On the best constant for Gagliardo-Nirenberg interpolation inequalities

Original · EN

In this paper we derive the best constant for the following Gagliardo-Nirenberg interpolation inequality eqnarray* uₗᵐ⁺¹≤ Cq,ₘ,ₚ u¹⁻θLq⁺¹∇ uθₗₚ, θ=pd(m-q)/(m+1)[d(p-q-1)+p(q+1)], eqnarray* where parameters q,m,p respectively belong to the following two ranges: (i) p>d≥ 1, q≥0 and m=∞. That shows L∞-type Gagliardo-Nirenberg interpolation inequality. (ii) p>{1,2d/d+2}, 0≤ q<σ-1, and q<m<σ, where σ is defined by σ:= (p-1)d+p /d-p if p<d; σ:=∞ if p≥ d. That gives Lᵐ-type Gagliardo-Nirenberg interpolation inequality. The best constant Cq,ₘ,ₚ is given by eqnarray* Cq,ₘ,ₚ:=θ-θp(1-θ)θp-1/m+1Mc-θd, Mc:=∫ᵣᵈuc,ₘq⁺¹dx, eqnarray* where uc,ₘ is the unique radial non-increasing solution to a generalized Lane-Emden equation. The case of equality holds when u=Auc,ₘ(λ(x-x₀)) for any real numbers A>0, λ>0 and x₀∈ Rᵈ. In particular, for the case m=+∞, the generalized Lane-Emden equation becomes a Thomas-Fermi type equation. For q=0, m=∞ or d=1, uc,ₘ are closed form solutions expressed in term of the incomplete Beta functions. Moreover, we show that uc,ₘ→ uc,∞ and Cq,ₘ,ₚ→ Cq,∞,ₚ as m→ +∞ for d=1.