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This is a Quantitative Study study.

September 17, 2025

Subdifferentials and Penalty Approximations of the Obstacle Problem

Puntos clave

The derivatives of the solution maps converge to elements of the strong-weak Bouligand subdifferential, enhancing our understanding of the obstacle problem's structures.
This approach can handle both smooth and nonsmooth penalty terms, broadening the scope of applicable methods for various optimization problems.
The framework's versatility allows for multiple specific penalty functions commonly used in literature, facilitating adoption in diverse mathematical contexts.
The implications extend to the theory of optimal control, showcasing how approximation techniques can enhance control strategies for nonlinear problems.

Resumen

. We consider a framework for approximating the obstacle problem through a penalty approach by nonlinear PDEs. By using tools from capacity theory, we show that derivatives of the solution maps of the penalized problems converge in the weak operator topology to an element of the strong-weak Bouligand subdifferential. We are able to treat smooth penalty terms as well as nonsmooth ones involving, for example, the positive part function \ ( (0, ) \). Our abstract framework applies to several specific choices of penalty functions which are omnipresent in the literature. We conclude with consequences to the theory of optimal control of the obstacle problem. Keywordsvariational inequalityobstacle problemoptimal controlweak operator topologymeasuresMSC codes49J4065K1549K2149M4147B9228A12

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