v( x) u( y) dy for any r > 0, B r ( x) Ω, or equivalently u( w) ds for any r > 0, B r ( x) Ω, or ( not really) equivalently if v exists, v 0.

Size: px
Start display at page:

Download "v( x) u( y) dy for any r > 0, B r ( x) Ω, or equivalently u( w) ds for any r > 0, B r ( x) Ω, or ( not really) equivalently if v exists, v 0."

Transcription

1 Sep. 26 The Perron Method In this lecture we show that one can show existence of solutions using maximum principle alone.. The Perron method. Recall in the last lecture we have shown the existence of solutions to the Laplace equation u = 0 in Ω, u = g on Ω ( ) for the special case of Ω = B R by writing down the explicit formula for the solution. Unfortunately, for general Ω this is not possible. The Perron method show the existence of the general problem by combining the explicit formula and the maximum principle. First we recall the following property of subharmonic functions. Let Ω R n and f: Ω [, ) be upper semicontinuous in Ω and f. The function f is called subharmonic in Ω if for all Ω Ω, the following property holds: If u is harmonic in Ω, and f u on Ω, then f u in Ω. or equivalently, v( x) B r v( x) B r B r( x ) B r( x ) if v exists, v 0. u( y) dy for any r > 0, B r ( x) Ω, or equivalently u( w) ds for any r > 0, B r ( x) Ω, or ( not really) equivalently The idea of the Perron method can be best illustrated via the D example: solving u = 0 in ( 0, ) ; u( 0) = u 0, u( ) = u. ( 2) The solution is a straight line connecting ( 0, u 0 ) and (, u ). Now let U 0 be any function connecting these two points. And we modify U 0 as follows. Take any subinterval ( a, b), and replace U 0 ( a, b) by the solution of u = 0 in ( a, b) ; u( a) = U 0 ( a), u( b) = U 0 ( b). ( 3) Doing this again and again, we see that the sequence approaches the straight line, which is the solution we need. It is hard to make this argument rigorous for general U 0. But if we take U 0 to be subharmonic, or roughly speaking U 0 0, then an important observation is that every modification makes the function larger. On the other hand they are all bounded from above by max ( u 0, u ). Therefore there must be an upper bound and this upper bound is the solution. This argument can be generalized to higher dimensional problems. We see that we need to guarantee. Modifying any subharmonic function by patching it with harmonic functions; 2. The modified function is still subharmonic; 3. Subharmonic functions is upper-bounded by its boundary values. They are indeed true. Lemma. We have i. If v C 0 ( Ω) is subharmonic and B R ( y) Ω, then the harmonic replacement v of v, defiend by v( x) x Ω\B R ( y) v ( x) R 2 2 x y v( z) n ds z x B R ( y) d w d R z x B R( y) ( 4) is subharmonic in Ω ; Furthermore v v in Ω.

2 ii. Let v be subharmonic in Ω, if there is x 0 Ω with v( x 0 ) = sup Ω v( x), then v is constant. In particular, if v C 0 ( Ω), then v( x) max Ω v( y) for all x Ω. iii. If v,, v k are subharmonic, so is v max v,, v k }. Proof. i. It is clear that v v in Ω. Now we argue that v is be subharmonic. Take arbitrary Ω Ω, and let u be harmonic in Ω with v u on Ω. Since v v in Ω, v u on Ω. Since v is subharmonic, v u in the whole Ω ( this implies v u in Ω \B R ( y) ). In particular, v u on ( Ω B R ( y) ). Since both u and v are harmonic in Ω B R ( y), we have v u in there too. ii. Left as exercise. iii. Left as exercise. Now we turn to the existence of solutions to the problem u = 0 in Ω, u = g on Ω. ( 5) Let S g be the set of all subharmonic functions which are C 0 ( Ω) and are below g on the boundary. We claim that the upper bound is in fact the solution. First we show that the upper bound is harmonic. Theorem 2. Let then u is harmonic in Ω. u( x) sup v S g v( x), ( 6) Proof. We devide the argument into several steps.. u is well-defined. To see this, first note that v min Ω g is subharmonic, so S ϕ is not empty; Next note that any v S g, we have v( x) sup Ω g for any x Ω. Therefore the supreme is well-defined at every point. 2. Now consider any x Ω. Let v n be such that lim v n ( x) = u( x). By replacing v k by max v,, v k, inf Ω g}, we can assume that v k } is monotonically increasing and bounded from below. Denote by B R the ball B R ( x) Ω. Now let v k be the harmonic replacements of v k. It is easy to see that v k is also monotonically increasing. Since v k v k, we must have lim k v k( x) = u( x). ( 7) 3. We show that the limit v = lim k v k is harmonic in. To see this, note that v k v l is harmonic in B R for any k, l. Thus Harnack inequality implies the existence of C such that v k( y) v l( y) C v k( x) v l( x) ( 8) for any y. As a consequence, the convergence v k v is uniform in and therefore v must be harmonic. 4. We need to show that v = u in. Since u is the supreme, v u. Now if there is x such that u( x ) > v( x ), since u = sup v Sg v, there is ṽ subharmonic such that u( x ) > ṽ ( x ) > v( x ). Set w k = max v k, ṽ }. Then the harmonic replacements w k convergens to a harmonic function w in satisfying which implies v w u, ( 9) v( x 0 ) = w( x 0 ). ( 0). To see this, recall that v is harmonic when it has the mean value property, which is kept by uniform convergence.

