Nonlinear Analysis–Theory and Methods, Springer Monographs in Mathematics, 558 References

Total Page:16

File Type:pdf, Size:1020Kb

Nonlinear Analysis–Theory and Methods, Springer Monographs in Mathematics, 558 References References 1. G. Acosta, R.G. Durán, An optimal Poincaré inequality in L1 for convex domains. Proc. Am. Math. Soc. 132, 195–202 (2004) 2. R.A. Adams, Sobolev Spaces (Academic Press, New York, 1975) 3. R.A. Adams, J. Fournier, Sobolev Spaces, 2nd edn. (Academic Press (Elsevier), Amsterdam, 2003) 4. D. Adams, L. Hedberg, Function Spaces and Potential Theory (Springer, Berlin, 1996) 5. S. Aizicovici, N.S. Papageorgiou, V. Staicu, Degree Theory for Operators of Monotone Type and Nonlinear Elliptic Equations with Inequality Constraints, Memoirs American Mathemat- ical Society, vol. 196, No. 915 (2008) 6. S. Aizicovici, N.S. Papageorgiou, V. Staicu, The spectrum and an index formula for the Neu- mann p-Laplacian and multiple solutions for problems with a crossing nonlinearity. Discret. Contin. Dyn. Syst. 25, 431–456 (2009) 7. N.I. Akhiezer, I.M. Glazman, Theory of Linear Operators in Hilbert Space,vol.IandII (Frederick Ungar Publishing Company, New York, 1961 and 1963) 8. V. Alexeev, V. Tikhomirov, S. Fomin, Commande Optimale (Mir, Moscou, 1982) 9. C. Aliprantis, K. Border, Infinite Dimensional Analysis. A Hitchhiker’s Guide (Springer, Berlin, 1994) 10. D. Alspach, A fixed point free nonexpansive map. Proc. Am. Math. Soc. 82, 423–424 (1981) 11. M. Altman, A fixed point theorem for completely continuous operators in Banach spaces. Bull. Acad. Polon. Sci. 3, 409–413 (1955) 12. H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces. SIAM Rev. 18, 620–709 (1976) 13. H. Amann, Ordered structures and fixed points, in Atti 2 Sem. Anal. Funz. Appl., Univ. Cosenza, Italy (1977), pp. 50 14. H. Amann, A note on degree theory for gradient mappings. Proc. Am. Math. Soc. 85, 591–595 (1982) 15. H. Amann, S. Weiss, On the uniqueness of the topological degree. Math. Z. 130, 39–54 (1973) 16. A. Ambrosetti, A. Malchiodi, Nonlinear Analysis and Semilinear Elliptic Problems (Cam- bridge University Press, Cambridge, UK, 2007) 17. A. Ambrosetti, P. Rabinowitz, Dual variational methods in critical point theory and applica- tions. J. Funct. Anal. 14, 349–381 (1973) 18. E. Asplund, Positivity of duality mappings. Bull. Am. Math. Soc. 73, 200–203 (1967) 19. E. Asplund, Averaged norms. Isr. J. Math. 5, 227–233 (1967) © Springer Nature Switzerland AG 2019 557 N. S. Papageorgiou et al., Nonlinear Analysis–Theory and Methods, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-03430-6 558 References 20. H. Attouch, G. Buttazzo, G. Michaille, Variational Analysis in Sobolev and BV Spaces. Ap- plications to PDEs and Optimization, MPS/SIAM Series on Optimization, 6. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA; Mathematical Programming Society (MPS), Philadelphia, PA, 2006 21. J.P. Aubin, I. Ekeland, Applied Nonlinear Analysis (Wiley, New York, 1984) 22. J.P. Aubin, H. Frankowska, Set-Valued Analysis (Birkhäuser, Boston, 1990) 23. T. Aubin, Problèmes isopérimétriques et espaces de Sobolev. J. Differ. Geom. 11, 573–598 (1976) 24. R. Aumann, Measurable utility and the measurable choice theorem (Aix-en-Provence, Actes Colloq. Internat., 1969), pp. 15–26 25. R. Baire, Sur les fonctions de variables réelles. Ann. Mat. Pura Appl. 3, 1–123 (1899) 26. R. Baire, Sur l’origine de la notion de semi-continuité. Bull. Soc. Math. Fr. 55, 141–142 (1927) 27. E.J. Balder, Necessary and sufficient conditions for L1-strong-weak lower semicontinuity of integral functionals. Nonlinear Anal. 11, 1399–1404 (1987) 28. E.J. Balder, Lectures on Young measure theory and its applications in economics, Workshop on Measure Theory and Real Analysis (Italian) (Grado, 1997). Rend. Istit. Mat. Univ. Trieste 31(suppl. 