MATH P.D.E. II
|
|
- Denis Cecil Johns
- 7 years ago
- Views:
Transcription
1 Homework # 1 (due Thursday, 4/16/215) 1. Let IR n be a bounded open set. Consider the hyperbolic equation utt u = c(x)u+g(u) x, u = x. (1) Construct a function φ(x,u) such that, for every smooth solution of (1), the total energy E = ( ut u 2 2 ) +φ(x,u) dx is constant in time. (Hint: write an identity that should be satisfied by the partial derivative φ u, in order that de(t) dt =. ) 2. On the space L 1 (IR), consider the semigroup S t ; t } generated by the operator Au = u xx. (i) Show that this is a contractive semigroup on the space L 1 (IR). (ii) Using Duhamel s formula, write out the mild solution to the Cauchy problem u t = u xx + u(,x) = e x. t 1+x 2, (2) (iii) Write an integral equation satisfied by the mild solution of the semilinear Cauchy problem u t = u xx +sinu, (3) u(,x) = f(x). Explain why this equation has a unique solution, defined for all t. 3. Decide which of the following maps is Lipschitz continuous from L 1 (IR) into itself. (Λ 1 u)(x) = sinu(x), (Λ 2 u)(x) = cosu(x), (Λ 3 u)(x) = u(y)dy 1+x 2, (Λ 4 u)(x) = u 2 (x). In the positive case, determine the Lipschitz constant. In other words, find C such that Λu Λv L 1 C u v L 1. 1
2 Homework # 9 (due Thursday, 4/9/215) 1. On the open interval =],3[, consider the boundary value problem uxx = 1 < x < 3, u() = u(3) =. (1) Consider the two linearly independent functions ϕ 1,ϕ 2 H 1 (],3[), defined by x if x [,1], if x [,1], ϕ 1 (x) = 2 x if x [1,2], ϕ 2 (x) = x 1 if x [1,2], if x [2,3], 3 x if x [2,3]. Explicitly compute the Galerkin approximation U(x) = c 1 ϕ 1 (x)+c 2 ϕ 2 (x) such that B[U,ϕ i ] = 3 U x ϕ i,x dx = Compare U with the exact solution of (1). 3 1 ϕ i dx = (1,ϕ i ) L 2 i = 1,2. 2. On the open interval =],3[, consider the parabolic problem u t = u xx < x < 3, u(t,) = u(t,3) =, u(,x) = 1. (2) Explicitly compute the Galerkin approximation U(t,x) = c 1 (t)ϕ 1 (x) + c 2 (t)ϕ 2 (x), choosing the functions c 1,c 2 so that ( U(, ) 1, ϕi ) L 2 = i = 1,2, t =, ( Ut, ϕ i ) L 2 +B[U, ϕ i ] = i = 1,2, t >. Compare U with the exact solution of (2). 3. Let = (x,y); x 2 1 +x2 2 < 1} be the open unit disc in IR2, and let u be a smooth solution to the equation u tt = 2u x1 x 1 +x 2 u x1 x 2 +3u x2 x u x 1 on [,T], (3) u = on [,T]. (i) Write the equation (3) in the form u tt + Lu =, showing that the operator L is symmetric (i.e., a 12 = a 21 ) and uniformly elliptic on the domain. (ii) Define a suitable energy e(t) = [kinetic energy] + [elastic potential energy], and check that it is constant in time. 2
3 Homework # 8 (due Thursday, 4/2/215) 1. Consider the space of square summable sequences of real numbers: } X =. x = (x 1,x 2,x 3,...); x =. ( x k 2) 1/2 < k (i) Show that the linear operator Ax = ( x 1, 2x 2, 3x 3,...) is not bounded on X. (ii) Explicitly write out the semigroup generated by A. In other words, for every x X, compute S t x. Prove that this semigroup is contractive. (iii) Prove that, for every x X and t > one has S t x Dom(A), even if x / Dom(A). Show that AS t sup λ> λe tλ <. 2. Let IR n be a bounded open set. A function u H 2 () is a weak solution of the biharmonic equation 2 u = f x, u = u ν = x, (1) if u vdx = f vdx for all v H 2 (). (2) Given f L 2 (), prove that the boundary value problem (1) has a unique weak solution. Hint: show that the bilinear form B[u,v] on H 2 () defined by the left hand side of (2) is strictly positive definite on H(). 2 Notice that, if u Cc (), integrating by parts one obtains u xi x j u xi x j dx = u xi x i u xi x j dx 1 2 (u 2 x i x i +u 2 x i x j )dx. 3