3 On the other hand we have v( x 0 ) = u( x 0 ) which gives w( y) dy v( y) dy = v( x 0 ) = w( x 0 ) = w( y) dy. ( ) The only possibility is v w in. But this gives a contradiction as by assumption w( x ) ṽ ( x ) > v( x ). ( 2) Thus ends the proof. The next task is to show that u satisfies the boundary condition. That is lim u( x) = g( x 0). ( 3) x x 0 Ω We use the concept of a barrier. Recall that a function f is superharmonic if f is subharmonic. Note that we can replace all subharmonic in the above construction by superharmonic and obtain u( x) = inf v where the infimum is over all superharmonic functions whose boundary value g. Definition 3. a ) Let ξ Ω. A function β C 0 ( Ω) is called a barrier at ξ with respect to Ω if i. β > 0 in Ω\ ξ}, β( ξ) = 0. ii. β is superharmonic in Ω. b ) ξ Ω is called regular if there existes a barrier β at ξ with respect to Ω. Remark 4. If β is a local barrier at ξ with respect to U Ω, then for any B ρ ( ξ) U, m inf β x Ω \B ρ ( ξ) β U \ B ρ( ξ) min ( m, β) x Ω B ρ ( ξ) ( 4) is a barrier at ξ with respect to Ω. To see this, take any Ω Ω, and let u be harmonic in Ω with boundary value β. We need to show that u β in Ω. Since β m, u m in Ω. Next we show u β in Ω B ρ ( ξ). At the boundary, we have min ( m, β) β on Ω B ρ u = β m = inf β on B ρ Ω u β in Ω B ρ ( ξ). ( 5) U \ B ρ( ξ) Lemma 5. If ξ is a regular point of Ω and g is continuous at ξ, then u takes g as its boundary value at ξ, that is lim u( x) = g( ξ). ( 6) x ξ Proof. The idea is the construct super and subharmonic function from g and the barrier β, which gives lower and upper bounds of u. For any ε > 0, there is δ > 0 such that Now consider We have g( x) g( ξ) < ε for all x ξ < δ; c β( x) 2 M 2 sup ϕ for all x ξ δ. ( 7) Ω v g( ξ) + ε + c β( x), v g( ξ) ε c β( x). ( 8) v u v ( 9)

4 which gives the convergence estimate. Summarizing, we obtain Theorem 6. Let Ω R n be bounded. The Dirichlet problem u = 0 in Ω, u = g on Ω ( 20) is solvable for all continuous boundary values g if and only if all points ξ Ω are regular. Proof. We have already done the if part. For the only if part, we take g 0 be such that g( ξ) = 0 while g( x) > 0 for all other x Ω. The solution u 0 is a barrier due to strong maximum principle. 2. The alternating method of H. A. Schwartz. It is important from the numerical point of view and is the foundation of domain decomposition method. Theorem 7. Let Ω and Ω 2 be bounded domains all of whose boundary points are regular for the Dirichlet problem. Suppose that Ω Ω 2 φ and that Ω and Ω 2 are of class C in some neighborhood of Ω Ω 2, and that they intersect at a nonzero angle. Then the Dirichlet problem fo the Laplace equation on Ω Ω Ω 2 is solvable for any continuous boundary values. Proof. We denote γ = Ω Ω 2, γ 2 = Ω 2 Ω, Γ = Ω \ γ, Γ 2 = Ω 2 \ γ 2.. Let M sup Ω g and m inf Ω g. We solve the Dirichlet problem 2 and obtain u, defined on Ω. Next we solve u = 0 in Ω, u = u = 0 in Ω 2, u = and obtain u 2. This process can be continued to obtain u 3, u 4, Ω 2. g Γ M γ. ( 2 ) g Γ2 u γ 2. ( 22) with u 2 k + defined on Ω and u 2 k defined on 2. By strong maximum principle we have u 2 M and in particular u 2 u on γ 2. This implies u 2 u on Ω Ω 2. ( 23) Next comparing the boundary values of u, and u 3 and using maximum principle, we have Comparing the values of u 2 and u 3 on γ γ 2 we have Comparing the values of u 2 and u 4 on Ω 2 = Γ 2 γ 2 we obtain 3. In general, we have u 3 u on Ω. ( 24) u 3 u 2 on Ω Ω 2. ( 25) u 4 u 2 on Ω 2. ( 26) u u 3 on Ω ; u 2 u 4 on Ω 2 ; u u 2 on Ω Ω 2. ( 27) Therefore there is u on Ω Ω 2 such that lim u2 u = k + lim u 2 k Ω. Ω 2 ( 28) 2. Note that the boundary value here may not be continuous. But it is still bounded and the Perron construction still works with convergence to boundary values only fail at discontinuous points.

5 Using Harnack inequality we can conclude that the convergence is locally uniform which means u is harmonic. 4. Next we show that u is continuous up to the boundary except for those points in Ω Ω 2. We can start from g Γ v = 0 in Ω, v = ( 29) m γ and obtain a increasing sequence v n }. It is clear that v n u n and therefore u is between each pair of ( v n, u n ). The continuity of v n, u n up to boundary then gives the continuity of u, except for those points in Ω Ω To show continuity at Ω Ω 2, one has to modify the iteration process. Instead of using M on γ to obtain u, we use a continuous extension of g. This way we can still obtain u n } but the monotonicity relation does not hold anymore. Instead, we show that ( u 2 n+ 3 u 2 n+ ) max γ u 2 n+ 2 u 2 n q <, on γ 2 ( 30) for a uniform q. This, combined with the maximum principle, implies u n+ u n converges to 0 as fast as the geometric series q n, which in turn gives uniform convergence of the sequence. For details of the last step, see 3. 3, 3. 4 of J. Jost Partial Differential Equations, GTM Further readings. J. Jost, Partial Differential Equations, Chap. 3. Exercise. Exercise. Let v be subharmonic in Ω, if there is x 0 Ω with v( x 0) = sup Ω v( x), then v is constant ( If you are not comfortable with upper semicontinuity, just assume v to be continuous). In particular, if v C 0 ( Ω), then v( x) max Ω v( y) for all x Ω. Exercise 2. Prove: If v,, v k are subharmonic in Ω, so is v max v,, v k }. 3. The key lemma is the following: There exists 0 < q < depending on Ω, Ω 2, such that if w: Ω and continuous on the closure Ω, with w = 0 on Γ, w on γ, then R is harmonic in Ω, A corresponding result holds if Ω is replaced by Ω 2. w q on γ 2. ( 3 )

u( x) = g( y) ds y ( 1 ) U solves u = 0 in U; u = 0 on U. ( 3)

u( x) = g( y) ds y ( 1 ) U solves u = 0 in U; u = 0 on U. ( 3) M ath 5 2 7 Fall 2 0 0 9 L ecture 4 ( S ep. 6, 2 0 0 9 ) Properties and Estimates of Laplace s and Poisson s Equations In our last lecture we derived the formulas for the solutions of Poisson s equation