1), 1–69 (2000) 29. J.M. Ball, K.-W. Zhang, Lower semicontinuity of multiple integrals and the biting lemma. Proc. R. Soc. Edinb. Sect. A 114, 367–379 (1990) 30. S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations integrales. Fundam. Math. 3, 133–181 (1922) 31. S. Banach, Théorie des Opérations Linéaires, Monografje Matematyczne, Warsaw, 1932 (En- glish Translation: Theory of Linear Operators, North Holland, Amsterdam, 1987) 32. V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces (Noordhoff, Leiden, The Netherlands, 1975) 33. V. Barbu, T. Precupanu, Convexity and Optimization in Banach Spaces (Reidel Publishing Company, Dordrecht, 1986) 34. P. Bartolo, V. Benci, D. Fortunato, Abstract critical point theorems and applications to some nonlinear problems with “strong” resonance at infinity. Nonlinear Anal. 7, 981–1012 (1983) 35. T. Bartsch, Infinitely many solutions of a symmetric Dirichlet problem. Nonlinear Anal. 20, 1205–1216 (1993) 36. T. Bartsch, Topological Methods for Variational Problems with Symmetries, Lecture Notes in Mathematics, vol. 1560 (Springer, Berlin, 1993) 37. T. Bartsch, Critical point theory on partially ordered Hilbert spaces. J. Funct. Anal. 186, 117–152 (2001) 38. T. Bartsch, S. Li, Critical point theory for asymptotically quadratic functionals and applications to problems with resonance. Nonlinear Anal. 28, 419–441 (1997) 39. H. Bauschke, P. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces (Springer, New York, 2011) 40. M. Bebendorf, A note on the Poincaré inequality for convex domains. Z. Anal. Anwendungen 22, 751–756 (2003) 41. R. Bellman, Introduction to Matrix Analysis (McGraw-Hill Company, New York, 1970) 42. V. Benci, A new approach to the Morse-Conley theory and some applications. Ann. Mat. Pura Appl. 158, 231–305 (1991) 43. V. Benci, P. Rabinowitz, Critical point theorems for indefinite functionals. Invent. Math. 52, 241–273 (1979) 44. M. Berger, Nonlinearity and Functional Analysis (Academic Press, New York, 1977) 45. J. Berkovits, A note on the imbedding theorem of Browder and Ton. Proc. Am. Math. Soc. 131, 2963–2966 (2003) 46. J. Berkovits, V. Mustonen, On the topological degree for mappings of monotone type. Non- linear Anal. 10, 1373–1383 (1986) 47. J. Berkovits, V. Mustonen, Nonlinear mappings of monotone type: classification and degree theory, No. 2/1988 in Mathematics, Faculty of Science, University of Oulu, Finland, 1988 References 559 48. H. Berliocchi, J.