Mircea Sofonea, Domingo A. Tarzia, Well-posedness and convergence results for elliptic hemivariational inequalities

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DOI: 10.23952/asvao.7.2025.1.01
Volume 7, Issue 1, 1 April 2025, Pages 1-21

Abstract. We consider an elliptic hemivariational inequality in a real reflexive Banach space $X$ which, under appropriate assumptions on the data, has a unique solution $u\in X$ . We recall the concepts of well-posedness in the sense of Tykhonov and Levitin-Polyak for this inequality, and then we extend these concepts by introducing new well-posedness concepts, constructed with a larger set of approximating sequences. We also prove that, under additional assumptions, these new well-posedness concepts are optimal in the sense that all the sequences of elements of $X$ which converge to the solution $u$ are approximating sequences. This result, presented in Theorem 4.1, provides necessary and sufficient conditions for any sequence $\{u_n\}\subset X$ which guarantees that it converges to $u$ and, therefore, it represents a convergence criterion to the solution of the hemivariational inequality. This criterion can be used in various applications. To provide an example, we illustrate its use in the study of a penalty method associated to an elliptic hemivariational inequality which describes the equilibrium of an elastic membrane in contact with a obstacle, the so-called foundation.

How to Cite this Article:
M. Sofonea, D. A. Tarzia, Well-posedness and convergence results for elliptic hemivariational inequalities, Appl. Set-Valued Anal. Optim. 7 (2025), 1-21.