Extremal functions for Morrey’s inequality in convex domains

Abstract

For a bounded domain Ω ⊂ R n and p > n, Morrey's inequality implies that there is c > 0 such thatfor each u belonging to the Sobolev space W 1,p 0 (Ω). We show that the ratio of any two extremal functions is constant provided that Ω is convex. We also show with concrete examples why this property fails to hold in general and verify that convexity is not a necessary condition for a domain to have this feature. As a by product, we obtain the uniqueness of an optimization problem involving the Green's function for the p-Laplacian.

Minimization of quotients with variable exponents

“…u l,j jq(x) = Λ l,j and by making j → ∞ we obtain µ l ≤ lim inf j→∞ Λ l,j , concluding thus the proof of (19). We say that u ∈ W 1,lp(x) 0…”

“…We also show in Section 4, by using arguments developed in [19], that µ l is achieved at u if, and only if,…”

“…(Ω). Now, by adapting arguments of [19] we characterize of the extremal functions of µ l . For this, let us denote by Γ u the set of the points where a function u ∈ C(Ω) assumes its uniform norm, that is…”

On the symmetry and monotonicity of Morrey extremals

“…In recent work, we showed that there is a smallest C = C * > 0 so that Morrey's inequality holds with C * and that there exist nonconstant functions for which equality is attained in (1.2) [19]. We will call these functions Morrey extremals.…”

“…While C * and the corresponding Morrey extremals are not explicitly known, many qualitative properties of these functions have been identified. In particular, Morrey extremals which satisfy (1.3) and (1.4) are known to be unique, axially symmetric about the x n −axis and antisymmetric about the x n = 0 plane ( [19], Section 3 and 6). We established this in our previous work by relying on a uniqueness property of solutions of (1.5).…”