-M. Lasry, Intégrandes normales et mesures paramétrées en calcul des vari- ations. Bull. Soc. Math. Fr. 101, 129–184 (1973) 49. C. Bessaga, A. Pelczynski, Topics in Infinite Dimensional Topology (Polish Scientific Pub- lishers, Warsaw, 1975) 50. A. Beurling, A. Livingston, A theorem on duality mappings in Banach spaces. Ark. Mat. 4, 405–411 (1962) 51. A. Bielecki, Une remarque sur la méthode de Banach-Caccioppoli-Tikhonov dans la théorie des équations différentielles ordinaires. Bull. Acad. Polon. Sci. 4, 261–264 (1956) 52. H.F. Bohnenblust, S. Karlin, On a theorem of Ville, Contributions to the Theory of Games, Annals of Mathematics Studies, No. 24 (Princeton University Press, Princeton, 1950), pp. 155–160 53. P. Bohl, Über die Bewegung eines mechnischen Systems in der Nähe einer Gleichgewicht- slage. J. Reine Angew. Math. 127, 179–276 (1904) 54. K.C. Border, Fixed Point Theorems with Applications to Economics and Game Theory (Cam- bridge University Press, Cambridge, 1985) 55. E. Borel, Leçons sur la théorie des fonctions (Gauthier-Villars, Paris, 1928) 56. K. Borsuk, Sur les rétractes. Fund. Math. 17, 152–170 (1931) 57. K. Borsuk, Sur un espace de transformations continues et ses applications topologiques. Monatsh. Math. Phys. 38, 381–386 (1931) 58. K. Borsuk, Drei Sätze uber die n-dimensionale euklidische Sphäre. Fund. Math. 20, 177–190 (1933) 59. N. Bourbaki, Elements of Mathematics–General Topology, Part 1 (Addison-Wesley Publish- ing Company, Reading, Mass, 1966) 60. A. Bressan, G. Colombo, Extensions and selections of maps with decomposable values. Studia Math. 90, 69–86 (1988) 61. G.E. Bredon, Some examples for the fixed point property. Pacific J. Math. 38, 571–575 (1971) 62. H. Brezis, Équations et inéquations non linéaires dans les espaces vectoriels en dualité. Ann. Inst. Fourier (Grenoble) 18, 115–175 (1968) 63. H. Brezis, Monotonicity methods in Hilbert spaces and some applications to nonlinear differ- ential equations, in Contributions to Nonlinear Functional Analysis, ed. by E. Zarantonello (Academic Press, New York, 1971), pp. 101–156 64. H. Brezis, Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Es- paces de Hilbert (North Holland, London, 1973) 65. H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations (Springer, New York, 2011) 66. H. Brezis, F. Browder, A general principle on ordered sets in nonlinear functional analysis. Adv. Math. 21, 355–364 (1976) 67. H. Brezis, M. Crandall, A. Pazy, Perturbations of maximal monotone sets. Commun. Pure Appl. Math. 23, 123–144 (1970) 68. H. Brezis, L. Nirenberg, Characterization of the ranges of some nonlinear operators and applications to boundary value problems. Ann. Scuola Norm. Sup. Pisa Cl. Sci. Ser. IV 5, 225–326 (1978) 69. H. Brezis, L. Nirenberg, Remarks on finding critical points. Comm. Pure Appl. Math. 44, 939–963 (1991) 70. M.S. Brodskii, D.P. Milman, On the center of a convex set. Dokl. Akad. Nauk SSSR 59, 837–840 (1948) 71. A. Brondsted, Conjugate convex functions in topological function spaces. Mat. Fys. Medd. Danske Vid. Selsk. 34, 1–26 (1964) 72. A. Brondsted, R.T. Rockafellar, On the subdifferentiability of convex functions. Proc. Am. Math. Soc. 16, 605–611 (1965) 73. L.E.J. Brouwer, On continuous one-to-one transformations of surfaces into themselves. Proc. Kon Nederl. Akad. Wetensch. 11, 788–798 (1909) 74. L.E.J. Brouwer, Beweis der Invarianz der Dimensionenzahl. Math. Ann. 70, 161–165 (1911) 75. L.E.J. Brouwer, Zur Invarianz des n-dimensionalen Gebiets. Math. Ann. 72, 55–56 (1912) 560 References 76. L.E.J. Brouwer, Über Abbildung von Mannigfaltigkeiten. Math. Ann. 71, 598 (1912) 77. F. Browder, On a generalization of the Schauder fixed point theorem. Duke Math. J. 26, 291–304 (1959) 78. F. Browder, Nonlinear elliptic boundary value problems.
