DOCSLIB.ORG

Partial Differential Equations II Spring 2008 Prof. D. Slepcev∗

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Partial Differential Equations II Spring 2008 Prof. D. Slepcev∗ Partial Differential Equations II Spring 2008 Prof. D. Slepˇcev∗ Chris Almost† Disclaimer: These notes are not the official course notes for this class. These notes have been transcribed under classroom conditions and as a result accuracy (of the transcription) and correctness (of the mathematics) cannot be guaranteed. ∗[email protected] †[email protected] 1 Contents Contents2 1 Sobolev Spaces3 1.1 Introduction.................................. 3 1.2 Weak derivatives ............................... 3 1.3 Sobolev spaces ................................ 5 1.4 Approximation by smooth functions ................... 6 1.5 Extension.................................... 10 1.6 Traces...................................... 11 1.7 Sobolev inequalities ............................. 13 1.8 Compactness.................................. 17 1.9 Poincaré inequalities............................. 19 1.10 Difference quotients ............................. 20 1 1.11 The dual space H− ............................. 22 2 Elliptic PDE 23 2.1 Introduction.................................. 23 2.2 Existence of solutions and the Fredholm alternative.......... 24 2.3 Regularity of solutions............................ 32 2.4 Maximum principles............................. 36 2.5 Eigenvalues and eigenfunctions...................... 39 2.6 Non-symmetric elliptic operators ..................... 41 3 Parabolic PDE 42 3.1 Bochner Integral ............................... 42 3.2 Spaces involving time ............................ 43 3.3 Parabolic equations.............................. 46 3.4 Existence via Galerkin Approximation .................. 47 3.5 Maximum principles............................. 48 Index 51 2 Sobolev Spaces 3 1 Sobolev Spaces 1.1 Introduction This course is concerned with modern methods in the theory of PDE, in particular with linear parabolic and elliptic equations. It will be a technical course more than an applied course, and we will be learning the language required to understand research in PDE. For the purposes of illustration, consider the boundary value problem ¨ ∆u = f in U − u = 0 on @ U where U is connected and U is compact. Multiply by a test function ' Cc1(U) and integrate both sides 2 Z Z ∆u'd x = f 'd x. U − U Integrate by parts to get, Z Z u 'd x = f 'd x, U r · r U since ' @ U = 0. Define a bilinear form on some space H, with the property that u = 0 onj @ U for u H, by 2 Z u1, u2 = u1 u2d x. h i U r · r R For u H, the norm is u u 2d x, and we hope that H is a Hilbert space = U 2 R k k jr j and ' f 'd x is a bounded linear functional. If these hopes hold true then U by the Riesz7! representation theorem there is a unique u H such that 2 Z Z u 'd x = u, ' = f 'd x. U r · r h i U Sounds nice, but does it all work? 1.2 Weak derivatives 1 1 1.2.1 Definition. Let u Lloc(U). The function v Lloc(U) is a weak partial derivative of u in the ith coordinate2 if 2 Z Z u' d x 1 v'd x xi = ( ) U − U 4 PDE II for all ' C U . We write v u . c1( ) = xi Analogously,2 v is the αth weak partial derivative for a multiindex α if Z Z α α uD 'd x = ( 1)j j v'd x U − U α for all ' Cc1(U), and we write v = D u. 2 Of course, to define such notation we need the following. 1.2.2 Proposition. Weak derivatives are unique if they exist. th ROOF P : Suppose that w1 and w2 are α derivatives of u. Then, for all ' Cc1(U), 2 Z Z α α ( 1)j j w1'd x = ( 1)j j w2'd x. − U − U R So w w 'd x 0 for all ' C U , and it follows that w w a.e. U ( 1 2) = c1( ) 1 = 2 (Indeed,− let η be the standard2 mollifier, and η be the rescaled mollifier with R " support B 0, " . Then w y : w x η y x d x 0 for all " and all y, and ( ) "( ) = U ( ) "( ) = − this completes the argument since w" w a.e.) ! 