# 2 A Model, Harmonic Map, Problem

Size: px
Start display at page:

## Transcription

1 ELLIPTIC SYSTEMS JOHN E. HUTCHINSON Department of Mathematics School of Mathematical Sciences, A.N.U. 1 Introduction Elliptic equations model the behaviour of scalar quantities u, such as temperature or gravitational potential, which are in an equilibrium situation subject to certain imposed boundary conditions. In his first four lectures, John Urbas discussed linear 1 elliptic equations. In his lectures on the minimal surface equation, Graham Williams discussed the minimal surface equation, a quasilinear 2 elliptic equation in divergence form. Neil Trudinger and Tim Cranny will discuss fully nonlinear 3 elliptic equations. Elliptic systems model vector-valued quantities in an equilibrium situation subject to certain imposed boundary conditions. Examples are a vectorfield describing the molecular orientation of a liquid crystal, and the displacement of an elastic body under an external force. Solutions of elliptic equations are typically as smooth as the data allows (e.g. are C if the given data is C ). Solutions of elliptic systems typically have singularities. We use as reference [G] the book Multiple Integrals in the Calculus of Variations by M. Giaquinta. 2 A Model, Harmonic Map, Problem Suppose Ω IR n is an elastic membrane, stretched via the function w over a part of the n-dimensional sphere S n IR n+1, where w is specified on the boundary Ω. As a simple approximation to the physical situation, we can regard w as a minimiser of the Dirichlet energy 1 Dw 2, 4 (1) 2 Ω amongst all maps w :Ω IR n+1 such that w =1, w Ω specified. 1 The unknown function u and its first and second derivatives occur linearly. The coefficients of u and its derivatives may be nonlinear, but usually smooth, functions of the domain variables x 1,...,x n. 2 Linear in the second derivatives of u, but not necessarily linear in u or its first derivatives. 3 Not even linear in the second derivatives of u. 4 Where Dw 2 = i,α D iw α 2. The 1 2 is merely a convenient normalisation constant.

2 w Ω R Rn n+1 S n ψ u = ψ o w R n A simpler related problem, without the constraint w = 1, is obtained as follows. Let ψ : S n IR n be stereographic projection from the north pole. If w[ω] avoids a neighbourhood of the south pole then u = ψ w solves the problem: Minimise E(u) = 1 a(u) Du 2, 2 Ω amongst all maps u:ω IR n such that u Ω specified. Here a(u) is a smooth positive function (which is determined 5 by ψ). We will consider this simpler problem Euler Lagrange System We now derive the Euler Lagrange system for minimisers of E. Arguing formally, if u is a minimiser of E (subject to fixing the boundary values of u), then for all φ Cc 1 (Ω; IR n ) 6 0 = d 1 a(u + tφ) D(u + tφ) 2 dt t=0 2 Ω = a(u)d i u α D i φ α + 1 Ω 2 D αa(u)φ α Du 2 = a(u)dudφ + B(u) Du 2 φ. Ω 5 a(u) = ψ 2, where ψ is the tangential gradient, defined in a natural manner. 6 C 1 c (Ω; IR n ) consists of all compactly supported C 1 functions φ:ω IR n.

3 We sum over repeated indices in the second line, and in the last line we repress the indices. If u satisfies the above integral equation for all φ Cc 1 (Ω; IR n ), we say that u is a weak solution of the system for α =1,...,n. 7 We abbreviate this to D i (a(u)d i u α )= 1 2 D αa(u) Du 2 (2) D(a(u)Du) =B(u) Du 2. (3) If u is C 1 then being weak solution is the same as satisfying (3) in the usual sense. Important features to note are the positivity of a(u), which makes the system elliptic, 8 and the quadratic nature of Du 2 on the right. 9 Solutions with Singularities In the theory of elliptic P.D.E s, you considered the class of W 1,2 (Ω) functions largely for technical reasons. 10 It was simple to show the existence of weak solutions in this class, and then one considered the question of regularity of solutions. In the vector-valued setting, solutions need not be smooth, and it becomes even more natural to work in the W 1,2 setting. Thus we define W 1,2 (Ω; IR N ) to be the class of functions u :Ω IR N such that each component function belongs to W 1,2 (Ω). Note that the energy E(u) is well defined for arbitrary functions u W 1,2 (Ω; IR n ). In particular, the function x/ x has partial derivatives which behave like 1/ x, and so x/ x W 1,2 (B 1 (0); IR n )ifn 3. But note that x/ x has a singularity at the origin. Let Ω = B 1 (0). The function w(x) =(x/ x, 0) maps B 1 (0) radially onto the equator of S n IR n+1. The function x/ x, and hence w, isaw 1,2 function if n 3. One can show that if n 7 then w 7 The fact that the number of dependent variables u 1,...,u n and the number of independent variables x 1,...,x n are the same is just a consequence of this particular problem. It is not the case in general. 8 More generally, if instead of a(u)d i u α D i φ α we had i=1,...,n A αβ ij D iu α D j φ β, then we α=1,...,n say the system is elliptic if A αβ ij ξα i ξβ j λ ξ 2 for some constant λ>0 and all ξ IR n+n. In many physical problems it is important to have a weaker form of ellipticity, namely A αβ ij ξ iη α ξ j η β λ ξ 2 for some constant λ>0 and all ξ IR n, η IR N. 9 An exponent less than two is easier to handle; an exponent greater than two is more difficult. But two is the natural exponent for many problems, as is the case here. 10 See also my lectures on measure theory.