4 Homework # 7 (due Thursday, 3/26/215) 1. Let A be the generator of a contractive semigroup, on a Banach space X. Show that, for every λ >, the bounded linear operator A λ. = λa(λi A) 1 also generates a contractive semigroup. 2. On the space L 1 (IR), consider the linear operators S t defined by (S t f)(x) = e 2t f(x+t). (i) Prove that the family of linear operators S t ; t } is a strongly continuous, contractive semigroup on L 1 (IR). Find the generator A of this semigroup. What is Dom(A)? (ii) Show that the family of operators S t ; t } is NOT a strongly continuous semigroup on L (IR). 3. Let S t ; t } be a strongly continuous semigroup of linear operators on IR n. Prove that there exists an n n matrix A such that S t = e ta for every t. 4. On the space L 1 (IR), consider the operator Au = xu with domain } Dom(A) = u L 1 (IR); u is absolutely continuous, u x L 1 (IR). (i)] Describe the semigroup S t ; t } generated by A. (ii) For any u L 1 (IR) and h >, construct the backward Euler approximation E h u. 5 (extra credit) Fix a time T >. On the space X = L 1 ([,1]), construct a strongly continuous, contractive semigroup of linear operators S t ; t } such that S t = 1 for t < T but S t = for t T. 4
5 Homework # 6 (due Thursday, 3/5/215) 1. On the space L 2( [, [ ), consider the linear operator Lu(x). = u(x+1). (i) Find the adjoint operator L, such that (u,l v) L 2 = (Lu,v) L 2 for every u,v L 2 ([, [). (ii) Find the kernel and the range of L and L. Do the two kernels have the same dimension? 2. Assume f L 2( [,π] ). Show that the problem u +9u = f, u() = u(π) = has a solution if and only if Is the solution unique? π f(x) sin3xdx =. 3. Let IR n be a bounded open set, and consider the operator Lu. = ij (a ij (x)u xi ) xj + i b i u xi +c(x)u, where the coefficients b i are constant. Assume that a ij, c L (), with c(x), and that L is uniformly elliptic, so that ij aij ξ i ξ j θ i ξ2 i, for some θ >. (i) Write the corresponding bilinear form B[u, v]. Prove that it is strictly positive definite on H 1 (). (Hint: compute the divergence of the vector (b1,...,b n ) u2 2 ) (ii) Prove that, for every f L 2 (), the problem Lu = f x, u = x, has a unique weak solution. 5
6 Homework # 5 (due Thursday, 2/26/215) 1. Let = (x 1,x 2 ); x 2 1 +x 2 2 < 1 } be the unit disc in IR 2. Consider the operator ( ) Lu = u x1 x 1 +u x2 x 2 +x 1 u x1 x 2 +3x 2 2 u x1 (i) Check whether the operator L is elliptic on. (ii) Given f L 2 (), explain what it means for a function u to be a weak solution of the equation Lu = f x, u = x, 2. Let IR n be a bounded open set, and let a,c : [1,2] be smooth functions. Define H to be the Hilbert space H 1 () with the equivalent inner product. (u,v) H = a(x) u(x) v(x)dx+ c(x)u(x)v(x) dx. Let ι : H L 2 () be the identity map, and let ι : L 2 () H be the adjoint operator. Prove that, for every f L 2 (), the function ι f H provides the weak solution to an elliptic boundary value problem. Which equation does ι f solve? 3. Let IR n be a bounded open set, and let u be a weak solution to u+u = f x, u = x, (i) Prove that u L 2 f L 2. (ii) Prove that u L 2 f L 2. Hint: in both cases, use the definition of weak solution taking v = u as test function. 6
7 Homework # 4 (due Thursday, 2/19/215) 1. Let be an open ball in IR n. Show that there exists a constant C such that u 2 (x)dx C u(x) 2 dx+c u(x) dx Let IR n be a bounded open set and let c( ) be a continuous function on. (i) Show that there exists µ > such that, if c(x) > µ for all x, then the bilinear operator B[u,v] = u vdx+ c(x)u(x)v(x) dx satisfies the assumption of the Lax-Milgram theorem on the space H 1 () = W 1,2 (). (ii) By (i), for every f L 2 (), there exists u H) 1 () such that B[u,v] = (f,v) L 2 for every v H 1 (). What PDE + boundary conditions does u solve, in a weak sense? 3. Consider the open interval =] 1,1[, and the Hilbert-Sobolev space H 1 (). (i) Show that the Dirac measure δ : v v() (= a unit mass concentrated at the origin) is a continuous linear functional on H 1 (). Is the Dirac measure an element of the space H 1 ()? (ii) Explicitly determine a function u H 1 () such that (u,v) H 1 = v() for every v H 1 (). Hint: solve the boundary value problem u u = x ] 1,[ ],1[, u( 1) = u(1) =, u ( ) u (+) = 1. 7
