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A posteriori estimates for problems with monotone operators

Vladimir Bobkov, Svetlana Pastukhova

TL;DR

The paper develops a general, duality-free framework for a posteriori estimates in variational inequalities with monotone operators, applicable to nonlinear $p$-Laplacian-type problems. It introduces a residual-type measure $\|Av-h\|_{*,K}$ and derives constructive majorants that bound the error $\|u-v\|_X$ in terms of data and computable auxiliary objects, with consistency as $v\to u$. The method is demonstrated across a spectrum of problems—classical Poisson, obstacle, anisotropic, Neumann, vector, polyharmonic, and nonlocal (fractional) Poisson—deriving explicit bounds and flux-deviation controls, without relying on dual formulations or operator potentiality. The results provide practical, verifiable a posteriori tools for nonlinear monotone operators, with clear extensions to higher-order and nonlocal settings, and highlight the interplay between residuals, flux regularity, and domain geometry in error control.

Abstract

We propose a method of obtaining a posteriori estimates which does not use the duality theory and which applies to variational inequalities with monotone operators, without assuming the potentiality of operators. The effectiveness of the method is demonstrated on problems driven by nonlinear operators of the $p$-Laplacian type, including the anisotropic $p$-Laplacian, polyharmonic $p$-Laplacian, and fractional $p$-Laplacian.

A posteriori estimates for problems with monotone operators

TL;DR

The paper develops a general, duality-free framework for a posteriori estimates in variational inequalities with monotone operators, applicable to nonlinear -Laplacian-type problems. It introduces a residual-type measure and derives constructive majorants that bound the error in terms of data and computable auxiliary objects, with consistency as . The method is demonstrated across a spectrum of problems—classical Poisson, obstacle, anisotropic, Neumann, vector, polyharmonic, and nonlocal (fractional) Poisson—deriving explicit bounds and flux-deviation controls, without relying on dual formulations or operator potentiality. The results provide practical, verifiable a posteriori tools for nonlinear monotone operators, with clear extensions to higher-order and nonlocal settings, and highlight the interplay between residuals, flux regularity, and domain geometry in error control.

Abstract

We propose a method of obtaining a posteriori estimates which does not use the duality theory and which applies to variational inequalities with monotone operators, without assuming the potentiality of operators. The effectiveness of the method is demonstrated on problems driven by nonlinear operators of the -Laplacian type, including the anisotropic -Laplacian, polyharmonic -Laplacian, and fractional -Laplacian.

Paper Structure

This paper contains 11 sections, 13 theorems, 187 equations.

Table of Contents

  1. Concept of a posteriori estimates
  2. General approach to variational inequalities
  3. Poisson problem
  4. Obstacle problem
  5. Other variants of the Poisson problem
  6. Anisotropic Poisson problem
  7. Poisson problem with Neumann boundary conditions
  8. Vector Poisson problem
  9. Polyharmonic obstacle problem
  10. Nonlocal Poisson problem
  11. Appendix

Key Result

Theorem 3.2

Let $p \geq 2$. Then for the solution $u$ of eq:weak1 and any $v \in W_0^{1,p}(\Omega)$ the following estimate holds: where $\tau^* := |\nabla v|^{p-2} \nabla v$.

Theorems & Definitions (41)

  • Remark 3.1: Regularity
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof
  • Remark 3.4
  • Remark 3.5
  • Remark 3.6
  • Proposition 3.7
  • proof
  • ...and 31 more