4 has least energy amongst all functions mapping B 1 (0) onto the unit sphere and having the same boundary values as w. Similarly, if n 7, u = ψ w minimises E(u) in (2) amongst all maps having the same boundary values. In particular, u satisfies the system of equations (2), i.e. (3). If 3 n<7 then u is no longer a minimiser, but it still satisfies the system (3). If n =2 it turns out that solutions of (3), and in particular minimisers of E(u), are smooth. We have just noted that a solution of (3) may have a singularity. If u(x) is a solution, then clearly so is u(x a) for any a IR n. Since a sum of solutions is also solution, we obtain solutions with any finite number of singularities. In general, a solution of (3) is said to be stationary, or an equilibrium solution, for the energy E. Thus minimisers are solutions of the Euler Lagrange system, but not necessarily conversely. 11 Since the energy is the Dirichlet Energy (for w, and also for u if we choose the appropriate metric), stationary functions for this particular problem are called harmonic. 3 A Simpler Model Problem Our intention is to provide a reasonably complete analysis for solutions of systems of the form (3), but with zero right side. Thus we consider systems of the form D(a(u)Du) =0, (4) which may or may not be an Euler Lagrange system. Systems of the type (4) were the first type of nonlinear elliptic system to be analysed. (In the next Section we briefly remark on linear elliptic systems.) If the right side is nonzero, as in (3), then the problem is considerably more complicated. In particular, minimisers will have nicer properties than merely stationary solutions. See [G] for more details. We remark (4) may also have singular solutions. For example, x/ x is a weak solution of (4) if ( A αβ ij (u) =δ ij δ αβ + δ βj + 4 u j u β )( δ n 2 1+ u 2 αi + 4 u i u α ), n 2 1+ u 2 where n = N 3. See [G, p. 57]. Note that the A are C, in fact analytic. The system of equations in this case is an Euler Lagrange system for a certain energy functional. Moreover, for sufficiently large n, x/ x is the (unique) minimiser of this particular energy functional. 11 The analogy is that a function E defined on IR k can have equilibrium points which are not minimisers.

5 4 Linear Elliptic Systems For completeness, we briefly discuss linear elliptic systems. Suppose Ω IR n and u:ω IR N. We say u satisfies a linear elliptic system in integral form if for all φ Cc 1 (Ω; IR N ). condition Ω i=1,...,n α=1,...,n A αβ ij (x)d i u α D j φ β = 0 (5) The A αβ ij (x) are required to satisfy the ellipticity A αβ ij (x)ξ α i ξ β j λ ξ 2 for some constant λ>0 and all ξ IR nn. Note that the coefficients A αβ ij (x) depend only on x and not on u. The summation sign is usually dropped, and we even suppress all indices and write A(x)DuDφ =0. (6) The ellipticity condition is then written Ω Aξξ λ ξ 2. Assuming the A αβ ij (x) are bounded, it is straightforward to show by an approximation argument that we may take φ W 1,2 0 (Ω; IR N ) in (5). Recall that W 1,2 0 (Ω; IR N ) consists of those W 1,2 (Ω; IR N ) functions which are zero on Ω in a natural way. Motivated by integration by parts, we usually write the system as for β =1,...,N. This abbreviates to D j ( A αβ ij (x)d i u α) = 0 (7) D(A(x)Du) =0. (8) If u W 1,2 (Ω; IR N ) satisfies (5) (i.e. (6)) we say u is a weak solution of the system (7) (i.e. (8)). If A(x) and u are C 1, then it follows from integration by parts that a weak solution is a solution in the classical pointwise sense. The theory of linear elliptic systems is similar to the theory of linear equations. In particular, one obtains an analogous Schauder theory (for C k,α solutions) and Sobolev theory (for W k,2 solutions). 12 The main difference is that if the functions A αβ ij (x) are merely bounded, then there exist solutions with singularities. This is not the case for a single equation. See [G, p. 54] 12 Although the details can be considerably more complicated, at least when one considers other than second-order elliptic systems.