8 Homework # 3 (due Thursday, 2/12/215) 1. Prove that, for any 1 p <, one has W 1,p (IR n ) = W 1,p (IR n ). Hint: Let u W 1,p (IR n ). Given ε >, find ũ C (IR n ) such that ũ u W 1,p < ε. Then take a C function ψ : IR : [,1], with ψ(r) = 1 if x, and ψ(r) = if r > 1. Show that the functions u k (x) = ũ(x) ψ( x k) are C with compact support. Prove that u k ũ W 1,p as k. 2. Consider the sequence of functions u k (x) = sinkx For a fixed ε >, define the regularization u ε k (x) = 1 2ε x+ε x ε u k (y)dy. (i) Prove that, for a fixed ε >, all functions u ε k, k 1 are uniformly Lipschitz continuous. (ii) By (i), on the compact interval [, 2π], by Ascoli s theorem (possibly taking a subsequence), one has the uniform convergence lim k u ε k = ūε. Describe the function ū ε. What is ū = lim ε ū ε? (iii) Estimate the limit 2π lim u ε k (x) u k (x) dx. k Does this approach zero as ε? Is it true that u k ū L1 ([,π])? 3. (i) Let IR 2 be an open ball. Assume that f W 1.p (). For which values of p,q can one conclude that the product f D x1 f lies in L q ()? (ii) Answer the same question in the case where is a ball in IR 3. 8
9 Homework # 2 (due Thursday, 2/5/215) 1. Let ξ : IR IR be a C 1, strictly increasing function (with ξ (x) > for every x), mapping =]a,b[ onto =]c,d[. Let f W 1,p () and define the composite function g(x) = f ( ξ(x) ). Prove that g W 1,p ( ), and compute the weak derivative of g. 2. Verify that if n > 1 the unbounded function ( u = loglog 1+ 1 ) x is in the space W 1,n (). Here IR n is the unit ball centered at the origin. This shows that functions u W 1,n (IR n ) need not be continuous, or bounded. 3. Let IR n be the unit ball centered at the origin. Consider the problem of minimizing the integral u p dx among all continuous functions that vanish on and take the value 1 at the origin. Assuming that u(x) = w( x ) is radially symmetric, this leads to the standard problem in the Calculus of Variations: minimize 1 r n 1 w (r) p dr subject to: w() = 1, w(1) =. (1) Study whether this problem has solutions, depending on n, p. Hint: consider the Euler-Lagrange equations L w = d L dr w, L(r,w,w ) = r n 1 w p. (2) Prove that (i) If p > n, the second order ODE (2) has a solution with boundary data (1) (ii) If p < n, no solution exists. In this case, for any ε > one can find a function w satisfying (1), such that 1 r n 1 w (r) p dr < ε. As a consequence, the norm u C () = sup x u(x) cannot be controlled by the norm u W 1,p (). 9
10 Homework # 1 (due Thursday, 1/29/215) 1. Which of the following spaces ( X, ) is a Banach space? Motivate your answers. (i) X= set of all continuous functions on [,1], u = 1 u(x) dx. (ii) X= set of all polynomials of degree 3, u = 1 u(x) dx. (iii) X= set of all convex functions on [,1], u = sup u(x). x [,1] (iv) X= set of all absolutely continuous functions on [, 1], u = 1 u() + u (x) dx. 2. On the open interval ] 1, 1[, consider the function u(x) = x 2 sin 1 x 4 x, u() =. - Does this function have a weak derivative on the open set = ], [? - Does this function have a weak derivative on the open set = ] 1,1[? (If yes, compute it. If no, explain why). 3. Let =]a,b[ be an open interval. Prove that (after a possible modification on a set of measure zero): (i) W 1, (]a,b[) is the space of all functions which are Lipschitz continuous on ]a,b[ (ii) W 1,p (]a,b[) is the space of absolutely continuous functions f : ]a,b[ IR with Df L p (]a,b[), (iii) For 1 p < one has W,p (]a,b[) = W,p (]a,b[) (iv) (extra credit) What is W, (]a,b[)? lim f(x) = lim f(x) = x a+ x b. = L p (]a,b[). 1
UNIVERSITETET I OSLO
NIVERSITETET I OSLO Det matematisk-naturvitenskapelige fakultet Examination in: Trial exam Partial differential equations and Sobolev spaces I. Day of examination: November 18. 2009. Examination hours:
More informationIntroduction to the Finite Element Method
Introduction to the Finite Element Method 09.06.2009 Outline Motivation Partial Differential Equations (PDEs) Finite Difference Method (FDM) Finite Element Method (FEM) References Motivation Figure: cross
More informationHøgskolen i Narvik Sivilingeniørutdanningen STE6237 ELEMENTMETODER. Oppgaver
Høgskolen i Narvik Sivilingeniørutdanningen STE637 ELEMENTMETODER Oppgaver Klasse: 4.ID, 4.IT Ekstern Professor: Gregory A. Chechkin e-mail: chechkin@mech.math.msu.su Narvik 6 PART I Task. Consider two-point
More informationExtremal equilibria for reaction diffusion equations in bounded domains and applications.
