Stability for the Sobolev inequality: existence of a minimizer

2022/11/25 by König, Tobias · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2211.14185

Abstract

We prove that the stability inequality associated to Sobolev's inequality and its set of optimizers \mathcal M and given by \frac‖∇ f‖_{L²(\mathbb R^d)}² - S_d ‖f‖_{L^(2d)/(d-2)(\mathbb R^d)}² inf_{h ∈ \mathcal M} ‖∇ (f - h)‖_{L²(\mathbb R^d)}² ≥ c_BE gt; 0 for every f ∈ H¹(\mathbb R^d), which is due to Bianchi and Egnell, admits a minimizer for every d ≥ 3. Our proof consists in an appropriate refinement of a classical strategy going back to Brezis and Lieb. As a crucial ingredient, we establish the strict inequality c_BE < 2 - 2^(d-2)/(d), which means that a sequence of two asymptotically non-interacting bubbles cannot be minimizing. Our arguments cover in fact the analogous stability inequality for the fractional Sobolev inequality for arbitrary fractional exponent s ∈ (0, d/2) and dimension d ≥ 2.

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• Quantitative stability of critical points for the nonlocal-Sobolev inequality in Heisenberg group

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