6 5 Regularity Results, Summary We now consider the question of partial regularity (i.e. smoothness) of solutions of (4). More precisely, suppose u W 1,2 (Ω; IR N ) and where 1. A(z) M... z IR N, 2. Aξξ λ ξ 2... ξ IR nn, where λ>0, 3. A C 0 (IR N ) is uniformly continuous. D(A(u)Du) =0, (9) More precisely, we are using an abbreviated notation as in the previous section. By u satisfying the system (9) we mean that the corresponding integral equations (as in (5) or (6) but with A αβ ij (u) instead of A αβ ij (x)), are satisfied for all test functions φ W 1,2 0 (Ω; IR N ) We will see that u C0 α (Ω 0 ) for some open Ω 0 Ω, where Ω \ Ω 0 is a set of dimension n 2 (in a sense to be explained later). If A is smoother than C 0, then u is correspondingly smoother in Ω 0. In particular, if A is C then u C0 (Ω 0 ). More can be proved. It is only necessary that A be continuous, not uniformly continuous. Moreover, Ω \ Ω 0 is in fact a set of dimension p for some p<n 2, and is empty if n =2. The idea of the proof is that if the graph of a solution u is sufficiently flat in the L 2 sense near x 0 Ω, then in fact u is smooth in a neighbourhood of x 0. We will see that the flatness condition holds at all except a small set of points. The key technical point in the proof is to consider the quantity U(x 0,R)= u (u) x0,r 2, for Ω. This measures the L 2 mean oscillation of u in. Here denotes the average, and is obtained by dividing by the volume ω n R n of. The quantity (u) x0,r is the average of u in and is given by (u) x0,r = u. We will see that if U(x 0,R) is sufficiently small then in fact U(x 0,r) approaches zero like a power of r. From this, one deduces the Hölder continuity of u in a neighbourhood of x 0. One also shows that except for a set of x 0 of dimension n 2, U(x 0,R) is indeed small for some R = R(x 0 ).

7 6 Some Important Preliminaries We discuss a number of fundamental results that are used in the proof of partial regularity. 6.1 Integral Characterisation of Hölder Continuity Theorem If Ω has Lipschitz boundary, then u C 0,α (Ω) 13 u (u) x0,r 2 cr n+2α for all Ω, and some constant c. Remark More precisely, if the integral condition holds, then the precise representative u of u, defined by u (x 0 ) = lim u, R 0 satisfies u C 0,α (Ω). Since u = u a.e., and changing u on a set of measure zero does not change the integral, this is the best one can expect. Proof: If u is Hölder continuous, the integral inequality is straightforward. For the other direction, one works from the definition of u, see [G; Ch. III,1]. 6.2 Energy (or Caccioppoli) Inequality Theorem If u is a solution of (9) and Ω, then B R/2 (x 0 ) Du 2 c R 2 u 2. Philosophy The important point here is that we are bounding the L 2 norm of the derivative of u in some ball in terms of the L 2 norm of u in a larger ball. Such an estimate is not true for arbitrary functions u, but it is typical of solutions of elliptic equations or systems that we can often bound integrals of higher derivatives in terms of integrals of lower derivatives, usually over a slightly larger set. 13 Suppose 0 <α 1. Then u C 0,α (Ω) u(x) u(y) M x y α for some M>0 and all x, y Ω. Note that if α>1 then the derivative of u would be everywhere zero, and so u is constant!

