Sobolev ve logaritmik sobolev eşitsizlikleri
Sobolev and logarithmic sobolev inequalities
- Tez No: 14131
- Danışmanlar: PROF.DR. CEVDET KOÇAK
- Tez Türü: Doktora
- Konular: Matematik, Mathematics
- Anahtar Kelimeler: Sobolev eşitsizlikleri, Sobolev inequalities
- Yıl: 1990
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Belirtilmemiş.
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: Belirtilmemiş.
Özet
ÜZET Bu çalışmada, klasik Sobolev ve Gross logaritmik Sobolev eşitsizlikleri ve bu eşitsizliklerde ortaya çıkan en iyi sabitler incelenmiştir. Gross logaritmik Sobolev eşitsizliği bir teoremle genelleştirilmiş ve en genel uzaylar için en iyi sabitler konjektürü olarak bilinen bir konjektür için bazı yaklaşımlar önerilmiştir. Tez altı bölümden oluşmaktadır. Giriş bölümünde konu ile ilgili bazı önemli çalışmalar bir literatür taraması olarak kısaca tanıtılmıştır. İkinci bölümde Sobolev uzayları tanımlanmış ve Sobolev eşitsiz likleri ile yakından ilgili olan Sobolev daldırma teoremleri verilmiştir. Üçüncü bölümde klasik Sobolev eşitsizlikleri ve bu eşitsizliklerde ortaya çıkan en iyi sabitler incelenmiştir. Dördüncü bölümde Gross logaritmik Sobolev eşitsizlikleri ele alınmış, Nelson hiper kontraktif eşitsizliğine denkliği gösterilmiştir. Beşinci bölümde Adams ve Clarke tarafından geliştirilen ispat tekniği uygulanarak Gross logaritmik Sobolev eşitsizliği genelleş tirilmiş ve bu eşitsizliğe ilişkin en iyi sabit elde edilmiştir. Son bölümde ise en genel logaritmik Sobolev eşitsizliklerinde en iyi sabitler konjektürünün ispatı için literatürdeki çalışmalar gözönünde bulundurularak bazı öneriler verilmiştir.
Özet (Çeviri)
SOBOLEV AND LOGARITHMIC SOBOLF.V INEQUALITIES SUMMARY This thesis is concerned with standard Sobolev inequalities and a special class of inequalities called logarithmic Sobolev inequali ties which play a very important role in quantum field theory. This inequalities could be candidates for inclusion in some future successor to the classical work of Littlewood-Hardy-Polya [ 21 ;].. Another problem on which we focus our attention will.be about' best constants arising in these inequalities. In the case of logarithmic Sobolev inequality in R^ we are led to examine an open problem of proving uniqueness of solutions of some semilinear partial differen tial equations in Rn. We call any inequality expressing the boundedness of a map from L to L^ a Sobolev type inequality or more precisely, INIlV),c|1vu||lV) (1) where q= np/ n-p, l<p<n and C is a constant independent of u. This inequality, also known as a Sobolev embedding theorem, was first proved by Sobolev in 1939 by using a representation formula and applying a theorem of Hardy-Little;y»od concerning fractional integrals. The publication of Sobolev inequality has attracted a great deal of attention among mathematicians working in Partial differential equations and in the calculus of variations. After this there followed a remarkable number of papers including applications and development of these inequalities.A drawback of the Sobolev method is that it does not give the exact value of the best possible constant C or any estimates for C. To this end several authors have made attempts to get best constants. For instance, Rosen [39] obtained the best constant for the special case n=3, p-2, q-6 which leads to a sharp form of inequality (1). The other special case when p is equal to unity is known as Federer-Fleming-Rishel form of the Sobolev inequality which is sharp and can be written as follows:“ 1-J- {r(l- ”-)}1/“ { / |u!n_1 dx > n * -X / |vu| dx (2) The constant appearing in (2) is the best possible. For the general case l<p<n, q= ^£- best constant in Sobolev inequality (1) has been given at approximately same time independently by Talenti [32] and Aubin [4]. In the standard Sobolev inequality the equation q» -2§- implies q.*? p as the”dimension n goes to infinity. So there is a loss of information in the usual form of Sobolev inequality as the dimension gets bigger. The problem of finding infinite dimensional versions of classical Sobolev inequalities is first discussed by Gross [19]. These types of inequalities are nowadays called Gross's logarithmic Sobolev inequalities. In his paper Gross also demonstrated the equivalence of logarithmic Sobolev inequality and Nelson's hype rcon tractive inequality [26] arising in constructive quantum field theory. VIGross's logarithmic inequality asserts that the inequality /|f|2 In |f| du(.x) « / [Vf |2 du'(x) ? | |f | \\ 1 n | |f | \2 (3) Rn Rn holds for all continuously differentiate functions f. Herein y denotes Gauss measure on R and ||f|L denotes L (y) norm of f. The dimension dependent analogue of (3), called the weak form of the logarithmic Sobolev inequality can also be obtained through the use of Nirenberg's derivation of Sobolev's inequality and Holder's inequality. Although this form is not suitable for passage to infinite dimensions, it designates a connection between the classical Sobolev inequality and the Gaussian Sobolev inequality. The inequality (3) has been derived by using rather lengthy probabilistic methods. Three years after Gross's paper, Adams and Clarke [2] gave a very simple proof of Gross's logarithmic Sobolev inequality using only elementary calculus. The aim of