1.2.3 Examples. (i) Let u(x) = x for x R. Then ux (x) = 1 if x > 0 and = 1 if x < 0. j j 2 − Indeed, for ' Cc1(R), 2 Z Z 0 Z 1 ux 'd x = ux 'd x ux 'd x − R − − 0 Z 0 −∞ Z 1 = u'x d x u(0)'(0) + u'x d x + u(0)'(0) − 0 Z−∞ = x 'x d x R j j We will see later that the weak derivative corresponds to the usual general- ization of derivative for absolutely continuous functions. (ii) Let u(x) = 1(0, ). Then ux (x) = 0 is the only candidate for the derivative (why?), but 1 Z Z 1 u'x d x = 'x d x = '(0) R 0 − which is not necessarily zero. There is a more general notion (that of a “dis- tribution”) that would give this function a derivative, but for our purposes this function does not have a weak derivative. Sobolev spaces 5 1.3 Sobolev spaces k,p 1.3.1 Definition. For 1 p < , let W (U) be the collection of functions u 1 ≤ 1 α p 2 Lloc(U) such that for all α with α k, D u exists and is in L (U). For u k,p j j ≤ 2 W (U), 1 1 Z p p X α p X α p k,p u W (U) := D u d x = D u p k k α k U j j α k k k j |≤ j |≤ For p = , take instead the essential supremum. 1 k k,2 When p = 2 we may write H (U) := W (U). W k,p U W k,p U C U ( ) 1.3.2 Definition. 0 ( ) := c1( ) . 1.3.3 Example. Let U B 0, 1 n, α > 0, and u x 1 . Note that = ( ) R ( ) = x α ⊆ j j " Z 1 Z 1 " n 1 ˜ n α d x = C r − dr = C r − , α α 0 B 0," x 0 r ( ) j j 1 so u is in Lloc(U) if and only if α < n. 1,p For which p and n is u in W (U)? u is a.e. differentiable, so 1 xi αxi u α , xi = α 1 = α 2 − x + · x − x + j j j j j j and Du α . As above, u L1 U if and only if α < n 1. To check that = x α+1 xi loc( ) j j j j 2 − this is truly the weak derivative, let A" := B(0, 1) B(0, "). By integration by parts, n Z Z Z u' d x u 'd x u'ν dS. xi = xi + i A" − A" @ B(0,") As " 0, A" B(0, 1) and ! ! Z 1 n 1 α 'νi dS C" ' L 0. "α − − 1 @ B(0,") ≤ k k ! R Finally, Du p d x < only if U j j 1 1 Z 1 Z 1 n 1 (α+1)p n (α+1)p > d x = C r − − d x = C r − , (α+1)p 0 1 U x 0 j j which happens when α 1 p < n, or α < n 1. ( + ) p − k,p 1.3.4 Proposition. Let u, v W (U) and let α k. 2 j j ≤ 6 PDE II α k α ,p (i) D u W −| j (U) 2 β α α β α β (ii) D (D u) = D (D u) = D + u, for each β such that α + β k. j j j j ≤ (iii) Dα is a linear operator. k,p (iv) If V U is open then u V W (V ). ⊆ j 2 k,p (v) If ξ Cc1(U) then ξu W (U) and 2 2 X α Dα ξu Dβ ξDα β u ( ) = β − β α ≤ where α : α! . β = β!(α β)! − k,p 1.3.5 Theorem. W (U) is a Banach space. PROOF: If u = 0 then u p u = 0, so u = 0 a.e. Clearly the norm is homogeneous,k k so we are leftk k to≤ check k k the triangle inequality. For p < , 1 1 p X α α p u + v = D u + D v p k k α k k k ≤ 1 p X α α p D u p + D v p ≤ α k k k k k ≤ 1 1 p p X α p X α p D u p + D v p ≤ α k k k α k k k ≤ ≤ = u + v k k k k by Minkovski’s inequality in both cases. The case p = is clear. k,p 1 Let um be a Cauchy sequence in W (U). Since the Sobolev norm dominates p α p the L norm,f g all of the sequences um and D um are Cauchy in L , and hence p p convergent in L . Let u and uα bef theg limitsf of theseg sequences in L . Now Z Z α α uD 'd x lim um D 'd x = m U !1 U Z Z α α α lim 1 'D umd x 1 uα'd x = m ( )j j = ( )j j !1 − U − U α k,p Therefore uα = D u, and the sequence converges in W (U). 1.4 Approximation by smooth functions Mollifiers n A mollifier is a function η Cc1(R ) such that 2 Approximation by smooth functions 7 R (i) n η 1; R = (ii) η 0; ≥ (iii) η is radially symmetric; and (iv) supp(η) B(0, 1). ⊆ For any " > 0 define η x 1 η x . Then η is also a mollifier for " < 1. For "( ) = "n ( " ) " 1 n f Lloc(R ), define 2 Z Z " f (x) := η" f (x) = η"(x y)f (y)d y = η"(y)f (x y)d y.
