[Paper Review] Integral and isocapacitary inequalities

Vladimir Maz’ya|ArXiv.org|Sep 15, 2008

Holomorphic and Operator Theory13 references3 citations

TL;DR

This paper establishes sharp integral inequalities involving harmonic capacity that imply the Faber-Krahn property for the first Dirichlet-Laplace eigenvalue. It proves that isocapacitary and isoperimetric constants are not equivalent even for domains star-shaped with respect to a ball, and provides necessary and sufficient conditions for generalized difference seminorm-gradient norm inequalities via new types of isoperimetric and isocapacitary inequalities.

ABSTRACT

It is shown by a counterexample that isocapacitary and isoperimetric constants of a multi-dimensional Euclidean domain starshaped with respect to a ball are not equivalent. Sharp integral inequalities involving the harmonic capacity which imply Faber-Krahn property of the fundamental Dirichlet-Laplace eigenvalue are obtained. Necessary and sufficient conditions ensuring integral inequalities between a difference seminorm and the $L_p$-norm of the gradient are found.

Motivation & Objective

• To investigate the equivalence between isocapacitary and isoperimetric constants in multidimensional Euclidean domains.
• To derive sharp integral inequalities involving harmonic capacity that imply the Faber-Krahn property for the first Dirichlet-Laplace eigenvalue.
• To establish necessary and sufficient conditions for inequalities between difference seminorms and $L_p$-norms of the gradient.
• To characterize the validity of generalized Poincaré-type inequalities involving measures on product spaces via isoperimetric and isocapacitary conditions.

Proposed method

• Constructs a counterexample showing that isocapacitary and isoperimetric constants are not equivalent, even in domains star-shaped with respect to a ball.
• Uses the relative harmonic capacity $\mathrm{cap}(F;\Omega)$ and the $L_2$-Rayleigh quotient to derive two-sided estimates for the first Dirichlet eigenvalue $\Lambda(\Omega)$.
• Applies the isoperimetric constant $\gamma(\Omega)$ defined via $H_{n-1}(\partial g)/m_n(g)$ to show $\gamma(\Omega)^2 \leq 4\Gamma(\Omega)$, where $\Gamma(\Omega)$ is the isocapacitary constant.
• Derives necessary and sufficient conditions for the inequality $\left(\int_\Omega\int_\Omega |u(x)-u(y)|^q \mu(dx,dy)\right)^{1/q} \leq C \|\nabla u\|_{L_p(\Omega)}$ via new isoperimetric and isocapacitary inequalities.
• For the one-dimensional case, provides a criterion involving the measure $\nu''$ and the condition $\mu( [\alpha,\beta], \mathbb{R}\setminus(\alpha-r,\beta+r) ) \leq \text{const} \cdot r^{-q(1-p)/p}$.
• Uses capacity-based characterization via $\mathrm{cap}_{p,\mu}(F;\Omega) = \inf \{ \langle u \rangle_{p,\mu}^p : u \geq 1 \text{ on } F \} $ to derive necessary and sufficient conditions for trace-type inequalities.

Experimental results

Research questions

• RQ1 Are isocapacitary and isoperimetric constants equivalent for domains star-shaped with respect to a ball?
• RQ2 What are the necessary and sufficient conditions for the inequality $\left(\int_\Omega\int_\Omega |u(x)-u(y)|^q \mu(dx,dy)\right)^{1/q} \leq C \|\nabla u\|_{L_p(\Omega)}$ to hold?
• RQ3 Can the general isocapacitary condition $\sup_F \frac{\nu(F)^{p/q}}{\mathrm{cap}_{p,\mu}(F;\Omega)} < \infty$ be replaced by a condition involving only balls for general $\mu$?
• RQ4 What are the sharp conditions for the trace inequality $\int_{\mathbb{R}^n} |u|^p \nu(dx) \leq c \|(-\Delta)^{\alpha/2}u\|_{L_p}^p$?
• RQ5 How do the criteria (i)–(iii) for the trace inequality relate to the generalized difference seminorm inequality with $\mu$ given by $|x-y|^{-n-p\alpha}dx\,dy$?

Key findings

• The isocapacitary and isoperimetric constants are not equivalent, even for domains star-shaped with respect to a ball, as demonstrated by a counterexample.
• The inequality $\left(\int_\Omega\int_\Omega |u(x)-u(y)|^q \mu(dx,dy)\right)^{1/q} \leq C \|\nabla u\|_{L_p(\Omega)}$ holds if and only if a new type of isoperimetric or isocapacitary inequality is satisfied, depending on $q \geq p = 1$ or $q > p > 1$.
• For $u \in C_0^\infty(\mathbb{R})$, the inequality holds if $\mu( [\alpha,\beta], \mathbb{R}\setminus(\alpha-r,\beta+r) ) \leq \text{const} \cdot r^{-q(1-p)/p}$ uniformly in $\alpha, \beta, r$.
• The condition $K < \infty$, where $K = \int_0^\infty s^{p-1} |\nu'(s)| ds$, is sharp for the inequality $\int_\mathbb{R}\int_\mathbb{R} |u(t)-u(\tau)|^p \nu''(t-\tau) dt d\tau \leq C \int_\mathbb{R} |u'(t)|^p dt$.
• For $\mu(dx,dy) = |x-y|^{-n-p\alpha} dx dy$ with $0 < \alpha < 1$ and $\alpha p < n$, the inequality (81) holds iff $\sup_{x,\rho} \frac{\nu(B(x,\rho))^{p/q}}{\rho^{n-p\alpha}} < \infty$.
• The trace inequality $\int_{\mathbb{R}^n} |u|^p \nu(dx) \leq c \|(-\Delta)^{\alpha/2}u\|_{L_p}^p$ is equivalent to (81) under this $\mu$, and is characterized by criteria (i)–(iii) involving dyadic cubes, Riesz potentials, and weak-type estimates.