Thin Obstacle Problem in Orlicz Spaces

• Thin Obstacle Problem in Orlicz Spaces is defined as minimizing an energy functional with a unilateral constraint using convex Orlicz functions to handle non-polynomial growth.
• The approach employs variational formulations in Orlicz–Sobolev spaces and adapts De Giorgi iteration techniques to establish Lipschitz continuity and Hölder regularity for minimizers.
• The study provides a detailed geometric stratification of the nodal set, characterizing submanifold structures and boundary regularity through explicit variational inequalities.

The thin obstacle problem in Orlicz spaces addresses the minimization of an energy functional under a unilateral constraint posed by a thin set, where the growth of the integrand is governed by a convex Orlicz function rather than the classical $p$-power. The analysis in Orlicz–Sobolev spaces presents new technical challenges and extends the scope of classical variational methods. Central to this theory is the establishment of regularity results for minimizers, including Lipschitz continuity and Hölder continuity of their gradients, as well as a geometric characterization of the nodal set structure of solutions (Bessa et al., 1 Feb 2026).

1. Variational Formulation in Orlicz–Sobolev Spaces

Let $G:[0,\infty)\rightarrow [0,\infty)$ denote an N-function (Orlicz function), a convex $C^1(0,\infty)$ function with $G(0)=0$, derivative $G' = g$, and satisfying Lieberman’s growth conditions: there exist constants $0<\Delta_0 \leq g_0 <\infty$ such that for all $t>0$,

$\Delta_0 \leq \frac{t g'(t)}{g(t)} \leq g_0.$

The Orlicz space $L^G(\Omega)$ comprises measurable $h$ with finite modular

$\rho_G(h)=\int_\Omega G(|h|)\,dx,$

with the Luxemburg norm $$\|h\|_{L^G}=\inf\{\lambda>0 : \int_\Omega G(|h|/\lambda)\leq 1\}$$. The Orlicz–Sobolev space $W^{1,G}(\Omega)$ consists of $u\in L^G(\Omega)$ with weak derivatives $\partial_i u\in L^G(\Omega)$, normed by $\|u\|_{W^{1,G}}=\|u\|_{L^G}+\|\nabla u\|_{L^G}$.

Domain configuration uses the upper half-ball $B_1^+ = \{x\in B_1: x_n > 0\} \subset \mathbb{R}^n$, with flat boundary $T_1 = \{x \in B_1 : x_n = 0\}$ and boundary data $\varphi \in W^{1,G}(B_1^+) \cap C^0(\bar B_1^+ )$ satisfying $\varphi\geq0$ on $T_1$. The admissible class is

$$\mathcal{G} = \{v \in W^{1,G}(B_1^+): v = \varphi \text{ on } \partial B_1^+ \setminus T_1,\, v\geq 0 \text{ on } T_1 \}.$$

The variational objective is to minimize

$J(u) = \int_{B_1^+} G(|\nabla u|)\, dx$

over $\mathcal{G}$. Existence of minimizers $u$ follows by direct methods leveraging convexity and compactness (Bessa et al., 1 Feb 2026).

2. Euler–Lagrange Variational Inequality and Boundary Conditions

The presence of a unilateral constraint $u\geq 0$ on $T_1$ translates, upon taking the first variation, into a variational inequality. Minimizers $u$ are $g$-harmonic inside $B_1^+$:

$$\int_{B_1^+} \frac{g(|\nabla u|)}{|\nabla u|} \nabla u \cdot \nabla \phi = 0$$

for all $\phi\in C_0^\infty(B_1^+)$.

On the thin set $T_1$, the minimizer satisfies Signorini-type (complementarity) boundary conditions:

• $u \geq 0$,
• $\frac{g(|\nabla u|)}{|\nabla u|} u_{x_n} \leq 0$,
• $$u \cdot \left(\frac{g(|\nabla u|)}{|\nabla u|} u_{x_n}\right) = 0$$,

which encode the absence of flux through $T_1$ where $u>0$ and nonpositive normal derivative where $u = 0$ (Bessa et al., 1 Feb 2026).

