COMPOSITION IN FRACTIONAL SOBOLEV SPACES
|
|
- Melvin Bradley
- 6 years ago
- Views:
Transcription
1 COMPOSITION IN FRACTIONAL SOBOLEV SPACES HAIM BREZIS (1)() AND PETRU MIRONESCU (3) 1. Introduction A classical result about composition in Sobolev spaces asserts tat if u W k,p (Ω) L (Ω) and Φ C k (R), ten Φ u W k,p (Ω). Here Ω denotes a smoot bounded domain in R N, k 1 is an integer and 1 p <. Tis result was first proved in [13] wit te elp of te Gagliardo-Nirenberg inequality [14]. In particular if u W k,p (Ω) wit kp > N and Φ C k (R) ten Φ u W k,p since W k,p L by te Sobolev embedding teorem. Wen kp = N te situation is more delicate since W k,p is not contained in L. However te following result still olds (see [],[3]) Teorem 1. Assume u W k,p (Ω) were k 1 is an integer, 1 p <, and (1) kp = N. Let Φ C k (R) wit () D j Φ L (R) j k. Ten Φ u W k,p (Ω) Te proof is based on te following Lemma 1. Assume u W k,p (Ω) W 1,kp (Ω) were k 1 is an integer and 1 p <. Assume Φ C k (R) satisfies (). Ten Φ u W k,p (Ω). Acknowledgment: Te first autor (H.B.) is partially supported by a European Grant ERB FMRX CT He is also a member of te Institut Universitaire de France. Part of tis work was done wen te second autor (P.M.) was visiting Rutgers University; e tanks te Matematics Department for its invitation and ospitality. We tank S. Klainerman for useful discussions. 1 Typeset by AMS-TEX
2 HAIM BREZIS (1)() AND PETRU MIRONESCU (3) Proof of Teorem 1. Since u W k,p we ave Du W k 1,p L q by te Sobolev embedding wit 1 q = 1 p k 1 N. Applying assumption (1) we find q = N = kp and tus u W 1,kp. Lemma 1 tat Φ u W k,p. We deduce from Proof of Lemma 1. Note tat if u W k,p L wit k 1 integer and 1 p < ten u W 1,kp by te Gagliardo - Nirenberg inequality [14]. Tus, Lemma 1 is a generalization of te standard result about composition. In fact, it is proved exactly in te same way as in te standard case (wen u W k,p L ). Wen k = te conclusion is trivial. Assume, for example tat, k = 3, ten W 3,p W 1,3p W,3p/ by te Gagliardo - Nirenberg inequality. Ten D 3 (Φ u) = Φ (u)d 3 u + 3Φ (u)d udu + Φ (u)(du) 3, and tus Φ u W 3,p since D u p Du p ( D u 3p/ ) /3( Du 3p) 1/3 A simular argument olds for any k 4. C u p/ W 3,p u 3p/ W 1,3p. Starting in te mid-60 s a number of autors considered composition in various classes of Sobolev spaces W s,p, were s > 0 is a real number and 1 p <. Te most commonly used are te Bessel potential spaces L s,p (R N ) = {f = G s g; g L p (R N )} were Ĝs = (1 + ξ ) s/ and te Besov spaces Bp s,p (R N ) (wo s definition is recalled below wen s is not an integer). It is well-known (see e.g. [1],[19] and [0]) tat if k is an integer, L k,p coincides wit te standard Sobolev space W k,p ; also if p =, te Bessel potential spaces L s, and te Besov spaces B s, coincide for every s non-integer and tey are usually denoted by H s. Wen p te spaces L s,p and Bp s,p are distinct. Te first result about composition in fractional Sobolev spaces seems to be due to Mizoata [1] for H s,s > N/. In 1970 Peetre [15] considered Bp s,p L using interpolation
3 COMPOSITION IN FRACTIONAL SOBOLEV SPACES 3 tecniques; a very simple direct argument for te same class, B s,p p L, was given by M. Escobedo [10] (see te proof of Lemma below). Starting in 1980 tecniques of dyadic analysis and Littlewood-Paley decomposition à la Bony [5] were introduced. For example, Y. Meyer [11] considered composition in L s,p for sp > N; see also [16],[4],[9] for H s wit s > N/ or for H s L, any s > 0. We refer to [17],[6],[7],[18] and teir bibliograpies for oter directions of researc concerning composition in Sobolev spaces. In wat follow we denote by W s,p (Ω) te restriction of Bp s,p (R N ) to Ω wen s is not an integer. Our main result is te following Teorem. Assume u W s,p (Ω) were s > 1 is a real number, 1 < p <, and (3) sp = N. Let Φ C k (R), were k = [s] + 1, be suc tat (4) D j Φ L (R) j k. Ten Φ u W s,p (Ω). Te proof of Teorem relies on a variant of Lemma 1 for fractional Sobolev spaces. Lemma. Let u W s,p (Ω), were s > 1 is a real number and 1 < p <. Assume, in addition, tat u W σ,q for some σ (0, 1) wit (5) q = sp/σ. Let Φ C k (R), were k = [s] + 1, be suc tat (4) olds. Ten Φ u W s,p Proof of Teorem. By te Sobolev embedding teorem we ave W s,p W r,q wit r < s and In view of assumption (3) we find 1 q = 1 (s r) p N. q = N/r.