Recommended publications
  • Sobolev Spaces, Theory and Applications
    Sobolev spaces, theory and applications Piotr Haj lasz1 Introduction These are the notes that I prepared for the participants of the Summer School in Mathematics in Jyv¨askyl¨a,August, 1998. I thank Pekka Koskela for his kind invitation. This is the second summer course that I delivere in Finland. Last August I delivered a similar course entitled Sobolev spaces and calculus of variations in Helsinki. The subject was similar, so it was not posible to avoid overlapping. However, the overlapping is little. I estimate it as 25%. While preparing the notes I used partially the notes that I prepared for the previous course. Moreover Lectures 9 and 10 are based on the text of my joint work with Pekka Koskela [33]. The notes probably will not cover all the material presented during the course and at the some time not all the material written here will be presented during the School. This is however, not so bad: if some of the results presented on lectures will go beyond the notes, then there will be some reasons to listen the course and at the same time if some of the results will be explained in more details in notes, then it might be worth to look at them. The notes were prepared in hurry and so there are many bugs and they are not complete. Some of the sections and theorems are unfinished. At the end of the notes I enclosed some references together with comments. This section was also prepared in hurry and so probably many of the authors who contributed to the subject were not mentioned.
    [Show full text]
  • Introduction to Sobolev Spaces
    Introduction to Sobolev Spaces Lecture Notes MM692 2018-2 Joa Weber UNICAMP December 23, 2018 Contents 1 Introduction1 1.1 Notation and conventions......................2 2 Lp-spaces5 2.1 Borel and Lebesgue measure space on Rn .............5 2.2 Definition...............................8 2.3 Basic properties............................ 11 3 Convolution 13 3.1 Convolution of functions....................... 13 3.2 Convolution of equivalence classes................. 15 3.3 Local Mollification.......................... 16 3.3.1 Locally integrable functions................. 16 3.3.2 Continuous functions..................... 17 3.4 Applications.............................. 18 4 Sobolev spaces 19 4.1 Weak derivatives of locally integrable functions.......... 19 1 4.1.1 The mother of all Sobolev spaces Lloc ........... 19 4.1.2 Examples........................... 20 4.1.3 ACL characterization.................... 21 4.1.4 Weak and partial derivatives................ 22 4.1.5 Approximation characterization............... 23 4.1.6 Bounded weakly differentiable means Lipschitz...... 24 4.1.7 Leibniz or product rule................... 24 4.1.8 Chain rule and change of coordinates............ 25 4.1.9 Equivalence classes of locally integrable functions..... 27 4.2 Definition and basic properties................... 27 4.2.1 The Sobolev spaces W k;p .................. 27 4.2.2 Difference quotient characterization of W 1;p ........ 29 k;p 4.2.3 The compact support Sobolev spaces W0 ........ 30 k;p 4.2.4 The local Sobolev spaces Wloc ............... 30 4.2.5 How the spaces relate.................... 31 4.2.6 Basic properties { products and coordinate change.... 31 i ii CONTENTS 5 Approximation and extension 33 5.1 Approximation............................ 33 5.1.1 Local approximation { any domain............. 33 5.1.2 Global approximation on bounded domains.......
    [Show full text]
  • Five Lectures on Optimal Transportation: Geometry, Regularity and Applications
    FIVE LECTURES ON OPTIMAL TRANSPORTATION: GEOMETRY, REGULARITY AND APPLICATIONS ROBERT J. MCCANN∗ AND NESTOR GUILLEN Abstract. In this series of lectures we introduce the Monge-Kantorovich problem of optimally transporting one distribution of mass onto another, where optimality is measured against a cost function c(x, y). Connections to geometry, inequalities, and partial differential equations will be discussed, focusing in particular on recent developments in the regularity theory for Monge-Amp`ere type equations. An ap- plication to microeconomics will also be described, which amounts to finding the equilibrium price distribution for a monopolist marketing a multidimensional line of products to a population of anonymous agents whose preferences are known only statistically. c 2010 by Robert J. McCann. All rights reserved. Contents Preamble 2 1. An introduction to optimal transportation 2 1.1. Monge-Kantorovich problem: transporting ore from mines to factories 2 1.2. Wasserstein distance and geometric applications 3 1.3. Brenier’s theorem and convex gradients 4 1.4. Fully-nonlinear degenerate-elliptic Monge-Amp`eretype PDE 4 1.5. Applications 5 1.6. Euclidean isoperimetric inequality 5 1.7. Kantorovich’s reformulation of Monge’s problem 6 2. Existence, uniqueness, and characterization of optimal maps 6 2.1. Linear programming duality 8 2.2. Game theory 8 2.3. Relevance to optimal transport: Kantorovich-Koopmans duality 9 2.4. Characterizing optimality by duality 9 2.5. Existence of optimal maps and uniqueness of optimal measures 10 3. Methods for obtaining regularity of optimal mappings 11 3.1. Rectifiability: differentiability almost everywhere 12 3.2. From regularity a.e.