Load more

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]
  • Notes on Partial Differential Equations John K. Hunter
    Notes on Partial Differential Equations John K. Hunter Department of Mathematics, University of California at Davis Contents Chapter 1. Preliminaries 1 1.1. Euclidean space 1 1.2. Spaces of continuous functions 1 1.3. H¨olderspaces 2 1.4. Lp spaces 3 1.5. Compactness 6 1.6. Averages 7 1.7. Convolutions 7 1.8. Derivatives and multi-index notation 8 1.9. Mollifiers 10 1.10. Boundaries of open sets 12 1.11. Change of variables 16 1.12. Divergence theorem 16 Chapter 2. Laplace's equation 19 2.1. Mean value theorem 20 2.2. Derivative estimates and analyticity 23 2.3. Maximum principle 26 2.4. Harnack's inequality 31 2.5. Green's identities 32 2.6. Fundamental solution 33 2.7. The Newtonian potential 34 2.8. Singular integral operators 43 Chapter 3. Sobolev spaces 47 3.1. Weak derivatives 47 3.2. Examples 47 3.3. Distributions 50 3.4. Properties of weak derivatives 53 3.5. Sobolev spaces 56 3.6. Approximation of Sobolev functions 57 3.7. Sobolev embedding: p < n 57 3.8. Sobolev embedding: p > n 66 3.9. Boundary values of Sobolev functions 69 3.10. Compactness results 71 3.11. Sobolev functions on Ω ⊂ Rn 73 3.A. Lipschitz functions 75 3.B. Absolutely continuous functions 76 3.C. Functions of bounded variation 78 3.D. Borel measures on R 80 v vi CONTENTS 3.E. Radon measures on R 82 3.F. Lebesgue-Stieltjes measures 83 3.G. Integration 84 3.H. Summary 86 Chapter 4.
    [Show full text]
  • Lecture Notes for MATH 592A
    Lecture Notes for MATH 592A Vladislav Panferov February 25, 2008 1. Introduction As a motivation for developing the theory let’s consider the following boundary-value problem, u′′ + u = f(x) x Ω=(0, 1) − ∈ u(0) = 0, u(1) = 0, where f is a given (smooth) function. We know that the solution is unique, sat- isfies the stability estimate following from the maximum principle, and it can be expressed explicitly through Green’s function. However, there is another way “to say something about the solution”, quite independent of what we’ve done before. Let’s multuply the differential equation by u and integrate by parts. We get x=1 1 1 u′ u + (u′ 2 + u2) dx = f u dx. − x=0 0 0 h i Z Z The boundary terms vanish because of the boundary conditions. We introduce the following notations 1 1 u 2 = u2 dx (thenorm), and (f,u)= f udx (the inner product). k k Z0 Z0 Then the integral identity above can be written in the short form as u′ 2 + u 2 =(f,u). k k k k Now we notice the following inequality (f,u) 6 f u . (Cauchy-Schwarz) | | k kk k This may be familiar from linear algebra. For a quick proof notice that f + λu 2 = f 2 +2λ(f,u)+ λ2 u 2 0 k k k k k k ≥ 1 for any λ. This expression is a quadratic function in λ which has a minimum for λ = (f,u)/ u 2. Using this value of λ and rearranging the terms we get − k k (f,u)2 6 f 2 u 2.