Extremal equilibria for reaction diffusion equations in bounded domains and applications. Aníbal Rodríguez-Bernal Alejandro Vidal-López Departamento de Matemática Aplicada Universidad Complutense de Madrid,
More informationCollege of the Holy Cross, Spring 2009 Math 373, Partial Differential Equations Midterm 1 Practice Questions
College of the Holy Cross, Spring 29 Math 373, Partial Differential Equations Midterm 1 Practice Questions 1. (a) Find a solution of u x + u y + u = xy. Hint: Try a polynomial of degree 2. Solution. Use
More informationA PRIORI ESTIMATES FOR SEMISTABLE SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS. In memory of Rou-Huai Wang
A PRIORI ESTIMATES FOR SEMISTABLE SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS XAVIER CABRÉ, MANEL SANCHÓN, AND JOEL SPRUCK In memory of Rou-Huai Wang 1. Introduction In this note we consider semistable
More informationBANACH AND HILBERT SPACE REVIEW
BANACH AND HILBET SPACE EVIEW CHISTOPHE HEIL These notes will briefly review some basic concepts related to the theory of Banach and Hilbert spaces. We are not trying to give a complete development, but
More informationInner Product Spaces
Math 571 Inner Product Spaces 1. Preliminaries An inner product space is a vector space V along with a function, called an inner product which associates each pair of vectors u, v with a scalar u, v, and
More information1 if 1 x 0 1 if 0 x 1
Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or
More information1 Norms and Vector Spaces
008.10.07.01 1 Norms and Vector Spaces Suppose we have a complex vector space V. A norm is a function f : V R which satisfies (i) f(x) 0 for all x V (ii) f(x + y) f(x) + f(y) for all x,y V (iii) f(λx)
More informationMetric Spaces. Chapter 7. 7.1. Metrics
Chapter 7 Metric Spaces A metric space is a set X that has a notion of the distance d(x, y) between every pair of points x, y X. The purpose of this chapter is to introduce metric spaces and give some
More informationSystems with Persistent Memory: the Observation Inequality Problems and Solutions
Chapter 6 Systems with Persistent Memory: the Observation Inequality Problems and Solutions Facts that are recalled in the problems wt) = ut) + 1 c A 1 s ] R c t s)) hws) + Ks r)wr)dr ds. 6.1) w = w +
More informationIntroduction to Algebraic Geometry. Bézout s Theorem and Inflection Points
Introduction to Algebraic Geometry Bézout s Theorem and Inflection Points 1. The resultant. Let K be a field. Then the polynomial ring K[x] is a unique factorisation domain (UFD). Another example of a
More informationA QUICK GUIDE TO THE FORMULAS OF MULTIVARIABLE CALCULUS
A QUIK GUIDE TO THE FOMULAS OF MULTIVAIABLE ALULUS ontents 1. Analytic Geometry 2 1.1. Definition of a Vector 2 1.2. Scalar Product 2 1.3. Properties of the Scalar Product 2 1.4. Length and Unit Vectors
More informationMATH 425, PRACTICE FINAL EXAM SOLUTIONS.
MATH 45, PRACTICE FINAL EXAM SOLUTIONS. Exercise. a Is the operator L defined on smooth functions of x, y by L u := u xx + cosu linear? b Does the answer change if we replace the operator L by the operator
More informationHøgskolen i Narvik Sivilingeniørutdanningen
Høgskolen i Narvik Sivilingeniørutdanningen Eksamen i Faget STE6237 ELEMENTMETODEN Klassen: 4.ID 4.IT Dato: 8.8.25 Tid: Kl. 9. 2. Tillatte hjelpemidler under eksamen: Kalkulator. Bok Numerical solution
More informationChapter 5. Banach Spaces
9 Chapter 5 Banach Spaces Many linear equations may be formulated in terms of a suitable linear operator acting on a Banach space. In this chapter, we study Banach spaces and linear operators acting on
More informationClass Meeting # 1: Introduction to PDEs
MATH 18.152 COURSE NOTES - CLASS MEETING # 1 18.152 Introduction to PDEs, Fall 2011 Professor: Jared Speck Class Meeting # 1: Introduction to PDEs 1. What is a PDE? We will be studying functions u = u(x
More informationDifferential Operators and their Adjoint Operators
Differential Operators and their Adjoint Operators Differential Operators inear functions from E n to E m may be described, once bases have been selected in both spaces ordinarily one uses the standard
More informationNumerical Methods for Differential Equations
Numerical Methods for Differential Equations Course objectives and preliminaries Gustaf Söderlind and Carmen Arévalo Numerical Analysis, Lund University Textbooks: A First Course in the Numerical Analysis
More informationSIXTY STUDY QUESTIONS TO THE COURSE NUMERISK BEHANDLING AV DIFFERENTIALEKVATIONER I
Lennart Edsberg, Nada, KTH Autumn 2008 SIXTY STUDY QUESTIONS TO THE COURSE NUMERISK BEHANDLING AV DIFFERENTIALEKVATIONER I Parameter values and functions occurring in the questions belowwill be exchanged
More information1. Let P be the space of all polynomials (of one real variable and with real coefficients) with the norm
Uppsala Universitet Matematiska Institutionen Andreas Strömbergsson Prov i matematik Funktionalanalys Kurs: F3B, F4Sy, NVP 005-06-15 Skrivtid: 9 14 Tillåtna hjälpmedel: Manuella skrivdon, Kreyszigs bok
More informationtegrals as General & Particular Solutions
tegrals as General & Particular Solutions dy dx = f(x) General Solution: y(x) = f(x) dx + C Particular Solution: dy dx = f(x), y(x 0) = y 0 Examples: 1) dy dx = (x 2)2 ;y(2) = 1; 2) dy ;y(0) = 0; 3) dx
More informationSolutions for Review Problems
olutions for Review Problems 1. Let be the triangle with vertices A (,, ), B (4,, 1) and C (,, 1). (a) Find the cosine of the angle BAC at vertex A. (b) Find the area of the triangle ABC. (c) Find a vector
More information3. INNER PRODUCT SPACES
. INNER PRODUCT SPACES.. Definition So far we have studied abstract vector spaces. These are a generalisation of the geometric spaces R and R. But these have more structure than just that of a vector space.
More informationThe Heat Equation. Lectures INF2320 p. 1/88
The Heat Equation Lectures INF232 p. 1/88 Lectures INF232 p. 2/88 The Heat Equation We study the heat equation: u t = u xx for x (,1), t >, (1) u(,t) = u(1,t) = for t >, (2) u(x,) = f(x) for x (,1), (3)
More informationNumerical Methods for Differential Equations
Numerical Methods for Differential Equations Chapter 1: Initial value problems in ODEs Gustaf Söderlind and Carmen Arévalo Numerical Analysis, Lund University Textbooks: A First Course in the Numerical
More informationPUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5.