    [Show full text]
  • Contents 1. Introduction 1 2. Cones in Vector Spaces 2 2.1. Ordered Vector Spaces 2 2.2
    ORDERED VECTOR SPACES AND ELEMENTS OF CHOQUET THEORY (A COMPENDIUM) S. COBZAS¸ Contents 1. Introduction 1 2. Cones in vector spaces 2 2.1. Ordered vector spaces 2 2.2. Ordered topological vector spaces (TVS) 7 2.3. Normal cones in TVS and in LCS 7 2.4. Normal cones in normed spaces 9 2.5. Dual pairs 9 2.6. Bases for cones 10 3. Linear operators on ordered vector spaces 11 3.1. Classes of linear operators 11 3.2. Extensions of positive operators 13 3.3. The case of linear functionals 14 3.4. Order units and the continuity of linear functionals 15 3.5. Locally order bounded TVS 15 4. Extremal structure of convex sets and elements of Choquet theory 16 4.1. Faces and extremal vectors 16 4.2. Extreme points, extreme rays and Krein-Milman's Theorem 16 4.3. Regular Borel measures and Riesz' Representation Theorem 17 4.4. Radon measures 19 4.5. Elements of Choquet theory 19 4.6. Maximal measures 21 4.7. Simplexes and uniqueness of representing measures 23 References 24 1. Introduction The aim of these notes is to present a compilation of some basic results on ordered vector spaces and positive operators and functionals acting on them. A short presentation of Choquet theory is also included. They grew up from a talk I delivered at the Seminar on Analysis and Optimization. The presentation follows mainly the books [3], [9], [19], [22], [25], and [11], [23] for the Choquet theory. Note that the first two chapters of [9] contains a thorough introduction (with full proofs) to some basics results on ordered vector spaces.
    [Show full text]
  • Arxiv:1404.0456V2 [Math.DS] 25 Aug 2015 9.3
    ON DENSITY OF ERGODIC MEASURES AND GENERIC POINTS KATRIN GELFERT AND DOMINIK KWIETNIAK Abstract. We provide conditions which guarantee that ergodic measures are dense in the simplex of invariant probability measures of a dynamical sys- tem given by a continuous map acting on a Polish space. Using them we study generic properties of invariant measures and prove that every invariant measure has a generic point. In the compact case, density of ergodic measures means that the simplex of invariant measures is either a singleton of a measure concentrated on a single periodic orbit or the Poulsen simplex. Our properties focus on the set of periodic points and we introduce two concepts: closeability with respect to a set of periodic points and linkability of a set of periodic points. Examples are provided to show that these are independent properties. They hold, for example, for systems having the periodic specification prop- erty. But they hold also for a much wider class of systems which contains, for example, irreducible Markov chains over a countable alphabet, all β-shifts, all S-gap shifts, C1-generic diffeomorphisms of a compact manifold M, and certain geodesic flows of a complete connected negatively curved manifold. Contents 1. Introduction 2 2. Preliminaries 5 3. Symbolic dynamics and examples 8 3.1. S-gap shifts 9 3.2. β-shifts 10 4. Closeability and approximability of ergodic measures 10 5. Linkability 13 6. Generic points 17 7. Proof of Theorem 1.1 20 8. Applications 22 8.1. C1-generic diffeomorphisms 22 8.2. Flows 23 9. Counterexamples 24 9.1.
