Inequalities for Functions Vanishing at the Boundary

Abstract

The present chapter deals with the necessary and sufficient conditions for the validity of certain estimates for the norm $\|u\|_{L_{q}(\varOmega ,\mu)}$, where and μ is a measure in Ω. Here we consider inequalities with the integral

$$\int_\varOmega\bigl[\varPhi(x, \nabla u)\bigr]^p\,\mathrm{d}x,$$

on the right-hand side. The function Φ(x,ξ), defined for x∈Ω and ξ∈ℝⁿ, is positive homogeneous of degree one in ξ. The conditions are stated in terms of isoperimetric (for p=1 in Sect. 2.1) and isocapacitary (for p≥1, in Sects. 2.2–2.4) inequalities. For example, we give a complete answer to the question of validity of the inequality

$$\| u\|_{L_q(\varOmega, \mu)} \le C \biggl(\int_\varOmega\bigl[\varPhi(x, \nabla u)\bigr]^p\, \mathrm{d}x \biggr)^{1/p},$$

both for q≥p≥1 and 0<q<p, p≥1.