PUTNAM TRAINING POLYNOMIALS (Last updated: November 17, 2015) Remark. This is a list of exercises on polynomials. Miguel A. Lerma Exercises 1. Find a polynomial with integral coefficients whose zeros include
More informationTHE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS
THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS KEITH CONRAD 1. Introduction The Fundamental Theorem of Algebra says every nonconstant polynomial with complex coefficients can be factored into linear
More informationCritical Thresholds in Euler-Poisson Equations. Shlomo Engelberg Jerusalem College of Technology Machon Lev
Critical Thresholds in Euler-Poisson Equations Shlomo Engelberg Jerusalem College of Technology Machon Lev 1 Publication Information This work was performed with Hailiang Liu & Eitan Tadmor. These results
More informationLecture 10. Finite difference and finite element methods. Option pricing Sensitivity analysis Numerical examples
Finite difference and finite element methods Lecture 10 Sensitivities and Greeks Key task in financial engineering: fast and accurate calculation of sensitivities of market models with respect to model
More informationOn computer algebra-aided stability analysis of dierence schemes generated by means of Gr obner bases
On computer algebra-aided stability analysis of dierence schemes generated by means of Gr obner bases Vladimir Gerdt 1 Yuri Blinkov 2 1 Laboratory of Information Technologies Joint Institute for Nuclear
More information0 <β 1 let u(x) u(y) kuk u := sup u(x) and [u] β := sup
456 BRUCE K. DRIVER 24. Hölder Spaces Notation 24.1. Let Ω be an open subset of R d,bc(ω) and BC( Ω) be the bounded continuous functions on Ω and Ω respectively. By identifying f BC( Ω) with f Ω BC(Ω),
More information8 Hyperbolic Systems of First-Order Equations
8 Hyperbolic Systems of First-Order Equations Ref: Evans, Sec 73 8 Definitions and Examples Let U : R n (, ) R m Let A i (x, t) beanm m matrix for i,,n Let F : R n (, ) R m Consider the system U t + A
More informationAn Introduction to Partial Differential Equations
An Introduction to Partial Differential Equations Andrew J. Bernoff LECTURE 2 Cooling of a Hot Bar: The Diffusion Equation 2.1. Outline of Lecture An Introduction to Heat Flow Derivation of the Diffusion
More informationEXISTENCE AND NON-EXISTENCE RESULTS FOR A NONLINEAR HEAT EQUATION
Sixth Mississippi State Conference on Differential Equations and Computational Simulations, Electronic Journal of Differential Equations, Conference 5 (7), pp. 5 65. ISSN: 7-669. UL: http://ejde.math.txstate.edu
More information(a) We have x = 3 + 2t, y = 2 t, z = 6 so solving for t we get the symmetric equations. x 3 2. = 2 y, z = 6. t 2 2t + 1 = 0,
Name: Solutions to Practice Final. Consider the line r(t) = 3 + t, t, 6. (a) Find symmetric equations for this line. (b) Find the point where the first line r(t) intersects the surface z = x + y. (a) We
More informationHow To Prove The Dirichlet Unit Theorem
Chapter 6 The Dirichlet Unit Theorem As usual, we will be working in the ring B of algebraic integers of a number field L. Two factorizations of an element of B are regarded as essentially the same if
More informationFIELDS-MITACS Conference. on the Mathematics of Medical Imaging. Photoacoustic and Thermoacoustic Tomography with a variable sound speed
FIELDS-MITACS Conference on the Mathematics of Medical Imaging Photoacoustic and Thermoacoustic Tomography with a variable sound speed Gunther Uhlmann UC Irvine & University of Washington Toronto, Canada,
More informationEXISTENCE OF SOLUTIONS TO NONLINEAR, SUBCRITICAL HIGHER-ORDER ELLIPTIC DIRICHLET PROBLEMS WOLFGANG REICHEL AND TOBIAS WETH
EXISTENCE OF SOLUTIONS TO NONLINEAR, SUBCRITICAL HIGHER-ORDER ELLIPTIC DIRICHLET PROBLEMS WOLFGANG REICHEL AND TOBIAS WETH Abstract. We consider the -th order elliptic boundary value problem Lu = f(x,
More informationVector and Matrix Norms
Chapter 1 Vector and Matrix Norms 11 Vector Spaces Let F be a field (such as the real numbers, R, or complex numbers, C) with elements called scalars A Vector Space, V, over the field F is a non-empty
More informationLimits and Continuity
Math 20C Multivariable Calculus Lecture Limits and Continuity Slide Review of Limit. Side limits and squeeze theorem. Continuous functions of 2,3 variables. Review: Limits Slide 2 Definition Given a function
More informationTheory of Sobolev Multipliers
Vladimir G. Maz'ya Tatyana O. Shaposhnikova Theory of Sobolev Multipliers With Applications to Differential and Integral Operators ^ Springer Introduction Part I Description and Properties of Multipliers
More informationChapter 17. Orthogonal Matrices and Symmetries of Space
Chapter 17. Orthogonal Matrices and Symmetries of Space Take a random matrix, say 1 3 A = 4 5 6, 7 8 9 and compare the lengths of e 1 and Ae 1. The vector e 1 has length 1, while Ae 1 = (1, 4, 7) has length
More informationb i (x) u x i + F (x, u),
Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 14, 1999, 275 293 TOPOLOGICAL DEGREE FOR A CLASS OF ELLIPTIC OPERATORS IN R n Cristelle Barillon Vitaly A. Volpert
More informationFinite Difference Approach to Option Pricing
Finite Difference Approach to Option Pricing February 998 CS5 Lab Note. Ordinary differential equation An ordinary differential equation, or ODE, is an equation of the form du = fut ( (), t) (.) dt where
More informationLinear Algebra Notes for Marsden and Tromba Vector Calculus
Linear Algebra Notes for Marsden and Tromba Vector Calculus n-dimensional Euclidean Space and Matrices Definition of n space As was learned in Math b, a point in Euclidean three space can be thought of
More informationLINEAR ALGEBRA. September 23, 2010
LINEAR ALGEBRA September 3, 00 Contents 0. LU-decomposition.................................... 0. Inverses and Transposes................................. 0.3 Column Spaces and NullSpaces.............................