More information

The Dirichlet s P rinciple. In this lecture we discuss an alternative formulation of the Dirichlet problem for the Laplace equation:

The Dirichlet s P rinciple. In this lecture we discuss an alternative formulation of the Dirichlet problem for the Laplace equation: Oct. 1 The Dirichlet s P rinciple In this lecture we discuss an alternative formulation of the Dirichlet problem for the Laplace equation: 1. Dirichlet s Principle. u = in, u = g on. ( 1 ) If we multiply

More information

Some lecture notes for Math 6050E: PDEs, Fall 2016

Some lecture notes for Math 6050E: PDEs, Fall 2016 Some lecture notes for Math 65E: PDEs, Fall 216 Tianling Jin December 1, 216 1 Variational methods We discuss an example of the use of variational methods in obtaining existence of solutions. Theorem 1.1.

More information

n 2 xi = x i. x i 2. r r ; i r 2 + V ( r) V ( r) = 0 r > 0. ( 1 1 ) a r n 1 ( 1 2) V( r) = b ln r + c n = 2 b r n 2 + c n 3 ( 1 3)

n 2 xi = x i. x i 2. r r ; i r 2 + V ( r) V ( r) = 0 r > 0. ( 1 1 ) a r n 1 ( 1 2) V( r) = b ln r + c n = 2 b r n 2 + c n 3 ( 1 3) Sep. 7 The L aplace/ P oisson Equations: Explicit Formulas In this lecture we study the properties of the Laplace equation and the Poisson equation with Dirichlet boundary conditions through explicit representations

More information

S chauder Theory. x 2. = log( x 1 + x 2 ) + 1 ( x 1 + x 2 ) 2. ( 5) x 1 + x 2 x 1 + x 2. 2 = 2 x 1. x 1 x 2. 1 x 1.

S chauder Theory. x 2. = log( x 1 + x 2 ) + 1 ( x 1 + x 2 ) 2. ( 5) x 1 + x 2 x 1 + x 2. 2 = 2 x 1. x 1 x 2. 1 x 1. Sep. 1 9 Intuitively, the solution u to the Poisson equation S chauder Theory u = f 1 should have better regularity than the right hand side f. In particular one expects u to be twice more differentiable

More information

Laplace s Equation. Chapter Mean Value Formulas

Laplace s Equation. Chapter Mean Value Formulas Chapter 1 Laplace s Equation Let be an open set in R n. A function u C 2 () is called harmonic in if it satisfies Laplace s equation n (1.1) u := D ii u = 0 in. i=1 A function u C 2 () is called subharmonic

More information

VISCOSITY SOLUTIONS OF ELLIPTIC EQUATIONS

VISCOSITY SOLUTIONS OF ELLIPTIC EQUATIONS VISCOSITY SOLUTIONS OF ELLIPTIC EQUATIONS LUIS SILVESTRE These are the notes from the summer course given in the Second Chicago Summer School In Analysis, in June 2015. We introduce the notion of viscosity

More information

Perron method for the Dirichlet problem.

Perron method for the Dirichlet problem. Introduzione alle equazioni alle derivate parziali, Laurea Magistrale in Matematica Perron method for the Dirichlet problem. We approach the question of existence of solution u C (Ω) C(Ω) of the Dirichlet

More information

Elliptic PDEs of 2nd Order, Gilbarg and Trudinger

Elliptic PDEs of 2nd Order, Gilbarg and Trudinger Elliptic PDEs of 2nd Order, Gilbarg and Trudinger Chapter 2 Laplace Equation Yung-Hsiang Huang 207.07.07. Mimic the proof for Theorem 3.. 2. Proof. I think we should assume u C 2 (Ω Γ). Let W be an open

More information

Hartogs Theorem: separate analyticity implies joint Paul Garrett garrett/

Hartogs Theorem: separate analyticity implies joint Paul Garrett  garrett/ (February 9, 25) Hartogs Theorem: separate analyticity implies joint Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ (The present proof of this old result roughly follows the proof

More information

Math The Laplacian. 1 Green s Identities, Fundamental Solution

Math The Laplacian. 1 Green s Identities, Fundamental Solution Math. 209 The Laplacian Green s Identities, Fundamental Solution Let be a bounded open set in R n, n 2, with smooth boundary. The fact that the boundary is smooth means that at each point x the external

More information

C.7. Numerical series. Pag. 147 Proof of the converging criteria for series. Theorem 5.29 (Comparison test) Let a k and b k be positive-term series

C.7. Numerical series. Pag. 147 Proof of the converging criteria for series. Theorem 5.29 (Comparison test) Let a k and b k be positive-term series C.7 Numerical series Pag. 147 Proof of the converging criteria for series Theorem 5.29 (Comparison test) Let and be positive-term series such that 0, for any k 0. i) If the series converges, then also

More information

FUNCTIONAL ANALYSIS LECTURE NOTES: COMPACT SETS AND FINITE-DIMENSIONAL SPACES. 1. Compact Sets

FUNCTIONAL ANALYSIS LECTURE NOTES: COMPACT SETS AND FINITE-DIMENSIONAL SPACES. 1. Compact Sets FUNCTIONAL ANALYSIS LECTURE NOTES: COMPACT SETS AND FINITE-DIMENSIONAL SPACES CHRISTOPHER HEIL 1. Compact Sets Definition 1.1 (Compact and Totally Bounded Sets). Let X be a metric space, and let E X be

More information

Lemma 15.1 (Sign preservation Lemma). Suppose that f : E R is continuous at some a R.

Lemma 15.1 (Sign preservation Lemma). Suppose that f : E R is continuous at some a R. 15. Intermediate Value Theorem and Classification of discontinuities 15.1. Intermediate Value Theorem. Let us begin by recalling the definition of a function continuous at a point of its domain. Definition.

More information

We denote the space of distributions on Ω by D ( Ω) 2.