    [Show full text]
  • L P and Sobolev Spaces
    NOTES ON Lp AND SOBOLEV SPACES STEVE SHKOLLER 1. Lp spaces 1.1. Definitions and basic properties. Definition 1.1. Let 0 < p < 1 and let (X; M; µ) denote a measure space. If f : X ! R is a measurable function, then we define 1 Z p p kfkLp(X) := jfj dx and kfkL1(X) := ess supx2X jf(x)j : X Note that kfkLp(X) may take the value 1. Definition 1.2. The space Lp(X) is the set p L (X) = ff : X ! R j kfkLp(X) < 1g : The space Lp(X) satisfies the following vector space properties: (1) For each α 2 R, if f 2 Lp(X) then αf 2 Lp(X); (2) If f; g 2 Lp(X), then jf + gjp ≤ 2p−1(jfjp + jgjp) ; so that f + g 2 Lp(X). (3) The triangle inequality is valid if p ≥ 1. The most interesting cases are p = 1; 2; 1, while all of the Lp arise often in nonlinear estimates. Definition 1.3. The space lp, called \little Lp", will be useful when we introduce Sobolev spaces on the torus and the Fourier series. For 1 ≤ p < 1, we set ( 1 ) p 1 X p l = fxngn=1 j jxnj < 1 : n=1 1.2. Basic inequalities. Lemma 1.4. For λ 2 (0; 1), xλ ≤ (1 − λ) + λx. Proof. Set f(x) = (1 − λ) + λx − xλ; hence, f 0(x) = λ − λxλ−1 = 0 if and only if λ(1 − xλ−1) = 0 so that x = 1 is the critical point of f. In particular, the minimum occurs at x = 1 with value f(1) = 0 ≤ (1 − λ) + λx − xλ : Lemma 1.5.
    [Show full text]
  • The Logarithmic Sobolev Inequality Along the Ricci Flow
    The Logarithmic Sobolev Inequality Along The Ricci Flow (revised version) Rugang Ye Department of Mathematics University of California, Santa Barbara July 20, 2007 1. Introduction 2. The Sobolev inequality 3. The logarithmic Sobolev inequality on a Riemannian manifold 4. The logarithmic Sobolev inequality along the Ricci flow 5. The Sobolev inequality along the Ricci flow 6. The κ-noncollapsing estimate Appendix A. The logarithmic Sobolev inequalities on the euclidean space Appendix B. The estimate of e−tH Appendix C. From the estimate for e−tH to the Sobolev inequality 1 Introduction Consider a compact manifold M of dimension n 3. Let g = g(t) be a smooth arXiv:0707.2424v4 [math.DG] 29 Aug 2007 solution of the Ricci flow ≥ ∂g = 2Ric (1.1) ∂t − on M [0, T ) for some (finite or infinite) T > 0 with a given initial metric g(0) = g . × 0 Theorem A For each σ > 0 and each t [0, T ) there holds ∈ R n σ u2 ln u2dvol σ ( u 2 + u2)dvol ln σ + A (t + )+ A (1.2) ≤ |∇ | 4 − 2 1 4 2 ZM ZM 1 for all u W 1,2(M) with u2dvol =1, where ∈ M R 4 A1 = 2 min Rg0 , ˜ 2 n − CS(M,g0) volg0 (M) n A = n ln C˜ (M,g )+ (ln n 1), 2 S 0 2 − and all geometric quantities are associated with the metric g(t) (e.g. the volume form dvol and the scalar curvature R), except the scalar curvature Rg0 , the modified Sobolev ˜ constant CS(M,g0) (see Section 2 for its definition) and the volume volg0 (M) which are those of the initial metric g0.
