The Dirichlet problem for nonlocal elliptic equations

Abstract

We study a class of nonlocal elliptic equations $(-\mathrm{\Delta }{)}^{\alpha /2}\rho (x)-b(x)\cdot \mathrm{\nabla }\rho (x)=f(x)$ on a bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^d$. For $f\in L^q\left(\mathrm{\Omega }\right)$ ($q>d/\alpha$), by the Lax–Milgram theorem and the De Giorgi iteration, we prove the existence and uniqueness of $L^{\mathrm{\infty }}$ solution. Furthermore, we investigate the existence of densities for measure-valued solutions to nonhomogeneous measure-valued nonlocal elliptic equations.

COMMUNICATED BY:

• Y. Ou

Keywords:

• Vanishing viscosity solution
• existence
• uniqueness
• measure-valued nonlocal equation

2010 Mathematics Subject Classifications:

• 35R06
• 35S15
• 60H15

Acknowledgements

The authors sincerely thank the referees and the editors for their helpful comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Jinlong Wei http://orcid.org/0000-0002-9182-4587

Yanbin Tang http://orcid.org/0000-0001-7445-2817

Additional information

Funding

R.R. Tian is supported by the National Natural Science Foundation of China under grant no. 11901442. J.L. Wei is supported by the National Natural Science Foundation of China under grant no. 11501577. Y.B. Tang is supported by the National Natural Science Foundation of China under grant no. 11971188.