We denote the space of distributions on Ω by D ( Ω) 2. Sep. 1 0, 008 Distributions Distributions are generalized functions. Some familiarity with the theory of distributions helps understanding of various function spaces which play important roles in the study

More information

M17 MAT25-21 HOMEWORK 6

M17 MAT25-21 HOMEWORK 6 M17 MAT25-21 HOMEWORK 6 DUE 10:00AM WEDNESDAY SEPTEMBER 13TH 1. To Hand In Double Series. The exercises in this section will guide you to complete the proof of the following theorem: Theorem 1: Absolute

More information

Solutions to Problem Set 5 for , Fall 2007

Solutions to Problem Set 5 for , Fall 2007 Solutions to Problem Set 5 for 18.101, Fall 2007 1 Exercise 1 Solution For the counterexample, let us consider M = (0, + ) and let us take V = on M. x Let W be the vector field on M that is identically

More information

Economics 204 Summer/Fall 2011 Lecture 5 Friday July 29, 2011

Economics 204 Summer/Fall 2011 Lecture 5 Friday July 29, 2011 Economics 204 Summer/Fall 2011 Lecture 5 Friday July 29, 2011 Section 2.6 (cont.) Properties of Real Functions Here we first study properties of functions from R to R, making use of the additional structure

More information

LECTURE 15: COMPLETENESS AND CONVEXITY

LECTURE 15: COMPLETENESS AND CONVEXITY LECTURE 15: COMPLETENESS AND CONVEXITY 1. The Hopf-Rinow Theorem Recall that a Riemannian manifold (M, g) is called geodesically complete if the maximal defining interval of any geodesic is R. On the other

More information

Economics 204 Fall 2011 Problem Set 2 Suggested Solutions

Economics 204 Fall 2011 Problem Set 2 Suggested Solutions Economics 24 Fall 211 Problem Set 2 Suggested Solutions 1. Determine whether the following sets are open, closed, both or neither under the topology induced by the usual metric. (Hint: think about limit

More information

Proof. We indicate by α, β (finite or not) the end-points of I and call

Proof. We indicate by α, β (finite or not) the end-points of I and call C.6 Continuous functions Pag. 111 Proof of Corollary 4.25 Corollary 4.25 Let f be continuous on the interval I and suppose it admits non-zero its (finite or infinite) that are different in sign for x tending

More information

Continuity. Matt Rosenzweig

Continuity. Matt Rosenzweig Continuity Matt Rosenzweig Contents 1 Continuity 1 1.1 Rudin Chapter 4 Exercises........................................ 1 1.1.1 Exercise 1............................................. 1 1.1.2 Exercise

More information

Metric Spaces and Topology

Metric Spaces and Topology Chapter 2 Metric Spaces and Topology From an engineering perspective, the most important way to construct a topology on a set is to define the topology in terms of a metric on the set. This approach underlies

More information

Solution. 1 Solution of Homework 7. Sangchul Lee. March 22, Problem 1.1

Solution. 1 Solution of Homework 7. Sangchul Lee. March 22, Problem 1.1 Solution Sangchul Lee March, 018 1 Solution of Homework 7 Problem 1.1 For a given k N, Consider two sequences (a n ) and (b n,k ) in R. Suppose that a n b n,k for all n,k N Show that limsup a n B k :=

More information

106 CHAPTER 3. TOPOLOGY OF THE REAL LINE. 2. The set of limit points of a set S is denoted L (S)

106 CHAPTER 3. TOPOLOGY OF THE REAL LINE. 2. The set of limit points of a set S is denoted L (S) 106 CHAPTER 3. TOPOLOGY OF THE REAL LINE 3.3 Limit Points 3.3.1 Main Definitions Intuitively speaking, a limit point of a set S in a space X is a point of X which can be approximated by points of S other

More information

U e = E (U\E) e E e + U\E e. (1.6)

U e = E (U\E) e E e + U\E e. (1.6) 12 1 Lebesgue Measure 1.2 Lebesgue Measure In Section 1.1 we defined the exterior Lebesgue measure of every subset of R d. Unfortunately, a major disadvantage of exterior measure is that it does not satisfy

More information

1 Definition of the Riemann integral

1 Definition of the Riemann integral MAT337H1, Introduction to Real Analysis: notes on Riemann integration 1 Definition of the Riemann integral Definition 1.1. Let [a, b] R be a closed interval. A partition P of [a, b] is a finite set of

More information

NATIONAL UNIVERSITY OF SINGAPORE Department of Mathematics MA4247 Complex Analysis II Lecture Notes Part II

NATIONAL UNIVERSITY OF SINGAPORE Department of Mathematics MA4247 Complex Analysis II Lecture Notes Part II NATIONAL UNIVERSITY OF SINGAPORE Department of Mathematics MA4247 Complex Analysis II Lecture Notes Part II Chapter 2 Further properties of analytic functions 21 Local/Global behavior of analytic functions;

More information

Math 421, Homework #6 Solutions. (1) Let E R n Show that = (E c ) o, i.e. the complement of the closure is the interior of the complement.

Math 421, Homework #6 Solutions. (1) Let E R n Show that = (E c ) o, i.e. the complement of the closure is the interior of the complement. Math 421, Homework #6 Solutions (1) Let E R n Show that (Ē) c = (E c ) o, i.e. the complement of the closure is the interior of the complement. 1 Proof. Before giving the proof we recall characterizations

More information

POTENTIAL THEORY OF QUASIMINIMIZERS

POTENTIAL THEORY OF QUASIMINIMIZERS Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 28, 2003, 459 490 POTENTIAL THEORY OF QUASIMINIMIZERS Juha Kinnunen and Olli Martio Institute of Mathematics, P.O. Box 1100 FI-02015 Helsinki University

More information

Homework 4, 5, 6 Solutions. > 0, and so a n 0 = n + 1 n = ( n+1 n)( n+1+ n) 1 if n is odd 1/n if n is even diverges.