8 Conversely, bounding integrals of lower derivatives in terms of integrals of higher derivatives is something we can do for arbitrary functions, by means of Sobolev or Poincaré inequalities. In particular, note the Poincaré inequality u (u) x0,r 2 cr 2 Du 2. Proof: Since the proof is one of the simplest examples of a test function argument, we sketch it here. As is usual in P.D.E. s, in the following, c denotes a constant which may change from line to line. But it will only depend on the dimension and constants such as M and λ which appear at the beginning of Section 5. Let φ = η 2 u, where η is smooth, η 0, η =1onB R/2 (x 0 ), η = 0 outside, and Dη 3/R. Substituting this in the integral form of (9), 0 = ADuDφ = ADu(η 2 Du +2ηu Dη) Hence η 2 ADuDu= 2AηDηuDu. Hence λ η 2 Du 2 c ɛ η Dη u Du η 2 Du 2 + c(ɛ) Dη 2 u 2, by Young s inequality 14. Taking ɛ = λ/2, as required. B R/2 (x 0 ) Du 2 c R 2 u 2, 6.3 A Decay estimate for Solutions of Constant Coefficient Systems Theorem Suppose u satisfies (9) where the A are constant and Ω=B 1 (0) for simplicity of notation. Then for 0 <r 1, for some constant c. U(0,r) cr 2 U(0, 1) 14 See the last Section of my measure theory notes.

9 Proof: We may assume r 1/4, since if r>1/4 we can take c 4 n+2. Then r 2 U(0,r) = ωn 1 r 2 n u (u) r 2 B r(0) cr n Du 2 Poincaré s inequality B r(0) c sup Du 2 c c B r(0) B 1/2 (0) B 1 (0) Du 2 a standard elliptic estimate u (u) 1 2 by Caccioppoli s inequality The standard elliptic estimate above is that one can typically bound higher norms (here L ) of solutions and their derivatives in terms of lower norms (here L 2 ) over a larger domain. Caccioppoli s inequality is applied to the solution u (u) 1. This gives the result. 7 Outline of Proof of Partial Regularity Lemma Suppose u is a solution of (9). Then there exist constants ɛ > 0 and τ (0, 1) such that U(x 0,r)<ɛ implies U(x 0,τr)< 1 2 U(x 0,r). Proof: Suppose τ (0, 1) and the conclusion of the lemma is false for each ɛ>0 (the intention is to obtain a contradiction if τ is sufficiently small). Then there exist balls B rk (x k ) Ω such that U(x k,r k )=λ k 2 0 (10) but U(x k,τr k ) 1 2 λ k 2. (11) Rescale to the unit ball by setting for z B 1 (0), where a k =(u) xk,r k. v k (z) = u(x k+r k z) a k λ k

10 Then, using the integral form of (9), A(λ k v k + a k )Dv k Dφ =0 for all φ W 1,2 0 (B 1 (0); IR N ). Moreover, from (10) and (11), (v k ) 1 = 0 v k 2 = 1 B 1 v k (v k ) τ 2 1/2. B τ From Caccioppoli s inequality, B 1 Dv k 2 is bounded independently of k. This allows one to pass to a subsequence of the v k which converges weakly in W 1,2, strongly in L 2 and pointwise a.e., to some function v. Moreover, a k a, say. From this it is not difficult to show that v will satisfy the limit equation A(a) Dv Dφ =0 for all φ W 1,2 0 (B 1 (0); IR N ). From the decay estimate for constant coefficient equations, v (v) τ 2 <cτ 2 v (v) 1 2 =cτ 2. B τ B 1 On the other hand, v (v) τ 2 1 B τ 2, using continuity of the L 2 norm under L 2 convergence in both lines. This is a contradiction for sufficiently small τ. The Lemma is now used as follows. The inequality U(x 0,r) <ɛmust hold in an open subset of Ω. Moreover, it can be iterated to show ( ) 1 j U(x 0,τ j r)< U(x 0,r). 2 This, together with the integral characterisation of Hölder continuity and elementary arguments, shows u C 0,α (Ω 0 ) for some α. Higher smoothness follows by fairly standard iteration techniques (although C 0,α to C 1,α is not quite so standard). The estimate on the dimension of Ω \ Ω 0 follows from noting U(x 0,r) cr 2 n Du 2 B r(x 0 ) by Poincaré s inequality, and the fact (using a Vitali covering argument) that the right side approaches zero except on a set E with H n 2 (E) =0.

