Convergence and well-posedness analysis of a nonlinear elliptic system

Maria-Magdalena Boureanu, Loredana Dafinoiu, Mircea Sofonea

Abstract


We consider two real Hilbert spaces X and Y and a nonlinear elliptic system governed by two sets of constraints K ⊂ X and W ⊂ Y. We prove that, under appropriate assumptions, the system has a unique solution (u, φ) ∈ K × W. Then, we provide necessary and sufficient conditions which guarantee the convergence of an arbitrary sequence (uₙ, φₙ) ∈ X × Y to the solution (u, φ). The proofs are based on standard results on elliptic variational inequalities, various estimates and arguments of convex analysis. Next, we introduce two concepts of well-posedness for the system, compare them, and derive the corresponding well-posedness results. Our results can be applied in the study of various problems arising in Mechanics and Physics. To provide an example we consider a mathematical model which describes the frictionless unilateral contact of a piezoelectric body with an insulated foundation.


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DOI: https://doi.org/10.52846/ami.v52i2.2249