Homework 4, 5, 6 Solutions. > 0, and so a n 0 = n + 1 n = ( n+1 n)( n+1+ n) 1 if n is odd 1/n if n is even diverges. 2..2(a) lim a n = 0. Homework 4, 5, 6 Solutions Proof. Let ɛ > 0. Then for n n = 2+ 2ɛ we have 2n 3 4+ ɛ 3 > ɛ > 0, so 0 < 2n 3 < ɛ, and thus a n 0 = 2n 3 < ɛ. 2..2(g) lim ( n + n) = 0. Proof. Let ɛ >

More information

Lecture 3. Econ August 12

Lecture 3. Econ August 12 Lecture 3 Econ 2001 2015 August 12 Lecture 3 Outline 1 Metric and Metric Spaces 2 Norm and Normed Spaces 3 Sequences and Subsequences 4 Convergence 5 Monotone and Bounded Sequences Announcements: - Friday

More information

The uniformization theorem

The uniformization theorem 1 The uniformization theorem Thurston s basic insight in all four of the theorems discussed in this book is that either the topology of the problem induces an appropriate geometry or there is an understandable

More information

Solution of the 7 th Homework

Solution of the 7 th Homework Solution of the 7 th Homework Sangchul Lee December 3, 2014 1 Preliminary In this section we deal with some facts that are relevant to our problems but can be coped with only previous materials. 1.1 Maximum

More information

2. Introduction to commutative rings (continued)

2. Introduction to commutative rings (continued) 2. Introduction to commutative rings (continued) 2.1. New examples of commutative rings. Recall that in the first lecture we defined the notions of commutative rings and field and gave some examples of

More information

Set, functions and Euclidean space. Seungjin Han

Set, functions and Euclidean space. Seungjin Han Set, functions and Euclidean space Seungjin Han September, 2018 1 Some Basics LOGIC A is necessary for B : If B holds, then A holds. B A A B is the contraposition of B A. A is sufficient for B: If A holds,

More information

Course 212: Academic Year Section 1: Metric Spaces

Course 212: Academic Year Section 1: Metric Spaces Course 212: Academic Year 1991-2 Section 1: Metric Spaces D. R. Wilkins Contents 1 Metric Spaces 3 1.1 Distance Functions and Metric Spaces............. 3 1.2 Convergence and Continuity in Metric Spaces.........

More information

X.9 Revisited, Cauchy s Formula for Contours

X.9 Revisited, Cauchy s Formula for Contours X.9 Revisited, Cauchy s Formula for Contours Let G C, G open. Let f be holomorphic on G. Let Γ be a simple contour with range(γ) G and int(γ) G. Then, for all z 0 int(γ), f (z 0 ) = 1 f (z) dz 2πi Γ z

More information

Harmonic Functions and Brownian motion

Harmonic Functions and Brownian motion Harmonic Functions and Brownian motion Steven P. Lalley April 25, 211 1 Dynkin s Formula Denote by W t = (W 1 t, W 2 t,..., W d t ) a standard d dimensional Wiener process on (Ω, F, P ), and let F = (F

More information

2. The Concept of Convergence: Ultrafilters and Nets

2. The Concept of Convergence: Ultrafilters and Nets 2. The Concept of Convergence: Ultrafilters and Nets NOTE: AS OF 2008, SOME OF THIS STUFF IS A BIT OUT- DATED AND HAS A FEW TYPOS. I WILL REVISE THIS MATE- RIAL SOMETIME. In this lecture we discuss two

More information

Lecture 4: Completion of a Metric Space

Lecture 4: Completion of a Metric Space 15 Lecture 4: Completion of a Metric Space Closure vs. Completeness. Recall the statement of Lemma??(b): A subspace M of a metric space X is closed if and only if every convergent sequence {x n } X satisfying

More information

Consequences of Continuity

Consequences of Continuity Consequences of Continuity James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University October 4, 2017 Outline 1 Domains of Continuous Functions 2 The

More information

FIRST YEAR CALCULUS W W L CHEN

FIRST YEAR CALCULUS W W L CHEN FIRST YER CLCULUS W W L CHEN c W W L Chen, 994, 28. This chapter is available free to all individuals, on the understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied,

More information

MATH 51H Section 4. October 16, Recall what it means for a function between metric spaces to be continuous:

MATH 51H Section 4. October 16, Recall what it means for a function between metric spaces to be continuous: MATH 51H Section 4 October 16, 2015 1 Continuity Recall what it means for a function between metric spaces to be continuous: Definition. Let (X, d X ), (Y, d Y ) be metric spaces. A function f : X Y is

More information

THE DIRICHLET PROBLEM

THE DIRICHLET PROBLEM THE DIRICHLET PROBLEM TSOGTGEREL GANTUMUR Abstract. We present here two approaches to the Dirichlet problem: The classical method of subharmonic functions that culminated in the works of Perron and Wiener,

More information

Real Analysis Notes. Thomas Goller

Real Analysis Notes. Thomas Goller Real Analysis Notes Thomas Goller September 4, 2011 Contents 1 Abstract Measure Spaces 2 1.1 Basic Definitions........................... 2 1.2 Measurable Functions........................ 2 1.3 Integration..............................

More information

V (v i + W i ) (v i + W i ) is path-connected and hence is connected.

V (v i + W i ) (v i + W i ) is path-connected and hence is connected. Math 396. Connectedness of hyperplane complements Note that the complement of a point in R is disconnected and the complement of a (translated) line in R 2 is disconnected. Quite generally, we claim that

More information

The Uniformization Theorem. Donald E. Marshall

The Uniformization Theorem. Donald E. Marshall The Uniformization Theorem Donald E. Marshall The Koebe uniformization theorem is a generalization of the Riemann mapping Theorem. It says that a simply connected Riemann surface is conformally equivalent

More information

MAT 257, Handout 13: December 5-7, 2011.