### 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.

### 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

### REGULARITY FOR INFINITY HARMONIC FUNCTIONS IN TWO DIMENSIONS

C,α REGULARITY FOR INFINITY HARMONIC FUNCTIONS IN TWO DIMENSIONS LAWRENCE C. EVANS AND OVIDIU SAVIN Abstract. We propose a new method for showing C,α regularity for solutions of the infinity Laplacian

### 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

### Sobolev Spaces. Chapter 10

Chapter 1 Sobolev Spaces We now define spaces H 1,p (R n ), known as Sobolev spaces. For u to belong to H 1,p (R n ), we require that u L p (R n ) and that u have weak derivatives of first order in L p

### 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)

### R. M. Brown. 29 March 2008 / Regional AMS meeting in Baton Rouge. Department of Mathematics University of Kentucky. The mixed problem.

mixed R. M. Department of Mathematics University of Kentucky 29 March 2008 / Regional AMS meeting in Baton Rouge Outline mixed 1 mixed 2 3 4 mixed We consider the mixed boundary value Lu = 0 u = f D u

### is a weak solution with the a ij,b i,c2 C 1 ( )

Thus @u @x i PDE 69 is a weak solution with the RHS @f @x i L. Thus u W 3, loc (). Iterating further, and using a generalized Sobolev imbedding gives that u is smooth. Theorem 3.33 (Local smoothness).

### Partial Differential Equations

Part II Partial Differential Equations Year 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2015 Paper 4, Section II 29E Partial Differential Equations 72 (a) Show that the Cauchy problem for u(x,

### JUHA KINNUNEN. Harmonic Analysis

JUHA KINNUNEN Harmonic Analysis Department of Mathematics and Systems Analysis, Aalto University 27 Contents Calderón-Zygmund decomposition. Dyadic subcubes of a cube.........................2 Dyadic cubes

### SYMMETRY RESULTS FOR PERTURBED PROBLEMS AND RELATED QUESTIONS. Massimo Grosi Filomena Pacella S. L. Yadava. 1. Introduction

Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 21, 2003, 211 226 SYMMETRY RESULTS FOR PERTURBED PROBLEMS AND RELATED QUESTIONS Massimo Grosi Filomena Pacella S.

### Partial regularity for fully nonlinear PDE

Partial regularity for fully nonlinear PDE Luis Silvestre University of Chicago Joint work with Scott Armstrong and Charles Smart Outline Introduction Intro Review of fully nonlinear elliptic PDE Our result

### 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

### Sobolev Spaces. Chapter Hölder spaces

Chapter 2 Sobolev Spaces Sobolev spaces turn out often to be the proper setting in which to apply ideas of functional analysis to get information concerning partial differential equations. Here, we collect

### 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

### TD M1 EDP 2018 no 2 Elliptic equations: regularity, maximum principle

TD M EDP 08 no Elliptic equations: regularity, maximum principle Estimates in the sup-norm I Let be an open bounded subset of R d of class C. Let A = (a ij ) be a symmetric matrix of functions of class

### Everywhere differentiability of infinity harmonic functions

Everywhere differentiability of infinity harmonic functions Lawrence C. Evans and Charles K. Smart Department of Mathematics University of California, Berkeley Abstract We show that an infinity harmonic

### Friedrich symmetric systems

viii CHAPTER 8 Friedrich symmetric systems In this chapter, we describe a theory due to Friedrich [13] for positive symmetric systems, which gives the existence and uniqueness of weak solutions of boundary

### The harmonic map flow

Chapter 2 The harmonic map flow 2.1 Definition of the flow The harmonic map flow was introduced by Eells-Sampson in 1964; their work could be considered the start of the field of geometric flows. The flow

### 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

### ERRATUM TO AFFINE MANIFOLDS, SYZ GEOMETRY AND THE Y VERTEX

ERRATUM TO AFFINE MANIFOLDS, SYZ GEOMETRY AND THE Y VERTEX JOHN LOFTIN, SHING-TUNG YAU, AND ERIC ZASLOW 1. Main result The purpose of this erratum is to correct an error in the proof of the main result

### MINIMAL GRAPHS PART I: EXISTENCE OF LIPSCHITZ WEAK SOLUTIONS TO THE DIRICHLET PROBLEM WITH C 2 BOUNDARY DATA

MINIMAL GRAPHS PART I: EXISTENCE OF LIPSCHITZ WEAK SOLUTIONS TO THE DIRICHLET PROBLEM WITH C 2 BOUNDARY DATA SPENCER HUGHES In these notes we prove that for any given smooth function on the boundary of

