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Limit of quasilinear equations and related extremal problems

Yuanlong Ruan

TL;DR

This work analyzes the limiting behavior of a class of quasilinear Dirichlet problems with boundary data $g$ as $n\to\infty$, showing that the Lipschitz bound on $g$ governs the $\Gamma$-convergence of the natural energies while the limit equation is noncoercive in $u$ but admits a unique viscosity solution via a weak comparison principle. The limiting PDE involves the infinity-Laplacian type operator with an advection term $-|\nabla u|\langle \Lambda(x,|\nabla u|), \nabla u\rangle$, and the analysis is grounded in generalized Sobolev-Orlicz (Musielak-Orlicz) spaces, including a discontinuous $\Phi$-framework. A key auxiliary result is a Poincaré inequality for discontinuous $\Phi$, enabling the study of an extremal problem with a discontinuous operator in a subdomain, where the associated energy Γ-converges to a limiting functional that enforces a gradient bound on the subdomain. The paper also treats an unbounded-coefficient extremal problem in a convex subdomain, establishing a Γ-convergence to a limit with a natural interface condition and proving existence and uniqueness of the limiting solution via viscosity methods, with several technical inequalities proved in the appendix that may be of independent interest.

Abstract

We perform a complete analysis of the limiting behaviour of a class of quasilinear problems with Dirichlet boundary data g. We show that the Lipschitz constant of g plays a role in controlling the Gamma-convergence of the natural energies. However the solutions converge uniformly to solution of a limiting equation irrelevant to the Lipschitz constant of g. The limiting equation has no coercivity in u. We prove that the limiting equation admits a weak comparison principle and has a unique viscosity solution. We also obtain a Poincare inequality in the Sobolev-Orlicz space for discontinuous operator, which paves the way for our study of an extremal problem where its operator becomes unbounded in a subdomain. Upon giving proper meaning to its solution, we show that the extremal problem has a unique solution. It turns out the solution has sufficient continuity, although operator is discontinuous. In the appendix we provide some technical inequalities which play crucial roles in the proof of uniqueness and we believe will be of independent interest.

Limit of quasilinear equations and related extremal problems

TL;DR

This work analyzes the limiting behavior of a class of quasilinear Dirichlet problems with boundary data as , showing that the Lipschitz bound on governs the -convergence of the natural energies while the limit equation is noncoercive in but admits a unique viscosity solution via a weak comparison principle. The limiting PDE involves the infinity-Laplacian type operator with an advection term , and the analysis is grounded in generalized Sobolev-Orlicz (Musielak-Orlicz) spaces, including a discontinuous -framework. A key auxiliary result is a Poincaré inequality for discontinuous , enabling the study of an extremal problem with a discontinuous operator in a subdomain, where the associated energy Γ-converges to a limiting functional that enforces a gradient bound on the subdomain. The paper also treats an unbounded-coefficient extremal problem in a convex subdomain, establishing a Γ-convergence to a limit with a natural interface condition and proving existence and uniqueness of the limiting solution via viscosity methods, with several technical inequalities proved in the appendix that may be of independent interest.

Abstract

We perform a complete analysis of the limiting behaviour of a class of quasilinear problems with Dirichlet boundary data g. We show that the Lipschitz constant of g plays a role in controlling the Gamma-convergence of the natural energies. However the solutions converge uniformly to solution of a limiting equation irrelevant to the Lipschitz constant of g. The limiting equation has no coercivity in u. We prove that the limiting equation admits a weak comparison principle and has a unique viscosity solution. We also obtain a Poincare inequality in the Sobolev-Orlicz space for discontinuous operator, which paves the way for our study of an extremal problem where its operator becomes unbounded in a subdomain. Upon giving proper meaning to its solution, we show that the extremal problem has a unique solution. It turns out the solution has sufficient continuity, although operator is discontinuous. In the appendix we provide some technical inequalities which play crucial roles in the proof of uniqueness and we believe will be of independent interest.

Paper Structure

This paper contains 12 sections, 35 theorems, 354 equations.

Table of Contents

  1. Introduction
  2. Acknowledgement
  3. Preliminaries
  4. $\Phi$-function and Sobolev-Orlicz space
  5. Generalized $\Phi$-function and Musielak-Orlicz space
  6. $\Gamma$-convergence of the Natural Energies
  7. Limit of the Solutions with $\left\Vert g\right\Vert _{Lip\left( \partial\Omega\right) }\leqslant1$
  8. Limit of the Solutions without $\left\Vert g\right\Vert _{Lip\left( \partial\Omega\right) }\leqslant1$
  9. Comparison Principle and Uniqueness of the Limit Equation
  10. Poincare's Inequality for Discontinuous $\Phi$
  11. Unbounded coefficients in a convex subdomain
  12. Appendix: Some technical inequalities

Key Result

Lemma 1

Let $\varphi\in C^{1}\left( 0,\infty\right)$ satisfy $\left( assumption_phi\right)$. Then $\Phi\in C^{2}$ is convex and or equivalently

Theorems & Definitions (74)

  • Remark 1
  • Definition 1
  • Lemma 1
  • Lemma 2
  • proof
  • Definition 2
  • Lemma 3
  • proof
  • Definition 3
  • Remark 2
  • ...and 64 more