On some Sobolev and Pólya-Szegö type inequalities with weights and applications

This paper establishes new weighted Sobolev embedding theorems and a Pólya-Szegö type inequality derived from an isoperimetric problem to analyze a class of semilinear degenerate elliptic equations in a three-dimensional domain, thereby extending existing results to higher dimensions.

The Gist

Imagine you are an architect trying to build a very special, fragile structure inside a room. This room isn't a normal box; it's a space where the "rules of physics" change depending on where you are. Specifically, near the center of the room, things behave normally, but as you move away, the space gets "thicker" or "heavier," making it harder to move or stretch things. This is what mathematicians call a degenerate elliptic equation.

The paper you shared is a guidebook for architects (mathematicians) trying to solve problems in this weird, heavy room. Here is the breakdown of their journey, explained simply:

1. The Problem: The "Heavy" Room

The authors are studying a specific type of equation that describes how a surface (like a drumhead or a soap bubble) behaves in this 3D room. The catch? The room has a "weight" attached to it.

• The Weight: Imagine the floor of the room is made of lead near the center but gets lighter as you go out, or vice versa. In this paper, the weight depends on the distance from the center in a specific way ($|x|^{2\alpha}$).
• The Goal: They want to know two things:
  1. Existence: Can we build a stable shape (a solution) in this room?
  2. Non-existence: Is the room so weird that no stable shape can exist under certain conditions?

2. The New Tool: The "Magic Mirror" (Rearrangement)

To solve this, the authors needed a new tool. They used something called a Pólya-Szegő inequality.

• The Analogy: Imagine you have a lump of clay (your function) that is shaped weirdly in this heavy room. You want to know how much energy it takes to hold that shape.
• The Trick: The authors invented a "Magic Mirror." If you look at your weird clay lump in this mirror, it transforms into a perfect, smooth sphere (or a specific round shape) that has the same amount of "clay" (volume) but is arranged in the most efficient way possible.
• The Result: The magic mirror proves that the perfect sphere always requires less energy (or the same amount) than your weird, messy shape. This is a huge shortcut. Instead of analyzing every possible messy shape, you only need to analyze the perfect sphere to find the limits of what's possible.

3. The "Weighted Area" Puzzle (Isoperimetric Inequality)

To make the Magic Mirror work, they first had to solve a puzzle about "Weighted Area."

• The Classic Puzzle: In normal geometry, if you have a fixed amount of string (perimeter), the shape that encloses the most area is a circle.
• The Weighted Puzzle: In this heavy room, a circle isn't the best shape anymore. Because the "floor" is heavier in some spots, the best shape is a distorted, egg-like sphere.
• The Breakthrough: The authors proved that even in this 3D heavy room, there is a specific "best shape" (a weighted sphere) that is the most efficient. This was the missing piece they needed to build their Magic Mirror.

4. The Results: What Did They Find?

Part A: The "Best Constant" (The Efficiency Limit)
They calculated the absolute limit of efficiency for these shapes. Think of it like finding the "fuel efficiency" of a car. They found a number (a constant) that tells you the minimum amount of energy required to hold a shape in this room.

• Note: They didn't find the exact perfect number, but they found a very good "lower bound" (a safe minimum estimate). They admit that finding the exact perfect number is still a mystery for future mathematicians.

Part B: When Shapes Can't Exist (The Pohozaev Identity)
They looked at a specific scenario where the "force" pushing on the shape gets too strong (a high power $p$).

• The Analogy: Imagine trying to blow up a balloon in a room where the air pressure increases the more you blow. If you blow too hard, the balloon will pop or never form at all.
• The Finding: They proved that if the "push" is too strong and the room has a specific star-like shape, no stable solution exists. The structure simply cannot hold together.

Part C: When Shapes Do Exist (The Mountain Pass)
On the flip side, they looked at conditions where a solution does exist.

• The Analogy: Imagine a hiker trying to cross a mountain range. To get from one valley (zero energy) to another, they must climb a pass.
• The Finding: Using a famous mathematical tool called the "Mountain Pass Lemma," they showed that if the "push" behaves nicely (not too crazy, not too weak), there is definitely a stable shape that can exist. It's like proving there is a safe path over the mountain.

Summary

In plain English, this paper says:
"We took a difficult math problem about shapes in a weird, heavy 3D space. We invented a new way to compare messy shapes to perfect spheres (using a 'Magic Mirror'). We proved that in this heavy space, perfect spheres are the most efficient. Using this, we figured out exactly when you can build a stable shape and when the forces are too strong for anything to exist."

They extended previous work that was only done in 2D (flat paper) to 3D (real space), which is much harder because the "weight" behaves differently in higher dimensions.

Technical Summary

Here is a detailed technical summary of the paper "On Some Sobolev and Pólya-Szegö Type Inequalities with Weights and Applications" by Trung Hieu Giang, Nguyen Minh Tri, and Dang Anh Tuan.

1. Problem Statement

The paper investigates a boundary-value problem for a class of semilinear degenerate elliptic equations in a three-dimensional domain $\Omega \subset \mathbb{R}^3$ containing the origin. The specific problem $(P)$ is given by:
$$\begin{cases} -\Delta_x u - |x|^{2\alpha} \frac{\partial^2 u}{\partial y^2} = f(x, y, u) & \text{in } \Omega, \\ u = 0 & \text{on } \partial\Omega, \end{cases}$$
where $x = (x_1, x_2) \in \mathbb{R}^2$, $\alpha > 0$, and $\Delta_x$ is the Laplacian with respect to $x$. The operator involves a Grushin-type degeneracy where the diffusion coefficient vanishes at $x=0$.