### A LOWER BOUND FOR THE GRADIENT OF -HARMONIC FUNCTIONS Edi Rosset. 1. Introduction. u xi u xj u xi x j

Electronic Journal of Differential Equations, Vol. 1996(1996) No. 0, pp. 1 7. ISSN 107-6691. URL: http://ejde.math.swt.edu (147.6.103.110) telnet (login: ejde), ftp, and gopher access: ejde.math.swt.edu

### P(E t, Ω)dt, (2) 4t has an advantage with respect. to the compactly supported mollifiers, i.e., the function W (t)f satisfies a semigroup law:

Introduction Functions of bounded variation, usually denoted by BV, have had and have an important role in several problems of calculus of variations. The main features that make BV functions suitable

### Geometric PDE and The Magic of Maximum Principles. Alexandrov s Theorem

Geometric PDE and The Magic of Maximum Principles Alexandrov s Theorem John McCuan December 12, 2013 Outline 1. Young s PDE 2. Geometric Interpretation 3. Maximum and Comparison Principles 4. Alexandrov

### SOLUTION OF POISSON S EQUATION. Contents

SOLUTION OF POISSON S EQUATION CRISTIAN E. GUTIÉRREZ OCTOBER 5, 2013 Contents 1. Differentiation under the integral sign 1 2. The Newtonian potential is C 1 2 3. The Newtonian potential from the 3rd Green

### Partial Differential Equations, 2nd Edition, L.C.Evans The Calculus of Variations

Partial Differential Equations, 2nd Edition, L.C.Evans Chapter 8 The Calculus of Variations Yung-Hsiang Huang 2018.03.25 Notation: denotes a bounded smooth, open subset of R n. All given functions are

### MATH 263: PROBLEM SET 1: BUNDLES, SHEAVES AND HODGE THEORY

MATH 263: PROBLEM SET 1: BUNDLES, SHEAVES AND HODGE THEORY 0.1. Vector Bundles and Connection 1-forms. Let E X be a complex vector bundle of rank r over a smooth manifold. Recall the following abstract

### CONVERGENCE THEORY. G. ALLAIRE CMAP, Ecole Polytechnique. 1. Maximum principle. 2. Oscillating test function. 3. Two-scale convergence

1 CONVERGENCE THEOR G. ALLAIRE CMAP, Ecole Polytechnique 1. Maximum principle 2. Oscillating test function 3. Two-scale convergence 4. Application to homogenization 5. General theory H-convergence) 6.

### Existence, stability and instability for Einstein-scalar field Lichnerowicz equations by Emmanuel Hebey

Existence, stability and instability for Einstein-scalar field Lichnerowicz equations by Emmanuel Hebey Joint works with Olivier Druet and with Frank Pacard and Dan Pollack Two hours lectures IAS, October

### EXPOSITORY NOTES ON DISTRIBUTION THEORY, FALL 2018

EXPOSITORY NOTES ON DISTRIBUTION THEORY, FALL 2018 While these notes are under construction, I expect there will be many typos. The main reference for this is volume 1 of Hörmander, The analysis of liner

### CAPACITIES ON METRIC SPACES

[June 8, 2001] CAPACITIES ON METRIC SPACES V. GOL DSHTEIN AND M. TROYANOV Abstract. We discuss the potential theory related to the variational capacity and the Sobolev capacity on metric measure spaces.

### 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

### POINTWISE BOUNDS ON QUASIMODES OF SEMICLASSICAL SCHRÖDINGER OPERATORS IN DIMENSION TWO

POINTWISE BOUNDS ON QUASIMODES OF SEMICLASSICAL SCHRÖDINGER OPERATORS IN DIMENSION TWO HART F. SMITH AND MACIEJ ZWORSKI Abstract. We prove optimal pointwise bounds on quasimodes of semiclassical Schrödinger

### The Dirichlet boundary problems for second order parabolic operators satisfying a Carleson condition

The Dirichlet boundary problems for second order parabolic operators satisfying a Carleson condition Sukjung Hwang CMAC, Yonsei University Collaboration with M. Dindos and M. Mitrea The 1st Meeting of

### Regularity for Poisson Equation

Regularity for Poisson Equation OcMountain Daylight Time. 4, 20 Intuitively, the solution u to the Poisson equation u= f () should have better regularity than the right hand side f. In particular one expects

