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Monotone iterations of two obstacle problems with different operators

Irene Gonzalvez, Alfredo Miranda, Julio D. Rossi

Abstract

In this paper we analyze iterations of the obstacle problem for two different operators. We solve iteratively the obstacle problem from above or below for two different differential operators with obstacles given by the previous functions in the iterative process. When we start the iterations with a super or a subsolution of one of the operators this procedure generates two monotone sequences of functions that we show that converge to a solution to the two membranes problem for the two different operators. We perform our analysis in both the variational and the viscosity settings.

Monotone iterations of two obstacle problems with different operators

Abstract

In this paper we analyze iterations of the obstacle problem for two different operators. We solve iteratively the obstacle problem from above or below for two different differential operators with obstacles given by the previous functions in the iterative process. When we start the iterations with a super or a subsolution of one of the operators this procedure generates two monotone sequences of functions that we show that converge to a solution to the two membranes problem for the two different operators. We perform our analysis in both the variational and the viscosity settings.
Paper Structure (8 sections, 11 theorems, 142 equations)

This paper contains 8 sections, 11 theorems, 142 equations.

Table of Contents

  1. Introduction.
  2. The obstacle problem
  3. The two membranes problem
  4. Description of the main results
  5. Variational solutions
  6. Iterative method.
  7. Extension to the two membranes problem for nonlocal operators
  8. Viscosity solutions

Key Result

Proposition 1

Let be $h_{p}\in W^{-1,p}(\Omega)$ and $f\in W^{1,p}(\Omega)$. If $\underline{\Lambda}^{\,p}_{f,\varphi}\neq\emptyset$ there exists a unique $u\in\underline{\Lambda}^{\,p}_{f,\varphi}$ such that $E_{p}(u)\leq E_{p}(w)$ for all $w\in\underline{\Lambda}^{\,p}_{f,\varphi}$, i.e, there exists a unique $ Analogously, let $h_{q}\in W^{-1,q}(\Omega)$ and $g\in W^{1,q}(\Omega)$, if $\overline{\Lambda}^{\,

Theorems & Definitions (37)

  • Definition 1.1
  • Definition 2.1
  • Definition 2.2
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Remark 1
  • Definition 2.3
  • Remark 2
  • ...and 27 more