4 4 HAIM BREZIS (1)() AND PETRU MIRONESCU (3) In particular, u W σ,q for all σ (0, 1) wit q = N σ = sp σ. Tus we may apply Lemma and conclude tat Φ u W s,p. Remark 1. Teorem is known to be true wen te Sobolev spaces W s,p are replaced by te Bessel potential spaces L s,p wit sp = N; see D. Adams and M. Frazier [3]. Even toug te two results are closely related it does not seem possible to deduce one from te oter. Teir argument relies on a variant of Lemma for Bessel potential spaces: Let u L s,p L 1,sp were s > 1 is a real number and 1 < p <. Let Φ be as in Lemma. Ten Φ u L s,p. Remark. Te assumption in Lemma, u W s,p W σ,q, wit q = sp/σ for some σ (0, 1), is weaker tan te assumption u W s,p L but it is stronger tan te assumption u W 1,sp ; tis is a consequence of Gagliardo - Nirenberg type inequalities (see e.g. te proof of Lemma D.1 in te Appendix D of [8]). It is terefore natural to raise te following: Open Problem. Is te conclusion of Lemma valid if one assumes only u W s,p W 1,sp were s > 1 is a (non-integer) real number? Before giving te proof of Lemma we recall some properties of W s,p wen s is not an integer. Wen 0 < σ < 1 and 1 < p < te standard definition of W σ,p is W σ,p (Ω) = {f L p (Ω); f(x) f(y) p dxdy < }. x y N+σp If s > 1 is not an integer write s = [s] + σ were [s] denotes te integer part of s and 0 < σ < 1. Ten W s,p (Ω) = {f W [s],p (Ω), D α f W σ,p for α = [s]}. Tere is a very useful caracterization of W s,p in terms of finite differences (see Triebel [0], p.110). Here it is more convenient to work wit functions defined on all of R N and to consider teir restrictions to Ω. Set (δ u)(x) = u(x + ) u(x), R N,
5 COMPOSITION IN FRACTIONAL SOBOLEV SPACES 5 so tat (δ u)(x) = u(x + ) u(x + ) + u(x), etc... Given s > 0 not integer, fix any integer M > s. Ten W s,p = {f L p ; M f(x) p dxd < }. Proof of Lemma. It suffices to consider te case were s is not an integer. For simplicity we treat just te case were 1 < s <. Te same argument extends to general s >, s noninteger, using te same type of computations as in Escobedo [10]. Te key observation is tat δ (Φ u) can be expressed in terms of δ u and δ u. Tis is te purpose of our next computation. Set X = u(x + ) Y = u(x + ) Z = u(x). Since Φ L (R) we ave (6) Φ(X) Φ(Y ) = Φ (Y )(X Y ) + 0( X Y ) and since Φ L (R) we also ave (7) Φ(X) Φ(Y ) = Φ (Y )(X Y ) + 0( X Y ). Combining (6) and (7) we find Φ(X) Φ(Y ) = Φ (Y )(X Y ) + 0( X Y a ) for any 1 a ( we will coose a specific value of a later) Similarly Φ(Z) Φ(Y ) = Φ (Y )(Z Y ) + 0( Z Y a ) Since δ (Φ u)(x) = (Φ(X) Φ(Y )) + (Φ(Z) Φ(Y )), one finds (8) δ (Φ u)(x) C( δ u(x) + δ u(x + ) a + δ u(x) a ).
6 6 HAIM BREZIS (1)() AND PETRU MIRONESCU (3) Tis yields (9) (Φ u)(x) p dxd C u(x) p u(x) ap dxd + C dxd. Te first integral on te rigt-and side of (9) is finite since u W s,p. To andle te second integral we argue as follows. From te assumption u W s,p W σ,q wit σ (0, 1) and q given by (5) we know tat (10) u(x) p dxd < and u(x) q From (10) and Hölder s inequality we derive tat (11) u(x) r dxd < for all r [p, q], i.e., u W τ,r wit τ = sp/r. We now coose a = min{, s/σ}, so tat a [1, ] and r = ap [p, q]. It follows tat u(x) ap wic is te desired in equality. dxd <, dxd <. Remark 3. Tere could be anoter natural proof of Teorem by induction on [s]. One migt attempt to prove tat D(Φ u) = Φ (u)du W s 1,p. Note tat u W (s 1),N/(s 1) and tus (by induction) we would ave Φ (u) W (s 1),N/(s 1). On te oter and Du W s 1,p. In order to conclude we need a lemma about products, but we are not aware of any suc tool. Remark 4. Wen s (or equivalently p) is a rational number, and Φ C wit D j Φ L j, tere is a simple proof of Teorem based on trace teory and Teorem 1. Assume for simplicity tat Ω = R N. Suppose tat s is not an integer, but tat s 1 = s + 1/p is an integer. Ten u is te trace of some function u 1 W s1,p (R N+1 ). Ten s 1 p = N +1 and by Teorem 1 we deduce tat Φ u 1 W s1,p (R N+1 ). Taking traces we find Φ u W s,p (R N ). If s 1 is not an integer we keep extending u 1 to iger dimensions and stop at te first integer k suc tat s k = s + k/p is an integer ( tis is possible since p is rational and s + k/p = (N + k)/p becomes an integer for some integer k). We ave an extension u k W sk,p (R N+k ) of u. Ten Φ u k W sk,p (R N+k ) by Teorem 1. Taking back traces yields u W s,p.