The authors aim to:

1. Establish weighted Sobolev embedding theorems and Pólya-Szegö type inequalities in $\mathbb{R}^3$.
2. Apply these analytical tools to prove existence and nonexistence results for nontrivial solutions to problem $(P)$.

This work extends previous research by Luyen, Tri, and Tuan (which was limited to 2D) to the three-dimensional context.

2. Methodology

The paper employs a combination of geometric analysis, rearrangement techniques, and variational methods:

• Weighted Isoperimetric Inequality: The authors derive a new weighted isoperimetric inequality. They define a specific weighted volume $|E|_{2,\alpha}$ and weighted area $P_{2,\alpha}(E)$ involving the weight $|x|^{2\alpha}$.
• Change of Variables & Symmetrization: To prove the isoperimetric inequality, they utilize a specific change of variables (homeomorphism) that transforms the weighted problem into a standard Euclidean isoperimetric problem within specific angular sectors ($R^3_{s_j}$). This allows them to leverage classical results (specifically from [14]) to establish the inequality for the weighted case.
• Pólya-Szegö Inequality: Using the isoperimetric result, they prove a Pólya-Szegö type inequality for the Grushin rearrangement ($u^*$). This inequality states that the weighted Dirichlet energy of the rearranged function is less than or equal to that of the original function.
• Sobolev Embeddings: The Pólya-Szegö inequality is used to derive the optimal (or near-optimal) weighted Sobolev embedding for the critical exponent $q=6$.
• Variational Methods: For existence results, the authors define an energy functional $\Phi$ on a weighted Sobolev space $S^2_{1,0}(\Omega)$. They verify the Palais-Smale (C)$_c$ condition and apply the Mountain Pass Lemma to find critical points.
• Pohozaev Identity: For nonexistence results, they derive a Pohozaev-type identity adapted to the degenerate operator and the specific geometry of the domain.

3. Key Contributions and Results

A. New Inequalities (Sections 2 & 3)

1. Theorem 1 (Weighted Isoperimetric Inequality):
   Establishes that for a bounded open set $E \subset \mathbb{R}^3$, the ratio of the weighted area to the weighted volume is minimized by specific "weighted balls" ($B_{s_j}$).
   $$\frac{P_{2,\alpha,j}(B_{s_j})^{3/2}}{|B_{s_j}|_{2,\alpha}} \leq \frac{P_{2,\alpha,j}(E)^{3/2}}{|E|_{2,\alpha}}$$
   Note: The authors note that unlike the 2D case, this result currently holds specifically for the weight $|x|^{2\alpha}$ (where the power is 2) due to the structure of the change of variables required for the proof.

2. Theorem 2 (Pólya-Szegö Type Inequality):
   Proves that for $u \in C_0^\infty(\mathbb{R}^3)$, the weighted gradient norm decreases under rearrangement:
   $$\int_{\mathbb{R}^3} |\nabla_G u^*|^2 \, dx \leq \int_{\mathbb{R}^3} |\nabla_G u|^2 \, dx$$
   where $\nabla_G u = (\partial_{x_1}u, \partial_{x_2}u, |x|^\alpha \partial_y u)$.

3. Theorem 3 (Weighted Sobolev Inequality):
   Establishes the embedding $W^{1, 2\alpha, 6}_0(\mathbb{R}^3) \hookrightarrow L^{6}_{|x|^{2\alpha}}(\mathbb{R}^3)$:
   $$\left( \int_{\mathbb{R}^3} |x|^{2\alpha}|u|^6 \, dx \right)^{1/6} \leq C_{2\alpha, 6}^{-1} \left( \int_{\mathbb{R}^3} |\nabla_G u|^2 \, dx \right)^{1/2}$$
   The authors provide a lower bound for the best constant $C_{2\alpha, 6}$, though the exact value remains an open question.

B. Applications to Degenerate Equations (Section 4)

1. Nonexistence Result (Theorem 4):
   For the specific nonlinearity $f(x, y, u) = |x|^{2\alpha}|u|^{p-1}u$ with $p > 5$, the authors prove that if the domain $\Omega$ is $G_\alpha$-star-shaped with respect to the origin, there are no nontrivial solutions in the space $S^2(\Omega)$. This is derived using the Pohozaev identity.

2. Existence Result (Theorem 5):
   Under general assumptions (A1–A5) on the nonlinearity $f$ (including subcritical growth, behavior at zero and infinity, and monotonicity), the authors prove the existence of a nontrivial weak solution using the Mountain Pass Lemma. This relies on the compactness of the embedding established in Theorem 6.

4. Significance and Novelty

• Dimensional Extension: The primary contribution is the successful generalization of weighted Sobolev and Pólya-Szegö inequalities from 2D to 3D. This is non-trivial because the change of variables required to symmetrize the weight behaves differently in higher dimensions.
• Optimal Constants: The paper provides a method to estimate the best constants for weighted Sobolev embeddings, a crucial step in analyzing the qualitative behavior of solutions to degenerate PDEs.
• Unified Framework: By establishing the embedding theorems first, the paper creates a robust framework for studying a wide class of semilinear degenerate elliptic equations, moving beyond specific cases to general existence/nonexistence criteria.
• Limitations: The authors explicitly acknowledge that their proof technique for the isoperimetric inequality currently restricts the weight to the form $|x|^{2\alpha}$ (specifically $p=2$ in the weight $|x|^{p\alpha}$) and does not yet generalize to arbitrary powers $p$ in the 3D case, unlike the 2D case.

In summary, this paper provides the necessary analytical foundation (inequalities and embeddings) to treat Grushin-type operators in 3D, enabling rigorous proofs for the existence and nonexistence of solutions to associated semilinear problems.