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Regularity to Thin Obstacle Problem in Orlicz spaces

Junior da Silva Bessa, Paulo Henryque da Costa Silva, Alan Pio Sousa

TL;DR

This work extends the regularity theory for thin obstacle problems to energies with nonstandard growth in Orlicz spaces. By proving Lipschitz continuity of minimizers and developing a De Giorgi-type framework for the nonhomogeneous g-Laplacian, it obtains C^{1,\\gamma} regularity of minimizers up to the boundary and characterizes the structure of nodal sets. The analysis unifies and generalizes classic p-Laplacian results to broad G: N-function growth, demonstrating robust regularity under Lieberman-type conditions and yielding a precise nodal-set decomposition into C^{1,\\gamma} manifolds. These results advance the mathematical understanding of variational problems with nonuniform growth, with implications for elasticity models and image processing applications where nonquadratic growth is natural.

Abstract

In this work, we establish regularity results for minimizers of the energy functional associated with the thin obstacle problem in Orlicz spaces. More precisely, we prove the Lipschitz continuity and the Hölder continuity of the gradient of minimizers. The analysis is based on techniques from De Giorgi's classical regularity theory. As a byproduct of our results, we also provide a characterization of the structure of the nodal sets of the minimizers.

Regularity to Thin Obstacle Problem in Orlicz spaces

TL;DR

This work extends the regularity theory for thin obstacle problems to energies with nonstandard growth in Orlicz spaces. By proving Lipschitz continuity of minimizers and developing a De Giorgi-type framework for the nonhomogeneous g-Laplacian, it obtains C^{1,\\gamma} regularity of minimizers up to the boundary and characterizes the structure of nodal sets. The analysis unifies and generalizes classic p-Laplacian results to broad G: N-function growth, demonstrating robust regularity under Lieberman-type conditions and yielding a precise nodal-set decomposition into C^{1,\\gamma} manifolds. These results advance the mathematical understanding of variational problems with nonuniform growth, with implications for elasticity models and image processing applications where nonquadratic growth is natural.

Abstract

In this work, we establish regularity results for minimizers of the energy functional associated with the thin obstacle problem in Orlicz spaces. More precisely, we prove the Lipschitz continuity and the Hölder continuity of the gradient of minimizers. The analysis is based on techniques from De Giorgi's classical regularity theory. As a byproduct of our results, we also provide a characterization of the structure of the nodal sets of the minimizers.
Paper Structure (10 sections, 16 theorems, 171 equations)

This paper contains 10 sections, 16 theorems, 171 equations.

Table of Contents

  1. Introduction
  2. Assumptions and Main Results
  3. An application: Structure of the nodal sets
  4. Preliminaries
  5. g-Laplacian
  6. Some facts about thin obstacle problem in Orlicz spaces
  7. Lipschitz regularity of minimizers
  8. Review of De Giorgi’s Theory on Regularity
  9. Regularity lemmas
  10. Hölder regularity of the gradient: Proof of Theorem \ref{['T1']}

Key Result

Theorem 1.1

Assume the structural conditions (H1) and (H2) hold. Let $u \in W^{1,G}(\mathrm B_1^+)$ be a minimizer of the functional eq1 over the admissible class $\mathcal{G}$. Then there exists $\gamma\in(0,1)$, depending only on $n$, $\Delta_0$, and $g_0$, such that $u \in C^{1,\gamma}(\overline{\mathrm B_{1

Theorems & Definitions (33)

  • Theorem 1.1
  • Theorem 1.2: Structure of Nodal set
  • Definition 2.1
  • Lemma 2.2: MW
  • Lemma 2.3: MW
  • Remark 2.4
  • Definition 2.5
  • Remark 2.6
  • Remark 2.7: Scaling and normalization
  • Proposition 3.1
  • ...and 23 more