[Paper Review] Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces

Giampiero Palatucci, Adriano Pisante|arXiv (Cornell University)|Feb 24, 2013

Nonlinear Partial Differential Equations25 references4 citations

TL;DR

This paper establishes an improved Sobolev embedding inequality in fractional Sobolev spaces involving Morrey norms, which provides a direct proof of the existence of optimizers and compactness up to symmetry for the critical Sobolev embedding. The refined inequality enables a transparent derivation of profile decomposition and concentration-compactness in $\dot{H}^{s}$ spaces, with applications to the asymptotic behavior of subcritical problems and concentration of maximizers at a single point.

ABSTRACT

We obtain an improved Sobolev inequality in H^s spaces involving Morrey norms. This refinement yields a direct proof of the existence of optimizers and the compactness up to symmetry of optimizing sequences for the usual Sobolev embedding. More generally, it allows to derive an alternative, more transparent proof of the profile decomposition in H^s obtained in [P. Gerard, ESAIM 1998] using the abstract approach of dislocation spaces developed in [K. Tintarev & K. H. Fieseler, Imperial College Press 2007]. We also analyze directly the local defect of compactness of the Sobolev embedding in terms of measures in the spirit of [P. L. Lions, Rev. Mat. Iberoamericana 1985]. As a model application, we study the asymptotic limit of a family of subcritical problems, obtaining concentration results for the corresponding optimizers which are well known when s is an integer ([O. Rey, Manuscripta math. 1989; Z.-C. Han, Ann. Inst. H. Poincare Anal. Non Lineaire 1991], [K. S. Chou & D. Geng, Differential Integral Equations 2000]).

Motivation & Objective

• To derive a refined Sobolev inequality in $\dot{H}^{s}({\mathds{R}}^{N})$ involving Morrey norms to strengthen the classical critical embedding.
• To provide a direct, non-abstract proof of the existence of optimizers and compactness up to symmetry for the critical Sobolev embedding in fractional spaces.
• To re-derive the profile decomposition in $\dot{H}^{s}$ using the refined inequality, avoiding reliance on dislocation space theory.
• To analyze the local defect of compactness via concentration-compactness in terms of measures, extending results from integer-order cases.
• To study the asymptotic limit of subcritical problems and prove concentration of optimizers at a single point in bounded domains.

Proposed method

• Derive a new improved Sobolev inequality by incorporating Morrey norms into the $\cdot{H}^{s}$ norm, refining the classical $L^{2^*}$ embedding.
• Use the refined inequality to directly prove compactness up to symmetries (translation and dilation) for optimizing sequences in $\cdot{H}^{s}$.
• Apply the concentration-compactness principle in the spirit of [31, 32] to analyze the defect of compactness via weak* convergence of measures.
• Characterize the concentration of energy and mass using atomic measures $\mu_j$ and $\nu_j$ arising from weak* limits of $|(-\Delta)^{s/2}u_n|^2dx$ and $|u_n|^{2^*}dx$.
• Analyze the asymptotic behavior of maximizers for subcritical problems $S^*_\varepsilon$ as $\varepsilon \to 0$, showing convergence to the critical case.
• Use rescaling and compactness arguments to prove that maximizers concentrate at a single point $x_0 \in \overline{\Omega}$ under suitable normalization.

Experimental results

Research questions

• RQ1 Can the classical Sobolev embedding in fractional $\dot{H}^s$ spaces be refined using Morrey norms to yield stronger compactness properties?
• RQ2 Does the improved inequality allow for a direct, non-abstract proof of the existence of optimizers and profile decomposition in $\dot{H}^s$?
• RQ3 How does the local defect of compactness manifest in terms of measures, and can it be characterized via concentration-compactness?
• RQ4 What happens to the maximizers of subcritical problems as the exponent approaches the critical threshold $2^*$?
• RQ5 Can the concentration of optimizers at a single point be rigorously established in bounded domains using the refined inequality and measure-theoretic tools?

Key findings

• An improved Sobolev inequality is established that incorporates Morrey norms, refining the classical $\dot{H}^s \hookrightarrow L^{2^*}$ embedding.
• The refined inequality provides a direct proof of the existence of optimizers and compactness up to symmetry for the critical Sobolev embedding in $\dot{H}^s$.
• Profile decomposition in $\dot{H}^s$ is re-derived via the refined inequality, offering a more transparent and constructive alternative to the abstract dislocation space approach.
• The local defect of compactness is characterized through weak* convergence of measures: $|u_n|^{2^*}dx \rightharpoonup^* \nu = S^*\delta_{x_0}$ and $|(-\Delta)^{s/2}u_n|^2dx \rightharpoonup^* \delta_{x_0}$, indicating concentration at a single point.
• For subcritical problems with exponent $2^* - \varepsilon$, the maximizers $u_\varepsilon$ concentrate at a single point $x_0 \in \overline{\Omega}$ as $\varepsilon \to 0$, with $\|u_\varepsilon\|_{\dot{H}^s} \to 1$ and $|u_\varepsilon|^{2^*}dx \rightharpoonup^* S^*\delta_{x_0}$.
• The optimal constant $S^*$ in the critical Sobolev inequality is attained in the limit, and the concentration is shown to be stable under rescaling, with scaling parameters satisfying $x_n \to x_0$ and $\lambda_n \to 0$.