More informationNotes on metric spaces
Notes on metric spaces 1 Introduction The purpose of these notes is to quickly review some of the basic concepts from Real Analysis, Metric Spaces and some related results that will be used in this course.
More informationMean value theorem, Taylors Theorem, Maxima and Minima.
MA 001 Preparatory Mathematics I. Complex numbers as ordered pairs. Argand s diagram. Triangle inequality. De Moivre s Theorem. Algebra: Quadratic equations and express-ions. Permutations and Combinations.
More informationWalrasian Demand. u(x) where B(p, w) = {x R n + : p x w}.
Walrasian Demand Econ 2100 Fall 2015 Lecture 5, September 16 Outline 1 Walrasian Demand 2 Properties of Walrasian Demand 3 An Optimization Recipe 4 First and Second Order Conditions Definition Walrasian
More informationAdvanced Microeconomics
Advanced Microeconomics Ordinal preference theory Harald Wiese University of Leipzig Harald Wiese (University of Leipzig) Advanced Microeconomics 1 / 68 Part A. Basic decision and preference theory 1 Decisions
More informationAB2.5: Surfaces and Surface Integrals. Divergence Theorem of Gauss
AB2.5: urfaces and urface Integrals. Divergence heorem of Gauss epresentations of surfaces or epresentation of a surface as projections on the xy- and xz-planes, etc. are For example, z = f(x, y), x =
More information7.2 Calculus of Variations
7 CALCULUS OF VARIATIONS 006 c Gilbert Strang 7 Calculus of Variations One theme of this book is the relation of equations to minimum principles To minimize P is to solve P = 0 There may be more to it,
More informationScalar Valued Functions of Several Variables; the Gradient Vector
Scalar Valued Functions of Several Variables; the Gradient Vector Scalar Valued Functions vector valued function of n variables: Let us consider a scalar (i.e., numerical, rather than y = φ(x = φ(x 1,
More informationRAJALAKSHMI ENGINEERING COLLEGE MA 2161 UNIT I - ORDINARY DIFFERENTIAL EQUATIONS PART A
RAJALAKSHMI ENGINEERING COLLEGE MA 26 UNIT I - ORDINARY DIFFERENTIAL EQUATIONS. Solve (D 2 + D 2)y = 0. 2. Solve (D 2 + 6D + 9)y = 0. PART A 3. Solve (D 4 + 4)x = 0 where D = d dt 4. Find Particular Integral:
More information1. Let X and Y be normed spaces and let T B(X, Y ).
Uppsala Universitet Matematiska Institutionen Andreas Strömbergsson Prov i matematik Funktionalanalys Kurs: NVP, Frist. 2005-03-14 Skrivtid: 9 11.30 Tillåtna hjälpmedel: Manuella skrivdon, Kreyszigs bok
More informationGeneral Theory of Differential Equations Sections 2.8, 3.1-3.2, 4.1
A B I L E N E C H R I S T I A N U N I V E R S I T Y Department of Mathematics General Theory of Differential Equations Sections 2.8, 3.1-3.2, 4.1 Dr. John Ehrke Department of Mathematics Fall 2012 Questions
More informationNonlinear Algebraic Equations Example
Nonlinear Algebraic Equations Example Continuous Stirred Tank Reactor (CSTR). Look for steady state concentrations & temperature. s r (in) p,i (in) i In: N spieces with concentrations c, heat capacities
More information7 Gaussian Elimination and LU Factorization
7 Gaussian Elimination and LU Factorization In this final section on matrix factorization methods for solving Ax = b we want to take a closer look at Gaussian elimination (probably the best known method
More informationBindel, Spring 2012 Intro to Scientific Computing (CS 3220) Week 3: Wednesday, Feb 8
Spaces and bases Week 3: Wednesday, Feb 8 I have two favorite vector spaces 1 : R n and the space P d of polynomials of degree at most d. For R n, we have a canonical basis: R n = span{e 1, e 2,..., e
More informationSOLUTIONS. f x = 6x 2 6xy 24x, f y = 3x 2 6y. To find the critical points, we solve
SOLUTIONS Problem. Find the critical points of the function f(x, y = 2x 3 3x 2 y 2x 2 3y 2 and determine their type i.e. local min/local max/saddle point. Are there any global min/max? Partial derivatives
More informationT ( a i x i ) = a i T (x i ).