7 COMPOSITION IN FRACTIONAL SOBOLEV SPACES 7 References 1. R. Adams, Sobolev spaces, Acad. Press, D.R. Adams, On te existence of capacitary strong type estimates in R n, Ark. Mat. 14, (1976), D.R. Adams and M. Frazier, Composition operators on potential spaces, Proc. Amer. Mat. Soc., 114, (199), S. Alinac and P. Gérard, Opérateurs pseudo-différentiels et téorème de Nas-Moser, Interéditions, J.-M. Bony, Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles nonlinéaires, Ann. Sc. Ec. Norm Sup. 14, (1981), G. Bourdaud, Le calcul fonctionnel dans les espaces de Sobolev, Invent. Mat., 104, (1991), G. Bourdaud and Y. Meyer, Fonctions qui opèrent sur les espaces de Sobolev, J. Funct. Anal., 97, (1991), J. Bourgain, H.Brezis and P.Mironescu, Lifting in Sobolev spaces, J. d Analyse 80, (000), J.-Y. Cemin, Fluides parfaits incompressibles, Astérisque 30, M. Escobedo, Some remarks on te density of regular mappings in Sobolev classes of S M -valued functions, Rev. Mat. Univ. Complut. Madrid, 1, (1988), Y.Meyer, Remarques sur un téorème de J.-M. Bony,, Suppl. Rend. Circ. Mat. Palermo, Series II 1, (1981), S. Mizoata, Lectures on te Caucy problem, Tata Inst., Bombay, J. Moser, A rapidly convergent iteration metod and non-linear differential equations, Ann. Sc. Norm. Sup. Pisa 0, (1966), L. Nirenberg, On elliptic partial differential equations, Ann. Sc. Norm. Sup. Pisa, 13, (1959), J. Peetre, Interpolation of Lipscitz operators and metric spaces, Matematica (Cluj) 1, (1970), J. Rauc and M. Reed, Nonlinear microlocal analysis of semilinear yperbolic systems in one space dimension, Duke Mat J. 49, (198), T. Runst, Mapping properties of non-linear operators in spaces of Triebel-Lizorkin and Besov type, Analysis Matematica, 1, (1986), T. Runst and W. Sickel, Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial differential equations, Walter de Gruyter, Berlin and New York, H. Triebel, Interpolation teory, function spaces, differential operators, Nort-Holland, Amsterdam, H. Triebel, Teory of function spaces, Birkäuser, Basel and Boston, (1) ANALYSE NUMÉRIQUE UNIVERSITÉ P. ET M. CURIE, B.C PL. JUSSIEU 755 PARIS CEDEX 05 () RUTGERS UNIVERSITY DEPT. OF MATH., HILL CENTER, BUSCH CAMPUS 110 FRELINGHUYSEN RD, PISCATAWAY, NJ address: brezis@ccr.jussieu.fr; brezis@mat.rutgers.edu (3) DEPARTEMENT DE MATHÉMATIQUES UNIVERSITÉ PARIS-SUD ORSAY address: : Petru.Mironescu@mat.u-psud.fr
COMPOSITION IN FRACTIONAL SOBOLEV SPACES. 1. Introduction. A classical result about composition in Sobolev spaces asserts
DISCRETE AND CONTINUOUS Website: ttp://mat.smsu.edu/journal DYNAMICAL SYSTEMS Volume 7, Number, April 001 pp. 41 46 COMPOSITION IN FRACTIONAL SOBOLEV SPACES HAIM BREZIS (1)() AND PETRU MIRONESCU (3) 1.
More informationDEGREE AND SOBOLEV SPACES. Haïm Brezis Yanyan Li Petru Mironescu Louis Nirenberg. Introduction
Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 13, 1999, 181 190 DEGREE AND SOBOLEV SPACES Haïm Brezis Yanyan Li Petru Mironescu Louis Nirenberg Dedicated to Jürgen
More informationarxiv: v1 [math.ap] 12 Mar 2009
LIMITING FRACTIONAL AND LORENTZ SPACES ESTIMATES OF DIFFERENTIAL FORMS JEAN VAN SCHAFTINGEN arxiv:0903.282v [math.ap] 2 Mar 2009 Abstract. We obtain estimates in Besov, Lizorkin-Triebel and Lorentz spaces
More informationch (for some fixed positive number c) reaching c
GSTF Journal of Matematics Statistics and Operations Researc (JMSOR) Vol. No. September 05 DOI 0.60/s4086-05-000-z Nonlinear Piecewise-defined Difference Equations wit Reciprocal and Cubic Terms Ramadan
More informationA CLASS OF EVEN DEGREE SPLINES OBTAINED THROUGH A MINIMUM CONDITION
STUDIA UNIV. BABEŞ BOLYAI, MATHEMATICA, Volume XLVIII, Number 3, September 2003 A CLASS OF EVEN DEGREE SPLINES OBTAINED THROUGH A MINIMUM CONDITION GH. MICULA, E. SANTI, AND M. G. CIMORONI Dedicated to
More informationOverlapping domain decomposition methods for elliptic quasi-variational inequalities related to impulse control problem with mixed boundary conditions
Proc. Indian Acad. Sci. (Mat. Sci.) Vol. 121, No. 4, November 2011, pp. 481 493. c Indian Academy of Sciences Overlapping domain decomposition metods for elliptic quasi-variational inequalities related
More informationANALYTIC SMOOTHING EFFECT FOR THE CUBIC HYPERBOLIC SCHRÖDINGER EQUATION IN TWO SPACE DIMENSIONS
Electronic Journal of Differential Equations, Vol. 2016 2016), No. 34, pp. 1 8. ISSN: 1072-6691. URL: ttp://ejde.mat.txstate.edu or ttp://ejde.mat.unt.edu ftp ejde.mat.txstate.edu ANALYTIC SMOOTHING EFFECT
More informationPoisson Equation in Sobolev Spaces
Poisson Equation in Sobolev Spaces OcMountain Dayligt Time. 6, 011 Today we discuss te Poisson equation in Sobolev spaces. It s existence, uniqueness, and regularity. Weak Solution. u = f in, u = g on
More informationNumerical Experiments Using MATLAB: Superconvergence of Nonconforming Finite Element Approximation for Second-Order Elliptic Problems
Applied Matematics, 06, 7, 74-8 ttp://wwwscirporg/journal/am ISSN Online: 5-7393 ISSN Print: 5-7385 Numerical Experiments Using MATLAB: Superconvergence of Nonconforming Finite Element Approximation for
More informationBesov-type spaces with variable smoothness and integrability
Besov-type spaces with variable smoothness and integrability Douadi Drihem M sila University, Department of Mathematics, Laboratory of Functional Analysis and Geometry of Spaces December 2015 M sila, Algeria
More informationSeong Joo Kang. Let u be a smooth enough solution to a quasilinear hyperbolic mixed problem:
