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Local boundedness of weak solutions to elliptic equations under unbalanced Orlicz growth conditions

Gabriele Giannone

TL;DR

This work develops a local boundedness theory for weak solutions of nonlinear elliptic equations in divergence form under unbalanced Orlicz growth driven by two Young functions $A$ and $B$, in a non-variational setting. The authors build a De Giorgi–type framework using Caccioppoli-type energy estimates and sharp sphere-based Sobolev inequalities, under a balance condition between $A$ and $B$ when $B$ is super-linear, and allow a broader class of reaction terms via $E$ with controlled growth. A pivotal step is a redefinition lemma that replaces $A,B,E$ near zero without affecting infinity behavior, enabling global energy estimates and an iterative scheme that yields local boundedness of solutions. The results generalize and interpolate known boundedness results for variational and non-variational problems with unbalanced Orlicz growth, including $p,q$-growth as a special case, and provide a robust framework for further regularity analysis. The methodology offers precise integrability and growth conditions on the data ($\psi$, $\phi$, $E$) and leverages sphere-based Sobolev embeddings to handle non-homogeneous, non-polynomial growth.

Abstract

We establish sufficient conditions for the local boundedness of weak solutions to a broad class of nonlinear elliptic equations in divergence form, under unbalanced growth conditions on the stress field. Our analysis is carried out in a non-variational setting, with no symmetry or structural assumptions on the operator. The ellipticity and growth are prescribed via distinct Young functions, leading to a Orlicz-type setting that captures a wide class of nonstandard behaviors. As a special case, the theory encompasses and extends the known results on equations with $p,q$-growth.

Local boundedness of weak solutions to elliptic equations under unbalanced Orlicz growth conditions

TL;DR

This work develops a local boundedness theory for weak solutions of nonlinear elliptic equations in divergence form under unbalanced Orlicz growth driven by two Young functions and , in a non-variational setting. The authors build a De Giorgi–type framework using Caccioppoli-type energy estimates and sharp sphere-based Sobolev inequalities, under a balance condition between and when is super-linear, and allow a broader class of reaction terms via with controlled growth. A pivotal step is a redefinition lemma that replaces near zero without affecting infinity behavior, enabling global energy estimates and an iterative scheme that yields local boundedness of solutions. The results generalize and interpolate known boundedness results for variational and non-variational problems with unbalanced Orlicz growth, including -growth as a special case, and provide a robust framework for further regularity analysis. The methodology offers precise integrability and growth conditions on the data (, , ) and leverages sphere-based Sobolev embeddings to handle non-homogeneous, non-polynomial growth.

Abstract

We establish sufficient conditions for the local boundedness of weak solutions to a broad class of nonlinear elliptic equations in divergence form, under unbalanced growth conditions on the stress field. Our analysis is carried out in a non-variational setting, with no symmetry or structural assumptions on the operator. The ellipticity and growth are prescribed via distinct Young functions, leading to a Orlicz-type setting that captures a wide class of nonstandard behaviors. As a special case, the theory encompasses and extends the known results on equations with -growth.
Paper Structure (14 sections, 11 theorems, 247 equations)

This paper contains 14 sections, 11 theorems, 247 equations.

Table of Contents

  1. Introduction
  2. Organization of the paper
  3. Background
  4. Young functions
  5. Orlicz and Orlicz-Sobolev spaces
  6. Main result
  7. Setting of the problem
  8. Definition of weak solution
  9. Statement of the main result
  10. Technical lemmas
  11. Proof of Theorem \ref{['boundedness theorem']}
  12. Caccioppoli-type estimates
  13. Recursive formula
  14. Conclusion

Key Result

Theorem 2.1

Let $n\ge2$, let $\rho>0$, and let $A$ be a Young function fulfilling the condition first case main thm. $(i)$ Assume that Then, there exists a constant $\gamma=\gamma(n,\rho)>0$ such that for every $u\in W^1K^A(\mathop{\mathrm{\mathbb{B}}}\nolimits_\rho)$. $(ii)$ Assume that Then, there exists a constant $\gamma=\gamma(n,\rho,A)>0$ such that for every $u\in W^1K^A(\mathop{\mathrm{\mathbb{B}}}

Theorems & Definitions (22)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Remark 3.1
  • Definition 3.2
  • Theorem 3.3
  • Remark 3.4
  • Example 3.5
  • Example 3.6
  • Lemma 4.1
  • ...and 12 more