boundarydependent

Boundarydependent is an adjective used to describe a quantity, solution, or property that depends on the boundary conditions imposed on a domain. In mathematics, physics, and engineering, boundary conditions specify the behavior of a system at the boundary of its spatial domain, and many objects are boundary-dependent because their form or value is determined by those conditions.

Examples include the solution to partial differential equations, Green's functions, eigenvalues of differential operators, and energy functionals defined on a finite domain. For example, the Laplace equation on a bounded domain has solutions that vary with Dirichlet, Neumann, or Robin boundary conditions. The Green's function is boundary-dependent because it encodes the influence of the domain boundary on interior points. In quantum mechanics, the allowed energy levels of a particle in a confined region depend on the boundary conditions. In fluid dynamics, boundary-dependent effects occur near walls, such as boundary layers, where the flow profile is determined by the boundary.

Implications and methods: The analysis of boundary-dependent problems requires explicit specification of boundary conditions; numerical discretizations must implement boundary terms carefully. In shape and boundary optimization, changing the boundary can alter the solution, making the problem boundary-dependent. In contrast, boundary-independent phenomena refer to bulk properties that may be insensitive to boundary conditions in certain limits, though such cases are contextual and problem-specific.