MATRIX OF PLAN B: SOBOLEV INEQUALITIES

The Sobolev inequality is a fundamental result in the theory of partial differential equations and functional analysis. It establishes a relationship between the integrability of a function and its derivatives. The classical Sobolev inequality states that for functions in W^{1,p}(ℝ^n) (the Sobolev space of functions with derivatives in L^p), there exists a constant C such that: ‖u‖{L^q} ≤ C‖∇u‖{L^p} where q = np/(n-p) for 1 ≤ p < n, and the sharp constant C is crucial for applications. The sharp constant is related to the best constant in the Gagliardo-Nirenberg-Sobolev inequality.

In this study the relationship between the exponent ( p ) and the dimension ( d ) is crucial in determining the asymptotic behavior of the lowest eigenvalue of the p-Laplacian perturbed by a weakly coupled potential as the coupling parameter approaches zero.

When mathematicians and physicists talk about the Sobolev inequality, they're discussing something that might seem incredibly abstract, but has surprising connections to our physical world. Let me break down this particular form of the inequality:

$\lim_{\epsilon \to 0^+} \frac{1}{d-1} \log \left( \frac{1}{|\lambda(V)|} \right) = d \omega_d^{\frac{1}{d-1}} \left( \int_{\mathbb{R}^d} V(x) \, dx \right)^{\frac{1}{d-1}}$

where $\omega_d$ denotes the surface area of the unit sphere in $\mathbb{R}^d$.

Further analyses on the asymptotic behavior of the lowest eigenvalue of the p-Laplacian operator lies in the presence of weakly coupled potentials, exploring its relationship with Sobolev interpolation inequalities for different values of p and d.

Case ( p > d ):

When ( p ) is greater than ( d ), the lowest eigenvalue vanishes algebraically as the weak coupling parameter approaches zero.

The asymptotic behavior is closely related to the Sobolev interpolation inequality, and the lowest eigenvalue's decay rate is influenced by the sharp constant in this inequality.

Case ( p = d ):

When ( p ) equals ( d ), the situation is more delicate. The lowest eigenvalue vanishes exponentially fast as the weak coupling parameter approaches zero.

This case requires more regularity of the potential ( V ) because functions in ( W^{1,d}(\mathbb{R}^d) ) are not necessarily bounded. The asymptotic coefficient depends on the integral of the potential ( V ) over ( \mathbb{R}^d ).

In summary, the relationship between ( p ) and ( d ) determines whether the lowest eigenvalue decays algebraically or exponentially as the weak coupling parameter approaches zero, with ( p > d ) leading to algebraic decay and ( p = d ) leading to exponential decay.

As the weak coupling approaches zero, the lowest eigenvalue of the p-Laplacian perturbed by a weakly coupled potential approaches zero. The asymptotic behavior of the lowest eigenvalue depends on the relationship between the exponent ( p ) and the dimension ( d ):

For ( p > d ): The lowest eigenvalue vanishes algebraically as the weak coupling approaches zero. The precise asymptotic behavior is given by:

\lim_{\epsilon \to 0^+} -\epsilon^{\frac{p-d}{p}} \lambda(V) = -\left( \frac{p-d}{p} \right) \left( \frac{1}{S_{d,p}} \int_{\mathbb{R}^d} V(x) , dx \right)^{\frac{p-d}{p}}, where ( S_{d,p} ) is the sharp constant in the Sobolev inequality.

For ( p = d ): The lowest eigenvalue vanishes exponentially fast as the weak coupling approaches zero. The precise asymptotic behavior is given by:

\lim_{\epsilon \to 0^+} \frac{1}{d-1} \log \left( \frac{1}{|\lambda(V)|} \right) = d \omega_d^{\frac{1}{d-1}} \left( \int_{\mathbb{R}^d} V(x) , dx \right)^{\frac{1}{d-1}} where ( \omega_d ) denotes the surface area of the unit sphere in ( \mathbb{R}^d ).

These results indicate that the rate at which the lowest eigenvalue approaches zero is significantly influenced by the interplay between the exponent ( p ) and the dimension ( d ).

The p-Laplacian plays a central role in this study as it is the primary operator being analyzed under weak perturbations. The significance of the p-Laplacian in this context includes:

Nonlinear Generalization: The p-Laplacian, denoted as ( -\Delta_p(u) = -\nabla \cdot ( |\nabla u|^{p-2} \nabla u) ), is a nonlinear generalization of the standard Laplacian (which corresponds to ( p = 2 )). This makes it suitable for studying a broader class of differential equations, particularly those that are nonlinear.

Weak Coupling Analysis: The study focuses on the behavior of the p-Laplacian when it is perturbed by a weakly coupled potential ( V ). The goal is to understand how the lowest eigenvalue of the operator changes as the coupling strength approaches zero.

Asymptotic Expansions: The p-Laplacian allows for the derivation of asymptotic expansions of the lowest eigenvalue in the weak coupling limit. These expansions differ based on whether ( p ) is greater than, less than, or equal to the dimension ( d ).

Connection to Sobolev Inequalities: The analysis of the p-Laplacian is closely related to Sobolev interpolation inequalities. The sharp constants in these inequalities play a crucial role in determining the asymptotic behavior of the eigenvalues.

Overall, the p-Laplacian is significant in this study as it provides a framework for understanding the impact of weak perturbations on nonlinear differential operators and their eigenvalues, with implications for various mathematical and physical problems.

The main results regarding the p-Laplacian in this study are focused on the asymptotic behavior of the lowest eigenvalue of the p-Laplacian perturbed by a weakly coupled potential ( V ) as the coupling parameter ( \epsilon ) approaches zero. These results are presented separately for the cases ( p > d ) and ( p = d ): Case ( p > d ):

Theorem 2.1: For ( p > d \geq 1 ) and a potential ( V \in L^1(\mathbb{R}^d) ) such that ( \int_{\mathbb{R}^d} V(x) , dx > 0 ), the asymptotic behavior of the lowest eigenvalue ( \lambda(V) ) is given by:

[

\lim_{\epsilon \to 0^+} \epsilon^{\frac{p-d}{p}} \lambda(\epsilon V) = -\left( \frac{p-d}{p} \right) \left( \frac{1}{S_{d,p}} \right)^{\frac{p}{p-d}} \left( \int_{\mathbb{R}^d} V(x) , dx \right)^{\frac{p}{p-d}},

]

where ( S_{d,p} ) is the sharp constant in the Sobolev inequality.

Case ( p = d ):

Theorem 2.2: For ( p = d > 1 ), a potential ( V \in L^1(\mathbb{R}^d) \cap L^q(\mathbb{R}^d) ) for some ( q > 1 ), and ( \int_{\mathbb{R}^d} V(x) , dx > 0 ), the asymptotic behavior of the lowest eigenvalue ( \lambda(V) ) is given by:

[

\lim_{\epsilon \to 0^+} \epsilon^{d-1} \log \left( \frac{1}{|\lambda(\epsilon V)|} \right) = d \omega_d^{d-1} \left( \int_{\mathbb{R}^d} V(x) , dx \right),

]

where ( \omega_d ) denotes the surface area of the unit sphere in ( \mathbb{R}^d ).

These results highlight the different asymptotic behaviors of the lowest eigenvalue depending on the relationship between ( p ) and ( d ). For ( p > d ), the eigenvalue vanishes algebraically, while for ( p = d ), it vanishes exponentially fast. The study also emphasizes the role of the Sobolev inequality and the integral of the potential ( V ) in determining these asymptotic behaviors.