    [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]
  • Advanced Partial Differential Equations Prof. Dr. Thomas
    Advanced Partial Differential Equations Prof. Dr. Thomas Sørensen summer term 2015 Marcel Schaub July 2, 2015 1 Contents 0 Recall PDE 1 & Motivation 3 0.1 Recall PDE 1 . .3 1 Weak derivatives and Sobolev spaces 7 1.1 Sobolev spaces . .8 1.2 Approximation by smooth functions . 11 1.3 Extension of Sobolev functions . 13 1.4 Traces . 15 1.5 Sobolev inequalities . 17 2 Linear 2nd order elliptic PDE 25 2.1 Linear 2nd order elliptic partial differential operators . 25 2.2 Weak solutions . 26 2.3 Existence via Lax-Milgram . 28 2.4 Inhomogeneous bounday value problems . 35 2.5 The space H−1(U) ................................ 36 2.6 Regularity of weak solutions . 39 A Tutorials 58 A.1 Tutorial 1: Review of Integration . 58 A.2 Tutorial 2 . 59 A.3 Tutorial 3: Norms . 61 A.4 Tutorial 4 . 62 A.5 Tutorial 6 (Sheet 7) . 65 A.6 Tutorial 7 . 65 A.7 Tutorial 9 . 67 A.8 Tutorium 11 . 67 B Solutions of the problem sheets 70 B.1 Solution to Sheet 1 . 70 B.2 Solution to Sheet 2 . 71 B.3 Solution to Problem Sheet 3 . 73 B.4 Solution to Problem Sheet 4 . 76 B.5 Solution to Exercise Sheet 5 . 77 B.6 Solution to Exercise Sheet 7 . 81 B.7 Solution to problem sheet 8 . 84 B.8 Solution to Exercise Sheet 9 . 87 2 0 Recall PDE 1 & Motivation 0.1 Recall PDE 1 We mainly studied linear 2nd order equations – specifically, elliptic, parabolic and hyper- bolic equations. Concretely: • The Laplace equation ∆u = 0 (elliptic) • The Poisson equation −∆u = f (elliptic) • The Heat equation ut − ∆u = 0, ut − ∆u = f (parabolic) • The Wave equation utt − ∆u = 0, utt − ∆u = f (hyperbolic) We studied (“main motivation; goal”) well-posedness (à la Hadamard) 1.
    [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]
  • 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]
  • Delta Functions and Distributions
    When functions have no value(s): Delta functions and distributions Steven G. Johnson, MIT course 18.303 notes Created October 2010, updated March 8, 2017. Abstract x = 0. That is, one would like the function δ(x) = 0 for all x 6= 0, but with R δ(x)dx = 1 for any in- These notes give a brief introduction to the mo- tegration region that includes x = 0; this concept tivations, concepts, and properties of distributions, is called a “Dirac delta function” or simply a “delta which generalize the notion of functions f(x) to al- function.” δ(x) is usually the simplest right-hand- low derivatives of discontinuities, “delta” functions, side for which to solve differential equations, yielding and other nice things. This generalization is in- a Green’s function. It is also the simplest way to creasingly important the more you work with linear consider physical effects that are concentrated within PDEs, as we do in 18.303. For example, Green’s func- very small volumes or times, for which you don’t ac- tions are extremely cumbersome if one does not al- tually want to worry about the microscopic details low delta functions. Moreover, solving PDEs with in this volume—for example, think of the concepts of functions that are not classically differentiable is of a “point charge,” a “point mass,” a force plucking a great practical importance (e.g. a plucked string with string at “one point,” a “kick” that “suddenly” imparts a triangle shape is not twice differentiable, making some momentum to an object, and so on.
    [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]
  • 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]