniform Lp-regularity established
- Solution behavior analyzed as small parameter tends to zero

## Abstract

Boundary value problem for complete second order elliptic equation is considered in Banach space. The equation and boundary conditions involve a small and spectral parameter. The uniform L_{p}-regularity properties with respect to space variable and parameters are established. Here, the explicit formula for the solution is given and behavior of solution is derived when the small parameter approaches zero. It used to obtain singular perturbation result for abstract elliptic equation