Existence, Comparison, and Convergence Results for a Class of Elliptic Hemivariational Inequalities

• Published in: Applied mathematics & optimization Vol. 84; no. Suppl 2; pp. 1453 - 1475
• Main Authors: Gariboldi, Claudia M., Migórski, Stanisław, Ochal, Anna, Tarzia, Domingo A.
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.12.2021; Springer Nature B.V
• Subjects: Applied mathematics; Asymptotic properties; Boundary conditions; Calculus of Variations and Optimal Control; Optimization; Conduction heating; Conductive heat transfer; Control; Heat; Heat flux; Heat transfer coefficients; Hypotheses; Inequalities; Internal energy; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Numerical and Computational Physics; Operators (mathematics); Optimization; Original Paper; Simulation; Systems Theory; Theoretical; 35J65; 35J87; 49J45; Asymptotic behavior; Mixed problem; 35J05; Nonlinear elliptic equation; Elliptic hemivariational inequality; Clarke generalized gradient; Convergence
• ISSN: 0095-4616; 1432-0606
• DOI: 10.1007/s00245-021-09800-9

In this paper we study a class of elliptic boundary hemivariational inequalities which originates in the steady-state heat conduction problem with nonmonotone multivalued subdifferential boundary condition on a portion of the boundary described by the Clarke generalized gradient of a locally Lipschitz function. First, we prove a new existence result for the inequality employing the theory of pseudomonotone operators. Next, we give a result on comparison of solutions, and provide sufficient conditions that guarantee the asymptotic behavior of solution, when the heat transfer coefficient tends to infinity. Further, we show a result on the continuous dependence of solution on the internal energy and heat flux. Finally, some examples of convex and nonconvex potentials illustrate our hypotheses.