MAT 257, Handout 13: December 5-7, 2011. MAT 257, Handout 13: December 5-7, 2011. The Change of Variables Theorem. In these notes, I try to make more explicit some parts of Spivak s proof of the Change of Variable Theorem, and to supply most

More information

NOTE ON A REMARKABLE SUPERPOSITION FOR A NONLINEAR EQUATION. 1. Introduction. ρ(y)dy, ρ 0, x y

NOTE ON A REMARKABLE SUPERPOSITION FOR A NONLINEAR EQUATION. 1. Introduction. ρ(y)dy, ρ 0, x y NOTE ON A REMARKABLE SUPERPOSITION FOR A NONLINEAR EQUATION PETER LINDQVIST AND JUAN J. MANFREDI Abstract. We give a simple proof of - and extend - a superposition principle for the equation div( u p 2

More information

MAT 570 REAL ANALYSIS LECTURE NOTES. Contents. 1. Sets Functions Countability Axiom of choice Equivalence relations 9

MAT 570 REAL ANALYSIS LECTURE NOTES. Contents. 1. Sets Functions Countability Axiom of choice Equivalence relations 9 MAT 570 REAL ANALYSIS LECTURE NOTES PROFESSOR: JOHN QUIGG SEMESTER: FALL 204 Contents. Sets 2 2. Functions 5 3. Countability 7 4. Axiom of choice 8 5. Equivalence relations 9 6. Real numbers 9 7. Extended

More information

Division of the Humanities and Social Sciences. Supergradients. KC Border Fall 2001 v ::15.45

Division of the Humanities and Social Sciences. Supergradients. KC Border Fall 2001 v ::15.45 Division of the Humanities and Social Sciences Supergradients KC Border Fall 2001 1 The supergradient of a concave function There is a useful way to characterize the concavity of differentiable functions.

More information

Minimal Surface equations non-solvability strongly convex functional further regularity Consider minimal surface equation.

Minimal Surface equations non-solvability strongly convex functional further regularity Consider minimal surface equation. Lecture 7 Minimal Surface equations non-solvability strongly convex functional further regularity Consider minimal surface equation div + u = ϕ on ) = 0 in The solution is a critical point or the minimizer

More information

On the p-laplacian and p-fluids

On the p-laplacian and p-fluids LMU Munich, Germany Lars Diening On the p-laplacian and p-fluids Lars Diening On the p-laplacian and p-fluids 1/50 p-laplacian Part I p-laplace and basic properties Lars Diening On the p-laplacian and

More information

F (x) = P [X x[. DF1 F is nondecreasing. DF2 F is right-continuous

F (x) = P [X x[. DF1 F is nondecreasing. DF2 F is right-continuous 7: /4/ TOPIC Distribution functions their inverses This section develops properties of probability distribution functions their inverses Two main topics are the so-called probability integral transformation

More information

VISCOSITY SOLUTIONS. We follow Han and Lin, Elliptic Partial Differential Equations, 5.

VISCOSITY SOLUTIONS. We follow Han and Lin, Elliptic Partial Differential Equations, 5. VISCOSITY SOLUTIONS PETER HINTZ We follow Han and Lin, Elliptic Partial Differential Equations, 5. 1. Motivation Throughout, we will assume that Ω R n is a bounded and connected domain and that a ij C(Ω)

More information

1 Independent increments

1 Independent increments Tel Aviv University, 2008 Brownian motion 1 1 Independent increments 1a Three convolution semigroups........... 1 1b Independent increments.............. 2 1c Continuous time................... 3 1d Bad

More information

MAT1000 ASSIGNMENT 1. a k 3 k. x =

MAT1000 ASSIGNMENT 1. a k 3 k. x = MAT1000 ASSIGNMENT 1 VITALY KUZNETSOV Question 1 (Exercise 2 on page 37). Tne Cantor set C can also be described in terms of ternary expansions. (a) Every number in [0, 1] has a ternary expansion x = a

More information

NOTE ON A REMARKABLE SUPERPOSITION FOR A NONLINEAR EQUATION. 1. Introduction. ρ(y)dy, ρ 0, x y

NOTE ON A REMARKABLE SUPERPOSITION FOR A NONLINEAR EQUATION. 1. Introduction. ρ(y)dy, ρ 0, x y NOTE ON A REMARKABLE SUPERPOSITION FOR A NONLINEAR EQUATION PETER LINDQVIST AND JUAN J. MANFREDI Abstract. We give a simple proof of - and extend - a superposition principle for the equation div( u p 2

More information

6.2 Mean value theorem and maximum principles

6.2 Mean value theorem and maximum principles Hence g = u x iu y is analytic. Since is simply connected, R g(z) dz =forany closed path =) g has an integral function, G = g in. Call G = U + iv. For analytic functions d G = d G. Hence dz dx g = d dz

More information

Mid Term-1 : Practice problems

Mid Term-1 : Practice problems Mid Term-1 : Practice problems These problems are meant only to provide practice; they do not necessarily reflect the difficulty level of the problems in the exam. The actual exam problems are likely to

More information

Part V. 17 Introduction: What are measures and why measurable sets. Lebesgue Integration Theory

Part V. 17 Introduction: What are measures and why measurable sets. Lebesgue Integration Theory Part V 7 Introduction: What are measures and why measurable sets Lebesgue Integration Theory Definition 7. (Preliminary). A measure on a set is a function :2 [ ] such that. () = 2. If { } = is a finite

More information

Some Background Material

Some Background Material Chapter 1 Some Background Material In the first chapter, we present a quick review of elementary - but important - material as a way of dipping our toes in the water. This chapter also introduces important

More information

1. Supremum and Infimum Remark: In this sections, all the subsets of R are assumed to be nonempty.

1. Supremum and Infimum Remark: In this sections, all the subsets of R are assumed to be nonempty. 1. Supremum and Infimum Remark: In this sections, all the subsets of R are assumed to be nonempty. Let E be a subset of R. We say that E is bounded above if there exists a real number U such that x U for

More information

MAS221 Analysis Semester Chapter 2 problems

MAS221 Analysis Semester Chapter 2 problems MAS221 Analysis Semester 1 2018-19 Chapter 2 problems 20. Consider the sequence (a n ), with general term a n = 1 + 3. Can you n guess the limit l of this sequence? (a) Verify that your guess is plausible

More information

General Theory of Large Deviations

General Theory of Large Deviations Chapter 30 General Theory of Large Deviations A family of random variables follows the large deviations principle if the probability of the variables falling into bad sets, representing large deviations