### Nonlinear aspects of Calderón-Zygmund theory

Ancona, June 7 2011 Overture: The standard CZ theory Consider the model case u = f in R n Overture: The standard CZ theory Consider the model case u = f in R n Then f L q implies D 2 u L q 1 < q < with

### Partial Differential Equations, 2nd Edition, L.C.Evans Chapter 5 Sobolev Spaces

Partial Differential Equations, nd Edition, L.C.Evans Chapter 5 Sobolev Spaces Shih-Hsin Chen, Yung-Hsiang Huang 7.8.3 Abstract In these exercises always denote an open set of with smooth boundary. As

### Glimpses on functionals with general growth

Glimpses on functionals with general growth Lars Diening 1 Bianca Stroffolini 2 Anna Verde 2 1 Universität München, Germany 2 Università Federico II, Napoli Minicourse, Mathematical Institute Oxford, October

### Obstacle Problems Involving The Fractional Laplacian

Obstacle Problems Involving The Fractional Laplacian Donatella Danielli and Sandro Salsa January 27, 2017 1 Introduction Obstacle problems involving a fractional power of the Laplace operator appear in

### An introduction to Birkhoff normal form

An introduction to Birkhoff normal form Dario Bambusi Dipartimento di Matematica, Universitá di Milano via Saldini 50, 0133 Milano (Italy) 19.11.14 1 Introduction The aim of this note is to present an

### Optimal Transportation. Nonlinear Partial Differential Equations

Optimal Transportation and Nonlinear Partial Differential Equations Neil S. Trudinger Centre of Mathematics and its Applications Australian National University 26th Brazilian Mathematical Colloquium 2007

### Sobolev spaces. May 18

Sobolev spaces May 18 2015 1 Weak derivatives The purpose of these notes is to give a very basic introduction to Sobolev spaces. More extensive treatments can e.g. be found in the classical references

### 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

### LECTURE # 0 BASIC NOTATIONS AND CONCEPTS IN THE THEORY OF PARTIAL DIFFERENTIAL EQUATIONS (PDES)

LECTURE # 0 BASIC NOTATIONS AND CONCEPTS IN THE THEORY OF PARTIAL DIFFERENTIAL EQUATIONS (PDES) RAYTCHO LAZAROV 1 Notations and Basic Functional Spaces Scalar function in R d, d 1 will be denoted by u,

### ASYMPTOTIC BEHAVIOUR OF NONLINEAR ELLIPTIC SYSTEMS ON VARYING DOMAINS

ASYMPTOTIC BEHAVIOUR OF NONLINEAR ELLIPTIC SYSTEMS ON VARYING DOMAINS Juan CASADO DIAZ ( 1 ) Adriana GARRONI ( 2 ) Abstract We consider a monotone operator of the form Au = div(a(x, Du)), with R N and

### Centre for Mathematics and Its Applications The Australian National University Canberra, ACT 0200 Australia. 1. Introduction

ON LOCALLY CONVEX HYPERSURFACES WITH BOUNDARY Neil S. Trudinger Xu-Jia Wang Centre for Mathematics and Its Applications The Australian National University Canberra, ACT 0200 Australia Abstract. In this

### Asymptotic behavior of the degenerate p Laplacian equation on bounded domains

Asymptotic behavior of the degenerate p Laplacian equation on bounded domains Diana Stan Instituto de Ciencias Matematicas (CSIC), Madrid, Spain UAM, September 19, 2011 Diana Stan (ICMAT & UAM) Nonlinear

### 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

### BIHARMONIC WAVE MAPS INTO SPHERES

BIHARMONIC WAVE MAPS INTO SPHERES SEBASTIAN HERR, TOBIAS LAMM, AND ROLAND SCHNAUBELT Abstract. A global weak solution of the biharmonic wave map equation in the energy space for spherical targets is constructed.