Chapter 2 Defn 1. (p. 65) Let V and W be vector spaces (over F ). We call a function T : V W a linear transformation form V to W if, for all x, y V and c F, we have (a) T (x + y) = T (x) + T (y) and (b)
More informationRate of convergence towards Hartree dynamics
Rate of convergence towards Hartree dynamics Benjamin Schlein, LMU München and University of Cambridge Universitá di Milano Bicocca, October 22, 2007 Joint work with I. Rodnianski 1. Introduction boson
More informationr (t) = 2r(t) + sin t θ (t) = r(t) θ(t) + 1 = 1 1 θ(t) 1 9.4.4 Write the given system in matrix form x = Ax + f ( ) sin(t) x y 1 0 5 z = dy cos(t)
Solutions HW 9.4.2 Write the given system in matrix form x = Ax + f r (t) = 2r(t) + sin t θ (t) = r(t) θ(t) + We write this as ( ) r (t) θ (t) = ( ) ( ) 2 r(t) θ(t) + ( ) sin(t) 9.4.4 Write the given system
More informationMAT 200, Midterm Exam Solution. a. (5 points) Compute the determinant of the matrix A =
MAT 200, Midterm Exam Solution. (0 points total) a. (5 points) Compute the determinant of the matrix 2 2 0 A = 0 3 0 3 0 Answer: det A = 3. The most efficient way is to develop the determinant along the
More informationTaylor and Maclaurin Series
Taylor and Maclaurin Series In the preceding section we were able to find power series representations for a certain restricted class of functions. Here we investigate more general problems: Which functions
More information3 1. Note that all cubes solve it; therefore, there are no more
Math 13 Problem set 5 Artin 11.4.7 Factor the following polynomials into irreducible factors in Q[x]: (a) x 3 3x (b) x 3 3x + (c) x 9 6x 6 + 9x 3 3 Solution: The first two polynomials are cubics, so if
More informationPractice Final Math 122 Spring 12 Instructor: Jeff Lang
Practice Final Math Spring Instructor: Jeff Lang. Find the limit of the sequence a n = ln (n 5) ln (3n + 8). A) ln ( ) 3 B) ln C) ln ( ) 3 D) does not exist. Find the limit of the sequence a n = (ln n)6
More informationNumerical Verification of Optimality Conditions in Optimal Control Problems
Numerical Verification of Optimality Conditions in Optimal Control Problems Dissertation zur Erlangung des naturwissenschaftlichen Doktorgrades der Julius-Maximilians-Universität Würzburg vorgelegt von
More informationL 2 : x = s + 1, y = s, z = 4s + 4. 3. Suppose that C has coordinates (x, y, z). Then from the vector equality AC = BD, one has
The line L through the points A and B is parallel to the vector AB = 3, 2, and has parametric equations x = 3t + 2, y = 2t +, z = t Therefore, the intersection point of the line with the plane should satisfy:
More informationMATH 304 Linear Algebra Lecture 20: Inner product spaces. Orthogonal sets.
MATH 304 Linear Algebra Lecture 20: Inner product spaces. Orthogonal sets. Norm The notion of norm generalizes the notion of length of a vector in R n. Definition. Let V be a vector space. A function α
More informationLINEAR ALGEBRA W W L CHEN
LINEAR ALGEBRA W W L CHEN c W W L Chen, 1997, 2008 This chapter is available free to all individuals, on understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied,
More informationDifferentiation and Integration
This material is a supplement to Appendix G of Stewart. You should read the appendix, except the last section on complex exponentials, before this material. Differentiation and Integration Suppose we have
More informationChapter 7 Nonlinear Systems
Chapter 7 Nonlinear Systems Nonlinear systems in R n : X = B x. x n X = F (t; X) F (t; x ; :::; x n ) B C A ; F (t; X) =. F n (t; x ; :::; x n ) When F (t; X) = F (X) is independent of t; it is an example
More informationTwo Topics in Parametric Integration Applied to Stochastic Simulation in Industrial Engineering
Two Topics in Parametric Integration Applied to Stochastic Simulation in Industrial Engineering Department of Industrial Engineering and Management Sciences Northwestern University September 15th, 2014
More informationMathematical Methods of Engineering Analysis
Mathematical Methods of Engineering Analysis Erhan Çinlar Robert J. Vanderbei February 2, 2000 Contents Sets and Functions 1 1 Sets................................... 1 Subsets.............................
