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Article: Sobolev spaces in metric measure spaces: reflexivity and lower semicontinuity of slope
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ABSTRACT: In this paper we make a survey of some recent developments of the theory of Sobolev spaces $W^{1,q}(X,\sfd,\mm)$, $1<q<\infty$, in metric measure spaces $(X,\sfd,\mm)$. In the final part of the paper we provide a new proof of the reflexivity of the Sobolev space based on $\Gamma$-convergence; this result extends Cheeger's work because no Poincar\'e inequality is needed and the measure-theoretic doubling property is weakened to the metric doubling property of the support of $\mm$. We also discuss the lower semicontinuity of the slope of Lipschitz functions and some open problems.12/2012; -
Conference Proceeding: Sobolev spaces in metric measure spaces: reflexivity and lower semicontinuity of slope
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ABSTRACT: In this paper we make a survey of some recent developments of the theory of Sobolev spaces W 1,q (X, d, m), 1 < q < ∞, in metric measure spaces (X, d, m). In the final part of the paper we provide a new proof of the reflexivity of the Sobolev space based on Γ-convergence; this result extends Cheeger's work because no Poincaré inequality is needed and the measure-theoretic doubling property is weakened to the metric doubling property of the support of m. We also discuss the lower semicontinuity of the slope of Lipschitz functions and some open problems. Contents 1. Introduction 1 2. Preliminary notions 4 3. Hopf-Lax formula and Hamilton-Jacobi equation 9 4. Weak gradients 12 5. Gradient flow of C q and energy dissipation 17 6. Equivalence of gradients 22 7. Reflexivity of W 1,q (X, d, m), 1 < q < ∞ 23 8. Lower semicontinuity of the slope of Lipschitz functions 29 9. Appendix A: other notions of weak gradient 31 10. Appendix B: discrete gradients in general spaces 33 11. Appendix C: some open problems 33Sapporo 2012; 08/2012
