Carlemantype

Carlemantype is a term used in mathematics to describe a family of results and techniques that generalize Carleman inequalities. These are weighted integral estimates that play a central role in questions of unique continuation, stability, and control for partial differential equations (PDEs). The Carlemantype framework collects inequalities that share a common structure: weighted norms with exponential weights, typically involving a large parameter, and carefully chosen weight functions.

Origins and usage: The name derives from Carleman-type ideas in the tradition of Torsten Carleman, whose eponymous inequalities first enabled quantitative control of solutions to PDEs. In contemporary literature, “Carlemantype” or “Carleman-type” is used to describe generalized or extended forms of these estimates. The term emphasizes the shared methodological pattern rather than a single fixed statement.

Definition and scope: A Carlemantype inequality usually provides a bound of the form that involves an operator L acting on a function u, integrated against an exponential weight e^{2τφ}, where τ is a large parameter and φ is a weight function. The inequality relates the weighted norm of Lu to weighted norms of u and its derivatives, often with additional lower-order terms. The exact formulation depends on the operator, the domain, and boundary conditions, and can apply to linear and some nonlinear PDEs, as well as certain nonlocal or time-dependent problems.

Applications: Carlemantype methods are widely used to prove unique continuation (determining a function from partial information), to obtain stability estimates in inverse problems (such as coefficient identification), and to study controllability and observability in PDE systems. They also contribute to backward-in-time ill-posed problems and quantitative estimates in various physical and geometric settings.

Relation to Carleman inequalities: Carlemantype results are closely related to classical Carleman inequalities; the distinction lies in their generalization and broader applicability across operator classes and problem settings. See also Carleman inequality, unique continuation, and inverse problems.