Comm. Korean Math. Soc. 16 2001, No. 2, pp. 225 233 THE ENERGY INEQUALITY OF A QUASILINEAR HYPERBOLIC MIXED PROBLEM Seong Joo Kang Abstract. In this paper, e establish the energy inequalities for second
More informationarxiv: v1 [math.dg] 4 Feb 2015
CENTROID OF TRIANGLES ASSOCIATED WITH A CURVE arxiv:1502.01205v1 [mat.dg] 4 Feb 2015 Dong-Soo Kim and Dong Seo Kim Abstract. Arcimedes sowed tat te area between a parabola and any cord AB on te parabola
More informationERROR BOUNDS FOR THE METHODS OF GLIMM, GODUNOV AND LEVEQUE BRADLEY J. LUCIER*
EO BOUNDS FO THE METHODS OF GLIMM, GODUNOV AND LEVEQUE BADLEY J. LUCIE* Abstract. Te expected error in L ) attimet for Glimm s sceme wen applied to a scalar conservation law is bounded by + 2 ) ) /2 T
More informationUniversity Mathematics 2
University Matematics 2 1 Differentiability In tis section, we discuss te differentiability of functions. Definition 1.1 Differentiable function). Let f) be a function. We say tat f is differentiable at
More informationINNER FUNCTIONS IN THE HYPERBOLIC LITTLE BLOCH CLASS. Wayne Smith
INNER FUNCTIONS IN THE HYPERBOLIC LITTLE BLOCH CLASS Wayne Smit Abstract. An analytic function ϕ mapping te unit disk into itself is said to belong to te yperbolic little Bloc class if te ratio (1 z 2
More informationMath 161 (33) - Final exam
Name: Id #: Mat 161 (33) - Final exam Fall Quarter 2015 Wednesday December 9, 2015-10:30am to 12:30am Instructions: Prob. Points Score possible 1 25 2 25 3 25 4 25 TOTAL 75 (BEST 3) Read eac problem carefully.
More informationSuperconvergence of energy-conserving discontinuous Galerkin methods for. linear hyperbolic equations. Abstract
Superconvergence of energy-conserving discontinuous Galerkin metods for linear yperbolic equations Yong Liu, Ci-Wang Su and Mengping Zang 3 Abstract In tis paper, we study superconvergence properties of
More informationAMS 147 Computational Methods and Applications Lecture 09 Copyright by Hongyun Wang, UCSC. Exact value. Effect of round-off error.
Lecture 09 Copyrigt by Hongyun Wang, UCSC Recap: Te total error in numerical differentiation fl( f ( x + fl( f ( x E T ( = f ( x Numerical result from a computer Exact value = e + f x+ Discretization error
More informationFINITE DIFFERENCE APPROXIMATIONS FOR NONLINEAR FIRST ORDER PARTIAL DIFFERENTIAL EQUATIONS
UNIVERSITATIS IAGELLONICAE ACTA MATHEMATICA, FASCICULUS XL 2002 FINITE DIFFERENCE APPROXIMATIONS FOR NONLINEAR FIRST ORDER PARTIAL DIFFERENTIAL EQUATIONS by Anna Baranowska Zdzis law Kamont Abstract. Classical
More informationWhere Sobolev interacts with Gagliardo-Nirenberg
Where Sobolev interacts with Gagliardo-Nirenberg Haim Brezis, Petru Mironescu To cite this version: Haim Brezis, Petru Mironescu. Where Sobolev interacts with Gagliardo-Nirenberg. Journal of Functional
More informationLyapunov characterization of input-to-state stability for semilinear control systems over Banach spaces
Lyapunov caracterization of input-to-state stability for semilinear control systems over Banac spaces Andrii Mironcenko a, Fabian Wirt a a Faculty of Computer Science and Matematics, University of Passau,
More informationVolume 29, Issue 3. Existence of competitive equilibrium in economies with multi-member households
Volume 29, Issue 3 Existence of competitive equilibrium in economies wit multi-member ouseolds Noriisa Sato Graduate Scool of Economics, Waseda University Abstract Tis paper focuses on te existence of
More informationDifferentiation in higher dimensions
Capter 2 Differentiation in iger dimensions 2.1 Te Total Derivative Recall tat if f : R R is a 1-variable function, and a R, we say tat f is differentiable at x = a if and only if te ratio f(a+) f(a) tends
More informationΓ -convergence and Sobolev norms
C. R. Acad. Sci. Paris, Ser. I 345 (2007) 679 684 http://france.elsevier.com/direct/crass1/ Partial Differential Equations Γ -convergence and Sobolev norms Hoai-Minh Nguyen Rutgers University, Department
More informationA proof in the finite-difference spirit of the superconvergence of the gradient for the Shortley-Weller method.
A proof in te finite-difference spirit of te superconvergence of te gradient for te Sortley-Weller metod. L. Weynans 1 1 Team Mempis, INRIA Bordeaux-Sud-Ouest & CNRS UMR 551, Université de Bordeaux, France
More informationStability properties of a family of chock capturing methods for hyperbolic conservation laws
Proceedings of te 3rd IASME/WSEAS Int. Conf. on FLUID DYNAMICS & AERODYNAMICS, Corfu, Greece, August 0-, 005 (pp48-5) Stability properties of a family of cock capturing metods for yperbolic conservation
More informationISABELLE GALLAGHER AND MARIUS PAICU
REMARKS ON THE BLOW-UP OF SOLUTIONS TO A TOY MODEL FOR THE NAVIER-STOKES EQUATIONS ISABELLE GALLAGHER AND MARIUS PAICU Abstract. In [14], S. Montgomery-Smith provides a one dimensional model for the three
More informationInfluence of the Stepsize on Hyers Ulam Stability of First-Order Homogeneous Linear Difference Equations
International Journal of Difference Equations ISSN 0973-6069, Volume 12, Number 2, pp. 281 302 (2017) ttp://campus.mst.edu/ijde Influence of te Stepsize on Hyers Ulam Stability of First-Order Homogeneous
More informationA Product Property of Sobolev Spaces with Application to Elliptic Estimates
Rend. Sem. Mat. Univ. Padova Manoscritto in corso di stampa pervenuto il 23 luglio 2012 accettato l 1 ottobre 2012 A Product Property of Sobolev Spaces with Application to Elliptic Estimates by Henry C.