    [Show full text]
  • 27. Sobolev Inequalities 27.1
    ANALYSIS TOOLS WITH APPLICATIONS 493 27. Sobolev Inequalities 27.1. Morrey’s Inequality. d 1 d Notation 27.1. Let S − be the sphere of radius one centered at zero inside R . d 1 d For a set Γ S − ,x R , and r (0, ), let ⊂ ∈ ∈ ∞ Γx,r x + sω : ω Γ such that 0 s r . ≡ { ∈ ≤ ≤ } So Γx,r = x + Γ0,r where Γ0,r is a cone based on Γ, seeFigure49below. Γ Γ Figure 49. The cone Γ0,r. d 1 Notation 27.2. If Γ S − is a measurable set let Γ = σ(Γ) be the surface “area” of Γ. ⊂ | | Notation 27.3. If Ω Rd is a measurable set and f : Rd C is a measurable function let ⊂ → 1 fΩ := f(x)dx := f(x)dx. − m(Ω) Ω ZΩ Z By Theorem 8.35, r d 1 (27.1) f(y)dy = f(x + y)dy = dt t − f(x + tω) dσ(ω) Γx,r Γ0,r 0 Z Z Z ZΓ and letting f =1in this equation implies d (27.2) m(Γx,r)= Γ r /d. | | d 1 Lemma 27.4. Let Γ S − be a measurable set such that Γ > 0. For u 1 ⊂ | | ∈ C (Γx,r), 1 u(y) (27.3) u(y) u(x) dy |∇ d | 1 dy. − | − | ≤ Γ x y − ΓZx,r | |ΓZx,r | − | 494 BRUCE K. DRIVER† d 1 Proof. Write y = x + sω with ω S − , then by the fundamental theorem of calculus, ∈ s u(x + sω) u(x)= u(x + tω) ωdt − ∇ · Z0 and therefore, s u(x + sω) u(x) dσ(ω) u(x + tω) dσ(ω)dt | − | ≤ 0 Γ |∇ | ZΓ Z Z s d 1 u(x + tω) = t − dt |∇ d | 1 dσ(ω) 0 Γ x + tω x − Z Z | − | u(y) u(y) = |∇ d | 1 dy |∇ d | 1 dy, y x − ≤ x y − ΓZx,s | − | ΓZx,r | − | wherein the second equality we have used Eq.
    [Show full text]
  • Sobolev Inequalities in Familiar and Unfamiliar Settings
    Sobolev Inequalities in Familiar and Unfamiliar Settings Laurent Salo®-Coste Abstract The classical Sobolev inequalities play a key role in analysis in Euclidean spaces and in the study of solutions of partial di®erential equations. In fact, they are extremely flexible tools and are useful in many di®erent settings. This paper gives a glimpse of assortments of such applications in a variety of contexts. 1 Introduction There are few articles that have turned out to be as influential and truly im- portant as S.L. Sobolev 1938 article [93] (the American translation appeared in 1963), where he introduces his famed inequalities. It is the idea of a func- tional inequality itself that Sobolev brings to life in his paper, as well as the now so familiar notion of an a priori inequality, i.e., a functional inequality established under some strong hypothesis and that might be extended later, perhaps almost automatically, to its natural domain of de¯nition. (These ideas are also related to the theory of distributions which did not exist at the time and whose magni¯cent development by L. Schwartz was, in part, anticipated in the work of S.L. Sobolev.) The most basic and important applications of Sobolev inequalities are to the study of partial di®erential equations. Simply put, Sobolev inequalities provide some of the very basic tools in the study of the existence, regularity, and uniqueness of the solutions of all sorts of partial di®erential equations, lin- ear and nonlinear, elliptic, parabolic, and hyperbolic. I leave to others, much better quali¯ed than me, to discuss these beautiful developments.
