Lagrange multipliers and characteristic functions
Davide Azevedo, Lisa Santos
TL;DR
Addresses a stationary variational inequality with both obstacle and gradient constraints and derives a coupled Lagrange-multiplier/characteristic-function formulation. It constructs an approximating penalized scheme with $k_\varepsilon$ and $\theta_\varepsilon$, obtains uniform a priori estimates, and passes to the limit to obtain $(\lambda,u)$ with $u$ solving the VI and $\lambda$ as a multiplier satisfying $(\lambda-1)(|\nabla u|-g)=0$, along with $\boldsymbol\Lambda=\lambda\nabla u$ and a localized obstacle term $(-\Delta\psi-f)^+\chi_{*}$. It proves the existence of solutions for each $p\in[2,\infty)$ and establishes the VI solution's uniqueness, together with identities linking the LMCF formulation to the obstacle problem. It also proves continuous dependence: as $(f_n,g_n,\psi_n)\to (f,g,\psi)$, the LMCF solutions $(\lambda_n,u_n)$ converge to $(\lambda,u)$ in $L^p$ and $C^{1,1-d/p}$ norms, demonstrating stability under data perturbations. This work extends variational inequality theory to joint obstacle-gradient constrained problems and provides a rigorous framework for related transport and free-boundary analyses.
Abstract
We consider a stationary variational inequality with gradient constraint and obstacle. We prove that this problem can be described by an equation using a Lagrange multiplier and a characteristic function. The Lagrange multiplier contains information about the contact set of the modulus of the gradient of the solution with the gradient constraint, and the characteristic function is defined in the contact set of the solution with the obstacle. Moreover, given a convergent sequence of data, we prove the stability of the corresponding solutions.