# Mixed local-nonlocal operators: maximum principles, eigenvalue problems and their applications

@inproceedings{Biswas2021MixedLO, title={Mixed local-nonlocal operators: maximum principles, eigenvalue problems and their applications}, author={Anup Biswas and Mitesh Modasiya}, year={2021} }

In this article we consider a class of non-degenerate elliptic operators obtained by superpositioning the Laplacian and a general nonlocal operator. We study the existence-uniqueness results for Dirichlet boundary value problems, maximum principles and generalized eigenvalue problems. As applications to these results, we obtain Faber-Krahn inequality and a one-dimensional symmetry result related to the Gibbons’ conjecture. The latter results substantially extend the recent results of Biagi et…

## References

SHOWING 1-10 OF 63 REFERENCES

Semilinear elliptic equations involving mixed local and nonlocal operators

- Mathematics, PhysicsProceedings of the Royal Society of Edinburgh: Section A Mathematics
- 2021

In this paper, we consider an elliptic operator obtained as the superposition of a classical second-order differential operator and a nonlocal operator of fractional type. Though the methods that we…

Local versus nonlocal elliptic equations: short-long range field interactions

- 2020

Abstract In this paper we study a class of one-parameter family of elliptic equations which combines local and nonlocal operators, namely the Laplacian and the fractional Laplacian. We analyze…

A Faber-Krahn inequality for mixed local and nonlocal operators

- Mathematics
- 2021

We consider the first Dirichlet eigenvalue problem for a mixed local/nonlocal elliptic operator and we establish a quantitative Faber-Krahn inequality. More precisely, we show that balls minimize the…

Nonexistence Results for Nonlocal Equations with Critical and Supercritical Nonlinearities

- Physics, Mathematics
- 2013

We prove nonexistence of nontrivial bounded solutions to some nonlinear problems involving nonlocal operators of the form These operators are infinitesimal generators of symmetric Lévy processes. Our…

Hopf’s lemma for viscosity solutions to a class of non-local equations with applications

- Mathematics
- 2019

One-dimensional symmetry of bounded entire solutions of some elliptic equations

- Mathematics
- 2000

This paper is about one-dimensional symmetry properties for some entire and bounded solutions of ∆u + f(u) = 0 in IR. We consider solutions u such that −1 < u < 1 and u(x1, · · · , xn) → ±1 as xn →…

Maximum Principles and Aleksandrov-Bakelman-Pucci Type Estimates for NonLocal Schrödinger Equations with Exterior Conditions

- Mathematics, Computer ScienceSIAM J. Math. Anal.
- 2019

This work considers Dirichlet exterior value problems related to a class of non-local Schr\"odinger operators, whose kinetic terms are given in terms of Bernstein functions of the Laplacian, and proves a refined maximum principle in the sense of Berestycki-Nirenberg-Varadhan and a converse.

On the Strong Maximum Principle for Second Order Nonlinear Parabolic Integro-Differential Equations

- Mathematics
- 2010

This paper is concerned with the study of the Strong Maximum Principle for semicontinuous viscosity solutions of fully nonlinear, second-order parabolic integro-differential equations. We study…