[论文解读] Regularity to Thin Obstacle Problem in Orlicz spaces

Junior da S. Bessa, Paulo Henryque da Costa Silva+1 more|arXiv (Cornell University)|Feb 1, 2026

Nonlinear Partial Differential Equations被引用 0

一句话总结

论文在 Orlicz 空间中通过 De Giorgi 型方法证明薄障问题极小值的 Lipschitz 与 C1,γ-正则性，并刻画结点集结构，适用于非标准增长的情形。

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.

研究动机与目标

• 在 Orlicz 空间中研究具有非标准增长的变分问题及其薄障（Signorini）表述的动机。
• 在结构条件下建立带 N-函数 G 的能量泛函极小值的正则性结果。
• Develop a De Giorgi-type regularity framework suited to nonhomogeneous g-Laplacian operators.
• Characterize the nodal set structure of minimizers as a byproduct of the regularity results.

提出的方法

• Study minimizers of the energy functional J(u)=∫_{B1^+} G(|∇u|) dx over an admissible class with a thin obstacle on T1.
• Prove minimizers are g-harmonic in B1^+ and extend to a symmetric problem on B1 to leverage obstacle problem techniques.
• Obtain Lipschitz regularity via extension to a classical obstacle problem for the g-Laplacian and classical regularity results.
• Develop De Giorgi-type lemmas adapted to the nonstandard growth governed by G and g to derive C1,γ regularity.
• Show that the minimizers are C1,γ in a smaller half-ball B1/2^+ and provide a nodal-set structure result.

实验结果

研究问题

• RQ1 Under the structural hypotheses (H1) and (H2), do minimizers of the G-energy in the thin obstacle setting exhibit Lipschitz continuity and Hölder continuous gradients?
• RQ2 How does the nonhomogeneous (nonlinear) g-Laplacian influence De Giorgi-type regularity methods in Orlicz spaces?
• RQ3 Can the regularity theory yield information about the geometry of nodal sets of minimizers?
• RQ4 What is the relationship between the Orlicz space framework and classical thin obstacle results when G(t)=t^p?

主要发现

• Minimizers of the G-energy in the thin obstacle problem are g-harmonic in the positive half-ball.
• Minimizers are Lipschitz continuous in B3/4^+ with a bound depending on the L∞ norm of the minimizer.
• There exists γ∈(0,1) depending on n, δ0, g0 such that u∈C1,γ(closure of B1/2^+) (Theorem 1.1).
• The minimizer can be related to a classical obstacle problem via even extension and an auxiliary obstacle, enabling Hölder regularity results.
• The nodal set for order 1, n1(u), decomposes into a finite union of C1,γ manifolds of dimensions up to n (Theorem 1.2).
• The analysis provides a unified framework that extends known p-growth results to general Orlicz settings and accommodates nonhomogeneous operators.