More informationLecture 10: Carnot theorem
ecture 0: Carnot teorem Feb 7, 005 Equivalence of Kelvin and Clausius formulations ast time we learned tat te Second aw can be formulated in two ways. e Kelvin formulation: No process is possible wose
More informationA method of Lagrange Galerkin of second order in time. Une méthode de Lagrange Galerkin d ordre deux en temps
A metod of Lagrange Galerkin of second order in time Une métode de Lagrange Galerkin d ordre deux en temps Jocelyn Étienne a a DAMTP, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, Great-Britain.
More informationDIFFERENTIAL POLYNOMIALS GENERATED BY SECOND ORDER LINEAR DIFFERENTIAL EQUATIONS
Journal o Applied Analysis Vol. 14, No. 2 2008, pp. 259 271 DIFFERENTIAL POLYNOMIALS GENERATED BY SECOND ORDER LINEAR DIFFERENTIAL EQUATIONS B. BELAÏDI and A. EL FARISSI Received December 5, 2007 and,
More informationKato s inequality when u is a measure. L inégalité de Kato lorsque u est une mesure
Kato s inequality when u is a measure L inégalité de Kato lorsque u est une mesure Haïm Brezis a,b, Augusto C. Ponce a,b, a Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, BC 187, 4
More informationContinuity. Example 1
Continuity MATH 1003 Calculus and Linear Algebra (Lecture 13.5) Maoseng Xiong Department of Matematics, HKUST A function f : (a, b) R is continuous at a point c (a, b) if 1. x c f (x) exists, 2. f (c)
More informationMATH 155A FALL 13 PRACTICE MIDTERM 1 SOLUTIONS. needs to be non-zero, thus x 1. Also 1 +
MATH 55A FALL 3 PRACTICE MIDTERM SOLUTIONS Question Find te domain of te following functions (a) f(x) = x3 5 x +x 6 (b) g(x) = x+ + x+ (c) f(x) = 5 x + x 0 (a) We need x + x 6 = (x + 3)(x ) 0 Hence Dom(f)
More informationApproximation of the Viability Kernel
Approximation of te Viability Kernel Patrick Saint-Pierre CEREMADE, Université Paris-Daupine Place du Marécal de Lattre de Tassigny 75775 Paris cedex 16 26 october 1990 Abstract We study recursive inclusions
More informationTHE DISCRETE PLATEAU PROBLEM: CONVERGENCE RESULTS
MATHEMATICS OF COMPUTATION Volume 00, Number 0, Pages 000 000 S 0025-5718XX0000-0 THE DISCRETE PLATEAU PROBLEM: CONVERGENCE RESULTS GERHARD DZIUK AND JOHN E. HUTCHINSON Abstract. We solve te problem of
More information3.4 Worksheet: Proof of the Chain Rule NAME
Mat 1170 3.4 Workseet: Proof of te Cain Rule NAME Te Cain Rule So far we are able to differentiate all types of functions. For example: polynomials, rational, root, and trigonometric functions. We are
More informationHOMEWORK HELP 2 FOR MATH 151
HOMEWORK HELP 2 FOR MATH 151 Here we go; te second round of omework elp. If tere are oters you would like to see, let me know! 2.4, 43 and 44 At wat points are te functions f(x) and g(x) = xf(x)continuous,
More informationGlobal Existence of Classical Solutions for a Class Nonlinear Parabolic Equations
Global Journal of Science Frontier Researc Matematics and Decision Sciences Volume 12 Issue 8 Version 1.0 Type : Double Blind Peer Reviewed International Researc Journal Publiser: Global Journals Inc.
More informationMore on generalized inverses of partitioned matrices with Banachiewicz-Schur forms
More on generalized inverses of partitioned matrices wit anaciewicz-scur forms Yongge Tian a,, Yosio Takane b a Cina Economics and Management cademy, Central University of Finance and Economics, eijing,
More informationA method for solving first order fuzzy differential equation
Available online at ttp://ijim.srbiau.ac.ir/ Int. J. Industrial Matematics (ISSN 2008-5621) Vol. 5, No. 3, 2013 Article ID IJIM-00250, 7 pages Researc Article A metod for solving first order fuzzy differential
More informationNUMERICAL DIFFERENTIATION. James T. Smith San Francisco State University. In calculus classes, you compute derivatives algebraically: for example,
NUMERICAL DIFFERENTIATION James T Smit San Francisco State University In calculus classes, you compute derivatives algebraically: for example, f( x) = x + x f ( x) = x x Tis tecnique requires your knowing
More informationAn approximation method using approximate approximations
Applicable Analysis: An International Journal Vol. 00, No. 00, September 2005, 1 13 An approximation metod using approximate approximations FRANK MÜLLER and WERNER VARNHORN, University of Kassel, Germany,
More informationlecture 26: Richardson extrapolation
43 lecture 26: Ricardson extrapolation 35 Ricardson extrapolation, Romberg integration Trougout numerical analysis, one encounters procedures tat apply some simple approximation (eg, linear interpolation)
More informationConvexity and Smoothness
Capter 4 Convexity and Smootness 4.1 Strict Convexity, Smootness, and Gateaux Differentiablity Definition 4.1.1. Let X be a Banac space wit a norm denoted by. A map f : X \{0} X \{0}, f f x is called a
More informationMath Spring 2013 Solutions to Assignment # 3 Completion Date: Wednesday May 15, (1/z) 2 (1/z 1) 2 = lim
Mat 311 - Spring 013 Solutions to Assignment # 3 Completion Date: Wednesday May 15, 013 Question 1. [p 56, #10 (a)] 4z Use te teorem of Sec. 17 to sow tat z (z 1) = 4. We ave z 4z (z 1) = z 0 4 (1/z) (1/z
More informationAn L p di erentiable non-di erentiable function
An L di erentiable non-di erentiable function J. Marsall As Abstract. Tere is a a set E of ositive Lebesgue measure and a function nowere di erentiable on E wic is di erentible in te L sense for every
