Abstract

We study the stability of solutions to a class of variational inequalities posed on obstacle-type convex sets, under Mosco-convergence. Specifically, we consider problems of the form {u∈W01,p(Ω)∩L∞(Ω),u≥ψ in Ω,⟨A(u),v−u⟩+∫ΩH(x,u,Du)(v−u)≥0,∀  v∈W01,p(Ω)∩L∞(Ω),v≥ψ in Ω.\begin{cases} u \in W_0^{1,p}(\Omega) \cap L^\infty(\Omega), & u \geq \psi \quad \text{ in } \Omega, \\[1mm] \langle A(u), v - u \rangle + \displaystyle\int_{\Omega} H(x, u, D u)(v - u) \geq 0, & \\[3mm] \forall\; v \in W_0^{1,p}(\Omega) \cap L^\infty(\Omega), \quad &v \geq \psi \quad \text{ in } \Omega. \end{cases} Here, AA is a Leray–Lions type operator, mapping W01,p(Ω)W_0^{1,p}(\Omega) into its dual W−1,p′(Ω)W^{-1, p'}(\Omega), while H(x,u,Du)H(x, u, D u) grows like ∣Du∣p|D u|^p. The obstacle ψ\psi is a function in W01,p(Ω)∩L∞(Ω)W_0^{1,p}(\Omega) \cap L^\infty(\Omega). Our main result establishes that the solutions are stable under Mosco-convergence of the constraint sets. This extends classical stability results to natural growth problems

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Keywords

  • Mosco-convergence
  • variational inequalities
  • obstacle-type convex sets
  • Leray-Lions operators
  • operators with natural growth terms
  • stability of solutions

Mathematics Subject Classification

35J87, 47J20, 49J40

References

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Suggested citation

L. Boccardo, M. A. Palladino, M. Picerni. “Mosco-Convergence of Convex Sets and Unilateral Problems for Differential Operators with Lower Order Terms Having Natural Growth.” Journal of Convex Analysis 34 (2027), No. 1, 1–15. https://doi.org/10.68381/jca34001

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