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Global boundedness of weak solutions with finite energy to a general class of Dirichlet problems

Giovanni Cupini, Paolo Marcellini

TL;DR

The paper develops a general framework to establish global boundedness for weak solutions with finite energy to Dirichlet problems in divergence form under nonstandard growth. By coupling Δ2 conditions on a convex energy density f with a generalized vector field a and unilateral growth bounds on the right-hand side b, the authors prove that weak solutions are bounded on the closure of the domain, and they provide an epsilon variant with explicit L∞ estimates. The core methodology hinges on a carefully chosen test function, a Caccioppoli-type inequality, and a De Giorgi type iteration adapted to nonuniform ellipticity and p-q growth, including double phase and variable exponent scenarios. This work extends classical maximum principle results to broad nonuniform elliptic settings and yields quantitative bounds useful for compactness and regularity analysis of approximating sequences.

Abstract

As explained in detail in the prologue to this manuscript, boundedness of weak solutions for general classes of elliptic equations in divergence form is a classic tool for achieving higher regularity. We propose here some global boundedness results under general assumptions that can be applied to several cases studied in the recent and extensive literature on partial differential equations \textit{under general growth}. In particular, we propose the class of \textit{weak solutions with finite energy} in which to search for solutions and in which regularity can be studied and achieved. We emphasize that we are not limited to minimizers of certain integral functionals, as often considered recently in this context of general growth, but to the broader class of weak solutions to Dirichlet problems for general nonlinear elliptic equations in divergence form.

Global boundedness of weak solutions with finite energy to a general class of Dirichlet problems

TL;DR

The paper develops a general framework to establish global boundedness for weak solutions with finite energy to Dirichlet problems in divergence form under nonstandard growth. By coupling Δ2 conditions on a convex energy density f with a generalized vector field a and unilateral growth bounds on the right-hand side b, the authors prove that weak solutions are bounded on the closure of the domain, and they provide an epsilon variant with explicit L∞ estimates. The core methodology hinges on a carefully chosen test function, a Caccioppoli-type inequality, and a De Giorgi type iteration adapted to nonuniform ellipticity and p-q growth, including double phase and variable exponent scenarios. This work extends classical maximum principle results to broad nonuniform elliptic settings and yields quantitative bounds useful for compactness and regularity analysis of approximating sequences.

Abstract

As explained in detail in the prologue to this manuscript, boundedness of weak solutions for general classes of elliptic equations in divergence form is a classic tool for achieving higher regularity. We propose here some global boundedness results under general assumptions that can be applied to several cases studied in the recent and extensive literature on partial differential equations \textit{under general growth}. In particular, we propose the class of \textit{weak solutions with finite energy} in which to search for solutions and in which regularity can be studied and achieved. We emphasize that we are not limited to minimizers of certain integral functionals, as often considered recently in this context of general growth, but to the broader class of weak solutions to Dirichlet problems for general nonlinear elliptic equations in divergence form.
Paper Structure (9 sections, 8 theorems, 119 equations)

This paper contains 9 sections, 8 theorems, 119 equations.

Table of Contents

  1. Prologue
  2. Introduction and statements of main results
  3. Gradient growth of a convex function
  4. Proof of the global boundedness results
  5. The test function and the well definition of weak solution
  6. The Caccioppoli-type inequality
  7. The iteration scheme
  8. Conclusion of the proof of Theorem \ref{['main boundeness\nresult']}
  9. Sketch of the proof of Theorem \ref{['main boundeness\nresult with epsilon']}

Key Result

Theorem 1.1

Assume the coercivity condition for the vector field $a=a\left( x,u,\xi \right)$, $a:\Omega \times \mathbb{R}\times \mathbb{R}^{n}\rightarrow \mathbb{R}^{n}$, and the unilateral growth for $b=b\left( x,u,\xi \right)$, $b:\Omega \times \mathbb{R}\times \mathbb{R}^{n}\rightarrow \mathbb{R}\,$, for some nonnegative $c_{i}$, $i=0,1,2,3$, (in particular $c_{0}\left( \left\vert u\right\vert \right) \g

Theorems & Definitions (18)

  • Theorem 1.1: Ladyzhenskaya-Uraltseva Ladyzhenskaya-Uraltseva 1968
  • Theorem 1.2: Gilbarg-Trudinger Gilbarg-Trudinger 1977
  • Theorem 2.1
  • Theorem 2.2
  • Remark 2.3
  • Example 2.4
  • Example 2.5
  • Example 2.6
  • Example 2.7
  • Lemma 3.1
  • ...and 8 more