More informationOSCILLATION OF SOLUTIONS TO NON-LINEAR DIFFERENCE EQUATIONS WITH SEVERAL ADVANCED ARGUMENTS. Sandra Pinelas and Julio G. Dix
Opuscula Mat. 37, no. 6 (2017), 887 898 ttp://dx.doi.org/10.7494/opmat.2017.37.6.887 Opuscula Matematica OSCILLATION OF SOLUTIONS TO NON-LINEAR DIFFERENCE EQUATIONS WITH SEVERAL ADVANCED ARGUMENTS Sandra
More informationLEAST-SQUARES FINITE ELEMENT APPROXIMATIONS TO SOLUTIONS OF INTERFACE PROBLEMS
SIAM J. NUMER. ANAL. c 998 Society for Industrial Applied Matematics Vol. 35, No., pp. 393 405, February 998 00 LEAST-SQUARES FINITE ELEMENT APPROXIMATIONS TO SOLUTIONS OF INTERFACE PROBLEMS YANZHAO CAO
More informationA New Class of Zienkiewicz-Type Nonconforming Element in Any Dimensions
Numerisce Matematik manuscript No. will be inserted by te editor A New Class of Zienkiewicz-Type Nonconforming Element in Any Dimensions Wang Ming 1, Zong-ci Si 2, Jincao Xu1,3 1 LMAM, Scool of Matematical
More information232 Calculus and Structures
3 Calculus and Structures CHAPTER 17 JUSTIFICATION OF THE AREA AND SLOPE METHODS FOR EVALUATING BEAMS Calculus and Structures 33 Copyrigt Capter 17 JUSTIFICATION OF THE AREA AND SLOPE METHODS 17.1 THE
More informationREGULARITY OF SOLUTIONS OF PARTIAL NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATIONS WITH UNBOUNDED DELAY
Proyecciones Vol. 21, N o 1, pp. 65-95, May 22. Universidad Católica del Norte Antofagasta - Cile REGULARITY OF SOLUTIONS OF PARTIAL NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATIONS WITH UNBOUNDED DELAY EDUARDO
More informationMANY scientific and engineering problems can be
A Domain Decomposition Metod using Elliptical Arc Artificial Boundary for Exterior Problems Yajun Cen, and Qikui Du Abstract In tis paper, a Diriclet-Neumann alternating metod using elliptical arc artificial
More informationDedicated to the 70th birthday of Professor Lin Qun
Journal of Computational Matematics, Vol.4, No.3, 6, 4 44. ACCELERATION METHODS OF NONLINEAR ITERATION FOR NONLINEAR PARABOLIC EQUATIONS Guang-wei Yuan Xu-deng Hang Laboratory of Computational Pysics,
More informationDoubly Warped Products in Locally Conformal Almost Cosymplectic Manifolds
INTERNATIONAL ELECTRONIC JOURNAL OF GEOMETRY VOLUME 0 NO. 2 PAGE 73 8 207) Doubly Warped Products in Locally Conformal Almost Cosymplectic Manifolds Andreea Olteanu Communicated by Ion Miai) ABSTRACT Recently,
More informationHow to Find the Derivative of a Function: Calculus 1
Introduction How to Find te Derivative of a Function: Calculus 1 Calculus is not an easy matematics course Te fact tat you ave enrolled in suc a difficult subject indicates tat you are interested in te
More informationSection 3.1: Derivatives of Polynomials and Exponential Functions
Section 3.1: Derivatives of Polynomials and Exponential Functions In previous sections we developed te concept of te derivative and derivative function. Te only issue wit our definition owever is tat it
More informationEXISTENCE OF SOLUTIONS FOR A CLASS OF VARIATIONAL INEQUALITIES
Journal of Matematics and Statistics 9 (4: 35-39, 3 ISSN: 549-3644 3 doi:.3844/jmssp.3.35.39 Publised Online 9 (4 3 (ttp://www.tescipub.com/jmss.toc EXISTENCE OF SOLUTIONS FOR A CLASS OF ARIATIONAL INEQUALITIES
More informationOrder of Accuracy. ũ h u Ch p, (1)
Order of Accuracy 1 Terminology We consider a numerical approximation of an exact value u. Te approximation depends on a small parameter, wic can be for instance te grid size or time step in a numerical
More informationGlobal Output Feedback Stabilization of a Class of Upper-Triangular Nonlinear Systems
9 American Control Conference Hyatt Regency Riverfront St Louis MO USA June - 9 FrA4 Global Output Feedbac Stabilization of a Class of Upper-Triangular Nonlinear Systems Cunjiang Qian Abstract Tis paper
More informationLECTURE 14 NUMERICAL INTEGRATION. Find
LECTURE 14 NUMERCAL NTEGRATON Find b a fxdx or b a vx ux fx ydy dx Often integration is required. However te form of fx may be suc tat analytical integration would be very difficult or impossible. Use
More informationRIGIDITY OF TIME-FLAT SURFACES IN THE MINKOWSKI SPACETIME
RIGIDITY OF TIME-FLAT SURFACES IN THE MINKOWSKI SPACETIME PO-NING CHEN, MU-TAO WANG, AND YE-KAI WANG Abstract. A time-flat condition on spacelike 2-surfaces in spacetime is considered ere. Tis condition
More informationAnalysis of A Continuous Finite Element Method for H(curl, div)-elliptic Interface Problem
Analysis of A Continuous inite Element Metod for Hcurl, div)-elliptic Interface Problem Huoyuan Duan, Ping Lin, and Roger C. E. Tan Abstract In tis paper, we develop a continuous finite element metod for
More informationResearch Article New Results on Multiple Solutions for Nth-Order Fuzzy Differential Equations under Generalized Differentiability
Hindawi Publising Corporation Boundary Value Problems Volume 009, Article ID 395714, 13 pages doi:10.1155/009/395714 Researc Article New Results on Multiple Solutions for Nt-Order Fuzzy Differential Equations
More informationMath 1210 Midterm 1 January 31st, 2014
Mat 110 Midterm 1 January 1st, 01 Tis exam consists of sections, A and B. Section A is conceptual, wereas section B is more computational. Te value of every question is indicated at te beginning of it.
