Mathematica
Volumen 42, 2017, 211-226

# IMPROVED POINCARÉ INEQUALITIES AND SOLUTIONS OF THE DIVERGENCE IN WEIGHTED NORMS

## Gabriel Acosta, María E. Cejas, Ricardo G. Durán

Universidad de Buenos Aires and IMAS-UBA-CONICET
Facultad de Ciencias Exactas y Naturales, Departamento de Matemática
Pabellón I, Ciudad Universitaria, 1428 CABA, Argentina; gacosta 'at' dm.uba.ar

Universidad Nacional de La Plata and CONICET
Facultad de Ciencias Exactas, Departamento de Matemática
Calle 50 y 115, 1900 La Plata, Buenos Aires, Argentina; mec.eugenia 'at' gmail.com

Universidad de Buenos Aires and IMAS-UBA-CONICET
Facultad de Ciencias Exactas y Naturales, Departamento de Matemática
Pabellón I, Ciudad Universitaria, 1428 CABA, Argentina; rduran 'at' dm.uba.ar

Abstract. The improved Poincaré inequality

||φ - φΩ||Lp(Ω)C||dφ||Lp(Ω)

where Ω ⊂ Rn is a bounded domain and d(x) is the distance from x to the boundary of Ω, has many applications. In particular, it can be used to obtain a decomposition of functions with vanishing integral into a sum of locally supported functions with the same property. Consequently, it can be used to go from local to global results, i.e., to extend to very general bounded domains results which are known for cubes. For example, this methodology can be used to prove the existence of solutions of the divergence in Sobolev spaces. The goal of this paper is to analyze the generalization of these results to the case of weighted norms. When the weight is in Ap the arguments used in the un-weighted case can be extended without great difficulty. However, we will show that the improved Poincaré inequality, as well as its above mentioned applications, can be extended to a more general class of weights.

2010 Mathematics Subject Classification: Primary 26D10, 46E35.

Key words: Poincaré inequalities, weights, divergence operator.

Reference to this article: G. Acosta, M. E. Cejas, R. G. Durán: Improved Poincaré inequalities and solutions of the divergence in weighted norms. Ann. Acad. Sci. Fenn. Math. 42 (2017), 211-226.

https://doi.org/10.5186/aasfm.2017.4212