More informationFixed Point Theorems
Fixed Point Theorems Definition: Let X be a set and let T : X X be a function that maps X into itself. (Such a function is often called an operator, a transformation, or a transform on X, and the notation
More informationMetric Spaces. Chapter 1
Chapter 1 Metric Spaces Many of the arguments you have seen in several variable calculus are almost identical to the corresponding arguments in one variable calculus, especially arguments concerning convergence
More informationNOTES ON LINEAR TRANSFORMATIONS
NOTES ON LINEAR TRANSFORMATIONS Definition 1. Let V and W be vector spaces. A function T : V W is a linear transformation from V to W if the following two properties hold. i T v + v = T v + T v for all
More informationRANDOM INTERVAL HOMEOMORPHISMS. MICHA L MISIUREWICZ Indiana University Purdue University Indianapolis
RANDOM INTERVAL HOMEOMORPHISMS MICHA L MISIUREWICZ Indiana University Purdue University Indianapolis This is a joint work with Lluís Alsedà Motivation: A talk by Yulij Ilyashenko. Two interval maps, applied
More informationMoving Least Squares Approximation
Chapter 7 Moving Least Squares Approimation An alternative to radial basis function interpolation and approimation is the so-called moving least squares method. As we will see below, in this method the
More informationAdding vectors We can do arithmetic with vectors. We ll start with vector addition and related operations. Suppose you have two vectors
1 Chapter 13. VECTORS IN THREE DIMENSIONAL SPACE Let s begin with some names and notation for things: R is the set (collection) of real numbers. We write x R to mean that x is a real number. A real number
More information4.5 Linear Dependence and Linear Independence
4.5 Linear Dependence and Linear Independence 267 32. {v 1, v 2 }, where v 1, v 2 are collinear vectors in R 3. 33. Prove that if S and S are subsets of a vector space V such that S is a subset of S, then
More informationEECS 556 Image Processing W 09. Interpolation. Interpolation techniques B splines
EECS 556 Image Processing W 09 Interpolation Interpolation techniques B splines What is image processing? Image processing is the application of 2D signal processing methods to images Image representation
More informationA Transmission Problem for Euler-Bernoulli beam with Kelvin-Voigt. Damping
Applied Mathematics & Information Sciences 5(1) (211), 17-28 An International Journal c 211 NSP A Transmission Problem for Euler-Bernoulli beam with Kelvin-Voigt Damping C. A Raposo 1, W. D. Bastos 2 and
More informationCITY UNIVERSITY LONDON. BEng Degree in Computer Systems Engineering Part II BSc Degree in Computer Systems Engineering Part III PART 2 EXAMINATION
No: CITY UNIVERSITY LONDON BEng Degree in Computer Systems Engineering Part II BSc Degree in Computer Systems Engineering Part III PART 2 EXAMINATION ENGINEERING MATHEMATICS 2 (resit) EX2005 Date: August
More information4 Lyapunov Stability Theory
4 Lyapunov Stability Theory In this section we review the tools of Lyapunov stability theory. These tools will be used in the next section to analyze the stability properties of a robot controller. We
More informationt := maxγ ν subject to ν {0,1,2,...} and f(x c +γ ν d) f(x c )+cγ ν f (x c ;d).
1. Line Search Methods Let f : R n R be given and suppose that x c is our current best estimate of a solution to P min x R nf(x). A standard method for improving the estimate x c is to choose a direction
More informationModel order reduction via Proper Orthogonal Decomposition
Model order reduction via Proper Orthogonal Decomposition Reduced Basis Summer School 2015 Martin Gubisch University of Konstanz September 17, 2015 Martin Gubisch (University of Konstanz) Model order reduction
More informationMathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
More informationSolutions to old Exam 1 problems
Solutions to old Exam 1 problems Hi students! I am putting this old version of my review for the first midterm review, place and time to be announced. Check for updates on the web site as to which sections
More informationTOPIC 4: DERIVATIVES
TOPIC 4: DERIVATIVES 1. The derivative of a function. Differentiation rules 1.1. The slope of a curve. The slope of a curve at a point P is a measure of the steepness of the curve. If Q is a point on the
More informationReference: Introduction to Partial Differential Equations by G. Folland, 1995, Chap. 3.
5 Potential Theory Reference: Introduction to Partial Differential Equations by G. Folland, 995, Chap. 3. 5. Problems of Interest. In what follows, we consider Ω an open, bounded subset of R n with C 2
More informationFEGYVERNEKI SÁNDOR, PROBABILITY THEORY AND MATHEmATICAL
FEGYVERNEKI SÁNDOR, PROBABILITY THEORY AND MATHEmATICAL STATIsTICs 4 IV. RANDOm VECTORs 1. JOINTLY DIsTRIBUTED RANDOm VARIABLEs If are two rom variables defined on the same sample space we define the joint
More informationThe Method of Partial Fractions Math 121 Calculus II Spring 2015
Rational functions. as The Method of Partial Fractions Math 11 Calculus II Spring 015 Recall that a rational function is a quotient of two polynomials such f(x) g(x) = 3x5 + x 3 + 16x x 60. The method
More informationRecall that two vectors in are perpendicular or orthogonal provided that their dot
Orthogonal Complements and Projections Recall that two vectors in are perpendicular or orthogonal provided that their dot product vanishes That is, if and only if Example 1 The vectors in are orthogonal
More information1 Lecture: Integration of rational functions by decomposition
Lecture: Integration of rational functions by decomposition into partial fractions Recognize and integrate basic rational functions, except when the denominator is a power of an irreducible quadratic.
More informationParabolic Equations. Chapter 5. Contents. 5.1.2 Well-Posed Initial-Boundary Value Problem. 5.1.3 Time Irreversibility of the Heat Equation
7 5.1 Definitions Properties Chapter 5 Parabolic Equations Note that we require the solution u(, t bounded in R n for all t. In particular we assume that the boundedness of the smooth function u at infinity
More information