More information

EXISTENCE OF SOLUTIONS TO ASYMPTOTICALLY PERIODIC SCHRÖDINGER EQUATIONS

EXISTENCE OF SOLUTIONS TO ASYMPTOTICALLY PERIODIC SCHRÖDINGER EQUATIONS Electronic Journal of Differential Equations, Vol. 017 (017), No. 15, pp. 1 7. ISSN: 107-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EXISTENCE OF SOLUTIONS TO ASYMPTOTICALLY PERIODIC

More information

An introduction to Mathematical Theory of Control

An introduction to Mathematical Theory of Control An introduction to Mathematical Theory of Control Vasile Staicu University of Aveiro UNICA, May 2018 Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018

More information

= 2 x x 2 n It was introduced by P.-S.Laplace in a 1784 paper in connection with gravitational

= 2 x x 2 n It was introduced by P.-S.Laplace in a 1784 paper in connection with gravitational Lecture 1, 9/8/2010 The Laplace operator in R n is = 2 x 2 + + 2 1 x 2 n It was introduced by P.-S.Laplace in a 1784 paper in connection with gravitational potentials. Recall that if we have masses M 1,...,

More information

We will outline two proofs of the main theorem:

We will outline two proofs of the main theorem: Chapter 4 Higher Genus Surfaces 4.1 The Main Result We will outline two proofs of the main theorem: Theorem 4.1.1. Let Σ be a closed oriented surface of genus g > 1. Then every homotopy class of homeomorphisms

More information

The Inverse Function Theorem via Newton s Method. Michael Taylor

The Inverse Function Theorem via Newton s Method. Michael Taylor The Inverse Function Theorem via Newton s Method Michael Taylor We aim to prove the following result, known as the inverse function theorem. Theorem 1. Let F be a C k map (k 1) from a neighborhood of p

More information

4th Preparation Sheet - Solutions

4th Preparation Sheet - Solutions Prof. Dr. Rainer Dahlhaus Probability Theory Summer term 017 4th Preparation Sheet - Solutions Remark: Throughout the exercise sheet we use the two equivalent definitions of separability of a metric space

More information

MATH 117 LECTURE NOTES

MATH 117 LECTURE NOTES MATH 117 LECTURE NOTES XIN ZHOU Abstract. This is the set of lecture notes for Math 117 during Fall quarter of 2017 at UC Santa Barbara. The lectures follow closely the textbook [1]. Contents 1. The set

More information

MATH202 Introduction to Analysis (2007 Fall and 2008 Spring) Tutorial Note #7

MATH202 Introduction to Analysis (2007 Fall and 2008 Spring) Tutorial Note #7 MATH202 Introduction to Analysis (2007 Fall and 2008 Spring) Tutorial Note #7 Real Number Summary of terminology and theorems: Definition: (Supremum & infimum) A supremum (or least upper bound) of a non-empty

More information

On the upper bounds of Green potentials. Hiroaki Aikawa

On the upper bounds of Green potentials. Hiroaki Aikawa On the upper bounds of Green potentials Dedicated to Professor M. Nakai on the occasion of his 60th birthday Hiroaki Aikawa 1. Introduction Let D be a domain in R n (n 2) with the Green function G(x, y)

More information

Boot camp - Problem set

Boot camp - Problem set Boot camp - Problem set Luis Silvestre September 29, 2017 In the summer of 2017, I led an intensive study group with four undergraduate students at the University of Chicago (Matthew Correia, David Lind,

More information

Exercise 1. Let f be a nonnegative measurable function. Show that. where ϕ is taken over all simple functions with ϕ f. k 1.

Exercise 1. Let f be a nonnegative measurable function. Show that. where ϕ is taken over all simple functions with ϕ f. k 1. Real Variables, Fall 2014 Problem set 3 Solution suggestions xercise 1. Let f be a nonnegative measurable function. Show that f = sup ϕ, where ϕ is taken over all simple functions with ϕ f. For each n

More information

Lecture No 1 Introduction to Diffusion equations The heat equat

Lecture No 1 Introduction to Diffusion equations The heat equat Lecture No 1 Introduction to Diffusion equations The heat equation Columbia University IAS summer program June, 2009 Outline of the lectures We will discuss some basic models of diffusion equations and

More information

Real Analysis - Notes and After Notes Fall 2008

Real Analysis - Notes and After Notes Fall 2008 Real Analysis - Notes and After Notes Fall 2008 October 29, 2008 1 Introduction into proof August 20, 2008 First we will go through some simple proofs to learn how one writes a rigorous proof. Let start

More information

ξ,i = x nx i x 3 + δ ni + x n x = 0. x Dξ = x i ξ,i = x nx i x i x 3 Du = λ x λ 2 xh + x λ h Dξ,

ξ,i = x nx i x 3 + δ ni + x n x = 0. x Dξ = x i ξ,i = x nx i x i x 3 Du = λ x λ 2 xh + x λ h Dξ, 1 PDE, HW 3 solutions Problem 1. No. If a sequence of harmonic polynomials on [ 1,1] n converges uniformly to a limit f then f is harmonic. Problem 2. By definition U r U for every r >. Suppose w is a

More information

MA651 Topology. Lecture 10. Metric Spaces.

