Gene

Optics

Physics

Mathematical analysis

Symbol of a differential operator

Psychology

Repressor

Operator (biology)

Plane (geometry)

Cognitive psychology

Chemistry

Differential algebraic equation

Hypoelliptic operator

Focus (optics)

Pure mathematics

Semi-elliptic operator

Geometry

Superposition principle

Concreteness

Symmetry (geometry)

Conjecture

Mathematics

Ordinary differential equation

Biochemistry

Elliptic operator

Differential operator

Differential equation

Transcription factor

Proceedings of the Royal Society of Edinburgh Section A Mathematics

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 develop are quite general, for concreteness we focus on the case in which the operator takes the form − Δ + ( − Δ) s , with s ∈ (0, 1). We focus here on symmetry properties of the solutions and we prove a radial symmetry result, based on the moving plane method, and a one-dimensional symmetry result, related to a classical conjecture by G.W. Gibbons.

Semilinear elliptic equations involving mixed local and nonlocal operators