this study will mainly be centered on the possibility of whether Gross's logarithmic Sobolev inequalities might be further extented on the basis of Adams and Clarke method of proof. In this context it will be sbdwn that this, to some extent, is possible. Sobolev and logarithmic Sobolev inequalities have proved to be very useful for the study of boundary value problems of mathematical physics, particularly in qualitative studies of very complicated problems that are generally intractable for analytic solutions. For example when one wishes to establish operator estimates this approach usually leads to Sobolev type inequalities. This thesis is organized into five chapters. In the first chapter, a literature survey on the subject has been presented. In the second chapter we summarize a number of preliminary facts, some definitions, important theorems (Embedding theorems) and related results needed in the sequel. VIIThe third chapter deals with standard Sobolev inequalities and best constants arising in these inequalities. In a typical Sobolev inequality best constant is defined by an investigation of the following infimum: UVulL../tvujpdx. np S= inf- - 1 = - : - :, q =. (4) lu'Iq (/ |u[q dx)p/q n-p u20 uewJ»p(Q) This problem has been studied by several authors. The main features are the following: a) S does not depend on Q, b) S is never achieved if Q is bounded c) if a=Rn and W 'p(0) is replaced by functions in Lq(Q) such that VueLp(fl), then S is achieved. A standard method for the investigation of the above problem is one due to Schwarz, known as Schwarz symmetrization argument. In this approach the problem is first reduced to the class of functions which are positive, spherically symmetric and monotonic decreasing. Then it is shown that the infimum is actually attained by some minimizing function in the class and that there is a unique solution, that is, unique minimizer to the Euler equation for the variational problem (4). This then gives the desired minimum. Aubin's and Talenti's papers follow the above pattern for investigating the variational problem (4) which leads to best constant for the Sobolev inequality. In the fourth chapter, logarithmic Sobolev inequalities have been introduced. Also the equivalence of Gross's logarithmic Sobolev and Nelson's hype rcon tractive inequalities is demonstrated and the weak form of Gross's logarithmic Sobolev inequality in which the VIIIcoefficients depend on the space dimension is obtained from Nirenberg's form of Sobolev inequality and Holder's inequality. The Logarithmic Sobolev inequality of Gross is, in its Lebesgue measure form, the p=2 case of the general inequality. P'1 / |V<j>|P dx.»/ |*|P lnlol dx- ||ç^i|Pjp ln||<j>|lnjP Rn Rn (5) ? k ||<M|P n,p ' |Y| 'n,p where ||. | |n^p denotes the norm on the Banach space Lp(Rn, dx.) defined by l|u||p. / |u|p dx 11“n,p ' ' Rn with dx Lebesgue messure in R For n= 1 and p=2, Adams and Clarke gave a very elegant proof of this inequality using elementary calculus and one dimensional calculus of variations. By converting to the Lebesgue form and using. induction on n, the proof for general n then follows. In the fifth chapter we present a proof of this inequality for n=l and general p>l by adapting the method of Adams and Clarke. Unfortunately, the induction method fails to generalize when p^2. We have the following theorem: Let u: R -*? C be a continuously differentiable function. Then for p>l p-Vlu'lP dx * /|u|p ln|u| dx - 1{U|1P ln ||u|| *k ||u||p (6) R R IXwhere k = -U 1+ln ( (p-Dİ/P-L-rc-i^))] p p p p is the best possible constant. The equality holds in (6) if u has the form u(x) = xexp {-_l_|x+ujq } where p and q are pairs of conjugate indices, i.e. p + q » 1 9 2 1/? and A and y are positive constants and |x| * (x, + + y£ ). The last chapter concentrates on the investigation of the tricky uniqueness problem of the positive, radial solutions of the equation: A u ¥ up_1 (plnu + 1) r 0 (7) r A u being the p- Laplacian of u defined by V.div(|Vu|P-2vu) and u is a symmetric rearrangement. This is needed in order to complete the proof of conjectured best constants in logarithmic Sobolev inequalities. At the moment this remains an open problem. Even with p=2 in which case Au + u(2 Inu + 1) = 0, A being the usual Laplace-operator, it is still very hard to prove the uniqueness of solutions which would show the existence of the unique mininizer. The p-2 case has recently been worked out by several researchers [ 6,7,8,17,18,23,26, 28 j. Relying on a paper by Kttong[22] who proves the uniqueness of positive, radial solutions of Au ? u^ - u r C (p>l) it seems quite possible that the method of Kwong's proof can be applied to ensure the uniqueness for our particular problem replacing polynomial function up - u (p>l) by the logarithmic function u(2 Inu + 1). In recent years a remarkable nura&er of -importantpapers [ 12 ] have been published concerning the existence and.nonexistence of solutions of p-Laplace equations. But the general uniqueness problem in A u +? u^”(plnu + 1) ; 0 is still unsolved. This would lead us to the best constants in (5), consequently we would get the sharp form of the logarithmic Sobolev inequality in l-P (Rn, dx) with dx Lebesgue measure in Rn. XI
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