More informationA SPLITTING LEAST-SQUARES MIXED FINITE ELEMENT METHOD FOR ELLIPTIC OPTIMAL CONTROL PROBLEMS
INTERNATIONAL JOURNAL OF NUMERICAL ANALSIS AND MODELING Volume 3, Number 4, Pages 6 626 c 26 Institute for Scientific Computing and Information A SPLITTING LEAST-SQUARES MIED FINITE ELEMENT METHOD FOR
More informationConvexity and Smoothness
Capter 4 Convexity and Smootness 4. Strict Convexity, Smootness, and Gateaux Di erentiablity Definition 4... Let X be a Banac space wit a norm denoted by k k. A map f : X \{0}!X \{0}, f 7! f x is called
More information4. The slope of the line 2x 7y = 8 is (a) 2/7 (b) 7/2 (c) 2 (d) 2/7 (e) None of these.
Mat 11. Test Form N Fall 016 Name. Instructions. Te first eleven problems are wort points eac. Te last six problems are wort 5 points eac. For te last six problems, you must use relevant metods of algebra
More informationBLOWUP THEORY FOR THE CRITICAL NONLINEAR SCHRÖDINGER EQUATIONS REVISITED
BLOWUP THEORY FOR THE CRITICAL NONLINEAR SCHRÖDINGER EQUATIONS REVISITED TAOUFIK HMIDI AND SAHBI KERAANI Abstract. In this note we prove a refined version of compactness lemma adapted to the blowup analysis
More informationPacking polynomials on multidimensional integer sectors
Pacing polynomials on multidimensional integer sectors Luis B Morales IIMAS, Universidad Nacional Autónoma de México, Ciudad de México, 04510, México lbm@unammx Submitted: Jun 3, 015; Accepted: Sep 8,
More informationarxiv: v1 [math.na] 9 Sep 2015
arxiv:509.02595v [mat.na] 9 Sep 205 An Expandable Local and Parallel Two-Grid Finite Element Sceme Yanren ou, GuangZi Du Abstract An expandable local and parallel two-grid finite element sceme based on
More informationON A PROBLEM OF SHALLOW WATER TYPE
24-Fez conference on Differential Equations and Mecanics Electronic Journal of Differential Equations, Conference 11, 24, pp. 19 116. ISSN: 172-6691. URL: ttp://ejde.mat.txstate.edu or ttp://ejde.mat.unt.edu
More informationMVT and Rolle s Theorem
AP Calculus CHAPTER 4 WORKSHEET APPLICATIONS OF DIFFERENTIATION MVT and Rolle s Teorem Name Seat # Date UNLESS INDICATED, DO NOT USE YOUR CALCULATOR FOR ANY OF THESE QUESTIONS In problems 1 and, state
More informationAnalytic Functions. Differentiable Functions of a Complex Variable
Analytic Functions Differentiable Functions of a Complex Variable In tis capter, we sall generalize te ideas for polynomials power series of a complex variable we developed in te previous capter to general
More informationOn convergence of the immersed boundary method for elliptic interface problems
On convergence of te immersed boundary metod for elliptic interface problems Zilin Li January 26, 2012 Abstract Peskin s Immersed Boundary (IB) metod is one of te most popular numerical metods for many
More informationPOLYNOMIAL AND SPLINE ESTIMATORS OF THE DISTRIBUTION FUNCTION WITH PRESCRIBED ACCURACY
APPLICATIONES MATHEMATICAE 36, (29), pp. 2 Zbigniew Ciesielski (Sopot) Ryszard Zieliński (Warszawa) POLYNOMIAL AND SPLINE ESTIMATORS OF THE DISTRIBUTION FUNCTION WITH PRESCRIBED ACCURACY Abstract. Dvoretzky
More informationAn Elementary Proof of a Generalization of Bernoulli s Formula
Revision: August 7, 009 An Elementary Proof of a Generalization of Bernoulli s Formula Kevin J. McGown Harold R. Pars Department of Matematics Department of Matematics University of California at San Diego
More informationConvergence of Rothe s Method for Fully Nonlinear Parabolic Equations
Te Journal of Geometric Analysis Volume 15, Number 3, 2005 Convergence of Rote s Metod for Fully Nonlinear Parabolic Equations By Ivan Blank and Penelope Smit ABSTRACT. Convergence of Rote s metod for
More informationA Finite Element Primer
A Finite Element Primer David J. Silvester Scool of Matematics, University of Mancester d.silvester@mancester.ac.uk. Version.3 updated 4 October Contents A Model Diffusion Problem.................... x.
More informationMath 212-Lecture 9. For a single-variable function z = f(x), the derivative is f (x) = lim h 0
3.4: Partial Derivatives Definition Mat 22-Lecture 9 For a single-variable function z = f(x), te derivative is f (x) = lim 0 f(x+) f(x). For a function z = f(x, y) of two variables, to define te derivatives,
More informationOn convexity of polynomial paths and generalized majorizations
On convexity of polynomial pats and generalized majorizations Marija Dodig Centro de Estruturas Lineares e Combinatórias, CELC, Universidade de Lisboa, Av. Prof. Gama Pinto 2, 1649-003 Lisboa, Portugal
More informationERROR BOUNDS FOR FINITE-DIFFERENCE METHODS FOR RUDIN OSHER FATEMI IMAGE SMOOTHING
ERROR BOUNDS FOR FINITE-DIFFERENCE METHODS FOR RUDIN OSHER FATEMI IMAGE SMOOTHING JINGYUE WANG AND BRADLEY J. LUCIER Abstract. We bound te difference between te solution to te continuous Rudin Oser Fatemi
More informationConsider a function f we ll specify which assumptions we need to make about it in a minute. Let us reformulate the integral. 1 f(x) dx.