3. Regularity via De Giorgi Iteration: Lipschitz Estimates

The regularity analysis adapts De Giorgi’s iterative scheme to the nonhomogeneous Orlicz-growth setting. The approach involves an even extension $\tilde u(x',x_n) = u(x',|x_n|)$, which belongs to $W^{1,G}(B_1)$ and solves a related obstacle problem with continuous obstacle $\psi$ vanishing on $T_1$. Utilizing $C^{1,\alpha}$ boundary regularity for $g$-harmonic functions, it is established that $\psi$ is Lipschitz with constant proportional to $\|u\|_{L^\infty}$.

A barrier comparison and Harnack estimate for $u$—lifting techniques from the classical theory—result in uniform control:

$$\sup_{B_{r/2}}\big|\tilde u - \tilde u(x_0)\big|\leq C \|u\|_{L^\infty} r$$

for all balls $B_r(x_0)$ intersecting $T_1$. Covering $B^+_{3/4}$ yields

$$u \in C^{0,1}(B_{3/4}^+),\quad \|u\|_{C^{0,1}(B_{3/4}^+)} \leq C(n,\Delta_0, g_0, g(1)) \|u\|_{L^\infty(B_1^+)}.$$

Caccioppoli inequalities and a refined De Giorgi iteration—featuring level-set energy decay and Giusti-type abstract lemmas—give $L^\infty$ bounds for tangential derivatives $u_{x_m}$, ensuring global Lipschitz continuity of $\nabla u$ up to $T_1$ (Bessa et al., 1 Feb 2026).

4. Hölder Regularity of the Gradient

A “boundary improvement” lemma ensures that at any free–boundary point $x_0\in \partial\{u=0\} \cap T_1$, some partial derivative decreases by a geometric factor when passing to smaller scales after normalization. This leverages previously obtained De Giorgi estimates and compactness arguments at boundary points. An abstract covering and iteration lemma (of Andersson–Mikayelyan type) is applied to the supremum function $\omega(R) = \sup_{B_{R/2}}|\nabla u|$, yielding a decay

$\omega(R) \leq C (R/R_0)^\beta$

with explicit $\beta = \beta(n, \Delta_0, \lambda) > 0$. By patching with interior $C^{1,\alpha}$ regularity for $g$-harmonic functions, one concludes

$$u \in C^{1,\gamma}(\bar B^+_{1/2}),\qquad \gamma = \min\{\beta, \text{ interior }\alpha\},$$

so that $\nabla u$ is Hölder continuous with exponent $\gamma$ depending only on $n$, $\Delta_0$, $g_0$ (Bessa et al., 1 Feb 2026).

5. Structure of the Nodal Set for the Thin Obstacle

The geometry of the contact set $\{u=0\}$ on the thin set is described with k-th order nodal sets:

$$\mathfrak{n}_k(u) = \{x \in \bar B^+_{1/2}: D^\alpha u(x) = 0 \text{ for } |\alpha|<k,\, \exists |\beta|=k,\, D^\beta u(x)\neq 0\}.$$

Leveraging the $C^{1,\gamma}$ regularity and classical implicit-function theorems (Han's theorem for nodal sets), the first nodal set admits the decomposition

$\mathfrak{n}_1(u) = \bigcup_{j=0}^n \mathcal{M}_j,$

where each $\mathcal{M}_j$ is a finite union of $j$-dimensional $C^{1,\gamma}$-submanifolds of $T_1$. The highest-dimensional stratum $\mathcal{M}_{n-1}$ is $C^{1,\gamma}$; lower strata are of lower dimension. The constants involved in all regularity statements depend explicitly on $n, \Delta_0, g_0, g(1)$ and $\|\varphi\|$ (Bessa et al., 1 Feb 2026).

6. Technical Framework and Relevance

The analytical strategy fundamentally extends De Giorgi's regularity theory to variable, non-polynomial growth functionals. The results provide a detailed regularity theory for thin obstacle problems with Orlicz growth, establishing both optimal regularity bounds for minimizers and a geometric stratification of the nodal set, via a careful combination of classical and modern techniques in the calculus of variations and nonlinear PDE regularity. All foundational lemmas and iteration procedures appear with explicit dependencies, and the approach is fully detailed in the work of Bessa, Silva, and Sousa (Bessa et al., 1 Feb 2026).