    [Show full text]
  • Conditions for Choquet Integral Representation of the Comonotonically Additive and Monotone Functional
    CORE Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - Publisher Connector J. Math. Anal. Appl. 282 (2003) 201–211 www.elsevier.com/locate/jmaa Conditions for Choquet integral representation of the comonotonically additive and monotone functional Yasuo Narukawa a,∗ and Toshiaki Murofushi b a Toho Gakuen, 3-1-10 Naka, Kunitachi, Tokyo 186-0004, Japan b Department of Computational Intelligence and Systems Science, Tokyo Institute of Technology, 4259 Nagatuta, Midori-ku, Yokohama 226-8502, Japan Received 25 December 2000 Submitted by J. Horvath Abstract If the universal set X is not compact but locally compact, a comonotonically additive and mon- otone functional (for short c.m.) on the class of continuous functions with compact support is not represented by one Choquet integral, but represented by the difference of two Choquet integrals. The conditions for which a c.m. functional can be represented by one Choquet integral are discussed. 2003 Elsevier Science (USA). All rights reserved. Keywords: Nonadditive measure; Fuzzy measure; Cooperative game; Choquet integral; Comonotonic additivity 1. Introduction The Choquet integral with respect to a nonadditive measure is one of the nonlinear functionals defined on the class B of measurable functions on a measurable space (X, B). It was introduced by Choquet [1] in potential theory with the concept of capacity. Then, in the field of economic theory, it has been used for utility theory [17], and has been used for image processing and recognition [4,5], in the context of fuzzy measure theory [9,20]. Essential properties characterizing this functional are comonotonic additivity and monotonicity.
    [Show full text]
  • Sharp Log–Sobolev Inequalities
    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 126, Number 10, October 1998, Pages 2903{2904 S 0002-9939(98)04406-2 SHARP LOG{SOBOLEV INEQUALITIES OSCAR S. ROTHAUS (Communicated by Palle E. T. Jorgensen) Abstract. We show existence of a wide variety of Log{Sobolev inequalities in which the constant is exactly that required by the Poincar´e inequality which may be inferred from the Log{Sobolev. We are given a smooth compact Riemannian manifold M, intrinsic gradient , and volume element dµ — we assume W.L.O.G. that µ(M)=1—andasmooth∇ positive function m, giving rise to a probability measure mdµ = dm, for which we have a log–Sobolev inequality (LSI): (1) ρ f 2dm f 2 ln f 2dm f 2dm ln f 2dm: |∇ | ≥ | | | | − | | | | ZM ZM ZM ZM It is well known that ρ 2/λ,whereλis the first non–zero eigenvalue of the Schr¨odinger operator ≥ m f ∆f + f ∇ ; → ∇ · m (∆ the usual Laplacian) attached to the Dirichlet form in (1). We call the LSI sharp if ρ =2/λ. Many examples of sharp inequalities are known, the most familiar arising from M the sphere with the usual metric scaled to give M unit volume, and m =1. We will show here that for every compact homogeneous Riemannian manifold, there are a continuum of choices of m for which sharp LSI’s exist. We follow the notation and conclusions of [1], which we now briefly review. For every M as described initially there is a least constant ρ0(M), the hypercontractive constant for M, such that (2) ρ (M) f 2dµ f 2 ln f 2dµ f 2dµ ln f 2dµ.
    [Show full text]
  • Sobolev's Inequality Under a Curvature-Dimension Condition
    Sobolev’s inequality under a curvature-dimension condition Louis Dupaigne, Ivan Gentil, Simon Zugmeyer To cite this version: Louis Dupaigne, Ivan Gentil, Simon Zugmeyer. Sobolev’s inequality under a curvature-dimension condition. 2021. hal-03006155v3 HAL Id: hal-03006155 https://hal.archives-ouvertes.fr/hal-03006155v3 Preprint submitted on 19 Jan 2021 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Sobolev's inequality under a curvature-dimension condition Louis Dupaigne, Ivan Gentil, Simon Zugmeyer January 19, 2021 R´esum´e Dans cette note, nous proposons une nouvelle preuve de l'in´egalit´ede Sobolev sur les vari´et´es`a courbure de Ricci minor´eepar une constante positive. Le r´esultatavait ´et´eobtenu en 1983 par Ilias. Nous pr´esentons une preuve tr`escourte de ce th´eor`eme, dressons l'´etatde l'art pour cette fameuse in´egalit´eet expliquons en quoi notre m´ethode, qui repose sur un flot de gradient, est simple et robuste. En particulier, nous ´elucidonsles calculs utilis´esdans des travaux pr´ec´edents, `acommencer par un c´el`ebrearticle de Bidaut-V´eron et V´eronpubli´een 1991. Abstract In this note we present a new proof of Sobolev's inequality under a uniform lower bound of the Ricci curvature.
    [Show full text]