Capter 2 Integrals as sums and derivatives as differences We now switc to te simplest metods for integrating or differentiating a function from its function samples. A careful study of Taylor expansions
More informationNumerical Solution of One Dimensional Nonlinear Longitudinal Oscillations in a Class of Generalized Functions
Proc. of te 8t WSEAS Int. Conf. on Matematical Metods and Computational Tecniques in Electrical Engineering, Bucarest, October 16-17, 2006 219 Numerical Solution of One Dimensional Nonlinear Longitudinal
More informationEfficient algorithms for for clone items detection
Efficient algoritms for for clone items detection Raoul Medina, Caroline Noyer, and Olivier Raynaud Raoul Medina, Caroline Noyer and Olivier Raynaud LIMOS - Université Blaise Pascal, Campus universitaire
More informationFunction Composition and Chain Rules
Function Composition and s James K. Peterson Department of Biological Sciences and Department of Matematical Sciences Clemson University Marc 8, 2017 Outline 1 Function Composition and Continuity 2 Function
More informationarxiv: v1 [math.na] 28 Apr 2017
THE SCOTT-VOGELIUS FINITE ELEMENTS REVISITED JOHNNY GUZMÁN AND L RIDGWAY SCOTT arxiv:170500020v1 [matna] 28 Apr 2017 Abstract We prove tat te Scott-Vogelius finite elements are inf-sup stable on sape-regular
More informationCLEMSON U N I V E R S I T Y
A Fractional Step θ-metod for Convection-Diffusion Equations Jon Crispell December, 006 Advisors: Dr. Lea Jenkins and Dr. Vincent Ervin Fractional Step θ-metod Outline Crispell,Ervin,Jenkins Motivation
More information1. Introduction. We consider the model problem: seeking an unknown function u satisfying
A DISCONTINUOUS LEAST-SQUARES FINITE ELEMENT METHOD FOR SECOND ORDER ELLIPTIC EQUATIONS XIU YE AND SHANGYOU ZHANG Abstract In tis paper, a discontinuous least-squares (DLS) finite element metod is introduced
More informationAnalysis of a second order discontinuous Galerkin finite element method for the Allen-Cahn equation
Graduate Teses and Dissertations Graduate College 06 Analysis of a second order discontinuous Galerin finite element metod for te Allen-Can equation Junzao Hu Iowa State University Follow tis and additional
More informationarxiv: v2 [math.fa] 30 Aug 2016
O FORMULAE DECOUPLIG THE TOTAL VARIATIO OF BV FUCTIOS AUGUSTO C. POCE AD DAIEL SPECTOR arxiv:163.8663v2 [mat.fa] 3 Aug 216 To icola Fusco, a master of BV, on te occasion of is sixtiet birtday ABSTRACT.
More informationGradient Descent etc.
1 Gradient Descent etc EE 13: Networked estimation and control Prof Kan) I DERIVATIVE Consider f : R R x fx) Te derivative is defined as d fx) = lim dx fx + ) fx) Te cain rule states tat if d d f gx) )
More informationSTABILITY OF DISCRETE STOKES OPERATORS IN FRACTIONAL SOBOLEV SPACES
STABILITY OF DISCRETE STOKES OPERATORS IN FRACTIONAL SOBOLEV SPACES JEAN-LUC GUERMOND,, JOSEPH E PASCIAK Abstract Using a general approximation setting aving te generic properties of finite-elements, we
More informationNUMERICAL DIFFERENTIATION
NUMERICAL IFFERENTIATION FIRST ERIVATIVES Te simplest difference formulas are based on using a straigt line to interpolate te given data; tey use two data pints to estimate te derivative. We assume tat
More informationWeak Solutions for Nonlinear Parabolic Equations with Variable Exponents
Communications in Matematics 25 217) 55 7 Copyrigt c 217 Te University of Ostrava 55 Weak Solutions for Nonlinear Parabolic Equations wit Variable Exponents Lingeswaran Sangerganes, Arumugam Gurusamy,
More informationReflection Symmetries of q-bernoulli Polynomials
Journal of Nonlinear Matematical Pysics Volume 1, Supplement 1 005, 41 4 Birtday Issue Reflection Symmetries of q-bernoulli Polynomials Boris A KUPERSHMIDT Te University of Tennessee Space Institute Tullaoma,
More informationANALYTIC SMOOTHING EFFECT FOR NONLI TitleSCHRÖDINGER EQUATION IN TWO SPACE DIMENSIONS. Citation Osaka Journal of Mathematics.
ANALYTIC SMOOTHING EFFECT FOR NONLI TitleSCHRÖDINGER EQUATION IN TWO SPACE DIMENSIONS Author(s) Hoshino, Gaku; Ozawa, Tohru Citation Osaka Journal of Mathematics. 51(3) Issue 014-07 Date Text Version publisher
More informationA Hybrid Mixed Discontinuous Galerkin Finite Element Method for Convection-Diffusion Problems
A Hybrid Mixed Discontinuous Galerkin Finite Element Metod for Convection-Diffusion Problems Herbert Egger Joacim Scöberl We propose and analyse a new finite element metod for convection diffusion problems
More informationClick here to see an animation of the derivative
Differentiation Massoud Malek Derivative Te concept of derivative is at te core of Calculus; It is a very powerful tool for understanding te beavior of matematical functions. It allows us to optimize functions,
More information