MA651 Topology. Lecture 10. Metric Spaces. MA65 Topology. Lecture 0. Metric Spaces. This text is based on the following books: Topology by James Dugundgji Fundamental concepts of topology by Peter O Neil Linear Algebra and Analysis by Marc Zamansky

More information

Iowa State University. Instructor: Alex Roitershtein Summer Homework #5. Solutions

Iowa State University. Instructor: Alex Roitershtein Summer Homework #5. Solutions Math 50 Iowa State University Introduction to Real Analysis Department of Mathematics Instructor: Alex Roitershtein Summer 205 Homework #5 Solutions. Let α and c be real numbers, c > 0, and f is defined

More information

CHAPTER 7. Connectedness

CHAPTER 7. Connectedness CHAPTER 7 Connectedness 7.1. Connected topological spaces Definition 7.1. A topological space (X, T X ) is said to be connected if there is no continuous surjection f : X {0, 1} where the two point set

More information

Growth Theorems and Harnack Inequality for Second Order Parabolic Equations

Growth Theorems and Harnack Inequality for Second Order Parabolic Equations This is an updated version of the paper published in: Contemporary Mathematics, Volume 277, 2001, pp. 87-112. Growth Theorems and Harnack Inequality for Second Order Parabolic Equations E. Ferretti and

More information

THE GREEN FUNCTION. Contents

THE GREEN FUNCTION. Contents THE GREEN FUNCTION CRISTIAN E. GUTIÉRREZ NOVEMBER 5, 203 Contents. Third Green s formula 2. The Green function 2.. Estimates of the Green function near the pole 2 2.2. Symmetry of the Green function 3

More information

Part 2 Continuous functions and their properties

Part 2 Continuous functions and their properties Part 2 Continuous functions and their properties 2.1 Definition Definition A function f is continuous at a R if, and only if, that is lim f (x) = f (a), x a ε > 0, δ > 0, x, x a < δ f (x) f (a) < ε. Notice

More information

BOUNDARY VALUE PROBLEMS ON A HALF SIERPINSKI GASKET

BOUNDARY VALUE PROBLEMS ON A HALF SIERPINSKI GASKET BOUNDARY VALUE PROBLEMS ON A HALF SIERPINSKI GASKET WEILIN LI AND ROBERT S. STRICHARTZ Abstract. We study boundary value problems for the Laplacian on a domain Ω consisting of the left half of the Sierpinski

More information

02. Measure and integral. 1. Borel-measurable functions and pointwise limits

02. Measure and integral. 1. Borel-measurable functions and pointwise limits (October 3, 2017) 02. Measure and integral Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ [This document is http://www.math.umn.edu/ garrett/m/real/notes 2017-18/02 measure and integral.pdf]

More information

2.2 Some Consequences of the Completeness Axiom

2.2 Some Consequences of the Completeness Axiom 60 CHAPTER 2. IMPORTANT PROPERTIES OF R 2.2 Some Consequences of the Completeness Axiom In this section, we use the fact that R is complete to establish some important results. First, we will prove that

More information

Mathematical Methods for Physics and Engineering

Mathematical Methods for Physics and Engineering Mathematical Methods for Physics and Engineering Lecture notes for PDEs Sergei V. Shabanov Department of Mathematics, University of Florida, Gainesville, FL 32611 USA CHAPTER 1 The integration theory

More information

Seminorms and locally convex spaces

Seminorms and locally convex spaces (April 23, 2014) Seminorms and locally convex spaces Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ [This document is http://www.math.umn.edu/ garrett/m/fun/notes 2012-13/07b seminorms.pdf]

More information

h(x) lim H(x) = lim Since h is nondecreasing then h(x) 0 for all x, and if h is discontinuous at a point x then H(x) > 0. Denote

h(x) lim H(x) = lim Since h is nondecreasing then h(x) 0 for all x, and if h is discontinuous at a point x then H(x) > 0. Denote Real Variables, Fall 4 Problem set 4 Solution suggestions Exercise. Let f be of bounded variation on [a, b]. Show that for each c (a, b), lim x c f(x) and lim x c f(x) exist. Prove that a monotone function

More information

II - REAL ANALYSIS. This property gives us a way to extend the notion of content to finite unions of rectangles: we define

II - REAL ANALYSIS. This property gives us a way to extend the notion of content to finite unions of rectangles: we define 1 Measures 1.1 Jordan content in R N II - REAL ANALYSIS Let I be an interval in R. Then its 1-content is defined as c 1 (I) := b a if I is bounded with endpoints a, b. If I is unbounded, we define c 1

More information

Foundations of Mathematical Analysis

Foundations of Mathematical Analysis Foundations of Mathematical Analysis Fabio Bagagiolo Dipartimento di Matematica, Università di Trento email:fabio.bagagiolo@unitn.it Contents 1 Introduction 2 2 Basic concepts in mathematical analysis

More information

Regularity Theory. Lihe Wang

Regularity Theory. Lihe Wang Regularity Theory Lihe Wang 2 Contents Schauder Estimates 5. The Maximum Principle Approach............... 7.2 Energy Method.......................... 3.3 Compactness Method...................... 7.4 Boundary

More information

A Unified Analysis of Nonconvex Optimization Duality and Penalty Methods with General Augmenting Functions

A Unified Analysis of Nonconvex Optimization Duality and Penalty Methods with General Augmenting Functions A Unified Analysis of Nonconvex Optimization Duality and Penalty Methods with General Augmenting Functions Angelia Nedić and Asuman Ozdaglar April 16, 2006 Abstract In this paper, we study a unifying framework

More information

Fuchsian groups. 2.1 Definitions and discreteness

Fuchsian groups. 2.1 Definitions and discreteness 2 Fuchsian groups In the previous chapter we introduced and studied the elements of Mob(H), which are the real Moebius transformations. In this chapter we focus the attention of special subgroups of this

More information

Mathematics for Economists

Mathematics for Economists Mathematics for Economists Victor Filipe Sao Paulo School of Economics FGV Metric Spaces: Basic Definitions Victor Filipe (EESP/FGV) Mathematics for Economists Jan.-Feb. 2017 1 / 34 Definitions and Examples

More information

The De Giorgi-Nash-Moser Estimates

The De Giorgi-Nash-Moser Estimates The De Giorgi-Nash-Moser Estimates We are going to discuss the the equation Lu D i (a ij (x)d j u) = 0 in B 4 R n. (1) The a ij, with i, j {1,..., n}, are functions on the ball B 4. Here and in the following

More information

Lecture notes on Ordinary Differential Equations. S. Sivaji Ganesh Department of Mathematics Indian Institute of Technology Bombay

Lecture notes on Ordinary Differential Equations. S. Sivaji Ganesh Department of Mathematics Indian Institute of Technology Bombay Lecture notes on Ordinary Differential Equations S. Ganesh Department of Mathematics Indian Institute of Technology Bombay May 20, 2016 ii IIT Bombay Contents I Ordinary Differential Equations 1 1 Initial

More information