### Asymptotic Behavior of Infinity Harmonic Functions Near an Isolated Singularity

Savin, O., and C. Wang. (2008) Asymptotic Behavior of Infinity Harmonic Functions, International Mathematics Research Notices, Vol. 2008, Article ID rnm163, 23 pages. doi:10.1093/imrn/rnm163 Asymptotic

### 16 1 Basic Facts from Functional Analysis and Banach Lattices

16 1 Basic Facts from Functional Analysis and Banach Lattices 1.2.3 Banach Steinhaus Theorem Another fundamental theorem of functional analysis is the Banach Steinhaus theorem, or the Uniform Boundedness

### Euler Equations: local existence

Euler Equations: local existence Mat 529, Lesson 2. 1 Active scalars formulation We start with a lemma. Lemma 1. Assume that w is a magnetization variable, i.e. t w + u w + ( u) w = 0. If u = Pw then u

### 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

### THE HARDY LITTLEWOOD MAXIMAL FUNCTION OF A SOBOLEV FUNCTION. Juha Kinnunen. 1 f(y) dy, B(x, r) B(x,r)

Appeared in Israel J. Math. 00 (997), 7 24 THE HARDY LITTLEWOOD MAXIMAL FUNCTION OF A SOBOLEV FUNCTION Juha Kinnunen Abstract. We prove that the Hardy Littlewood maximal operator is bounded in the Sobolev

### Fifth Abel Conference : Celebrating the Mathematical Impact of John F. Nash Jr. and Louis Nirenberg

Fifth Abel Conference : Celebrating the Mathematical Impact of John F. Nash Jr. and Louis Nirenberg Some analytic aspects of second order conformally invariant equations Yanyan Li Rutgers University November

### Applied Math Qualifying Exam 11 October Instructions: Work 2 out of 3 problems in each of the 3 parts for a total of 6 problems.

Printed Name: Signature: Applied Math Qualifying Exam 11 October 2014 Instructions: Work 2 out of 3 problems in each of the 3 parts for a total of 6 problems. 2 Part 1 (1) Let Ω be an open subset of R

### Boundedness, Harnack inequality and Hölder continuity for weak solutions

Boundedness, Harnack inequality and Hölder continuity for weak solutions Intoduction to PDE We describe results for weak solutions of elliptic equations with bounded coefficients in divergence-form. The

### u xx + u yy = 0. (5.1)

Chapter 5 Laplace Equation The following equation is called Laplace equation in two independent variables x, y: The non-homogeneous problem u xx + u yy =. (5.1) u xx + u yy = F, (5.) where F is a function

### Simple Examples on Rectangular Domains

84 Chapter 5 Simple Examples on Rectangular Domains In this chapter we consider simple elliptic boundary value problems in rectangular domains in R 2 or R 3 ; our prototype example is the Poisson equation

### MATH 426, TOPOLOGY. p 1.

MATH 426, TOPOLOGY THE p-norms In this document we assume an extended real line, where is an element greater than all real numbers; the interval notation [1, ] will be used to mean [1, ) { }. 1. THE p

### Poisson Equation on Closed Manifolds

Poisson Equation on Closed anifolds Andrew acdougall December 15, 2011 1 Introduction The purpose of this project is to investigate the poisson equation φ = ρ on closed manifolds (compact manifolds without

### You should be able to...

Lecture Outline Gradient Projection Algorithm Constant Step Length, Varying Step Length, Diminishing Step Length Complexity Issues Gradient Projection With Exploration Projection Solving QPs: active set

### Nonlocal self-improving properties

Nonlocal self-improving properties Tuomo Kuusi (Aalto University) July 23rd, 2015 Frontiers of Mathematics And Applications IV UIMP 2015 Part 1: The local setting Self-improving properties: Meyers estimate

### Behaviour of Lipschitz functions on negligible sets. Non-differentiability in R. Outline

Behaviour of Lipschitz functions on negligible sets G. Alberti 1 M. Csörnyei 2 D. Preiss 3 1 Università di Pisa 2 University College London 3 University of Warwick Lars Ahlfors Centennial Celebration Helsinki,

### Measure and Integration: Solutions of CW2

Measure and Integration: s of CW2 Fall 206 [G. Holzegel] December 9, 206 Problem of Sheet 5 a) Left (f n ) and (g n ) be sequences of integrable functions with f n (x) f (x) and g n (x) g (x) for almost

### Mathematical Research Letters 4, (1997) HARDY S INEQUALITIES FOR SOBOLEV FUNCTIONS. Juha Kinnunen and Olli Martio

Mathematical Research Letters 4, 489 500 1997) HARDY S INEQUALITIES FOR SOBOLEV FUNCTIONS Juha Kinnunen and Olli Martio Abstract. The fractional maximal function of the gradient gives a pointwise interpretation

### Finite-dimensional spaces. C n is the space of n-tuples x = (x 1,..., x n ) of complex numbers. It is a Hilbert space with the inner product

Chapter 4 Hilbert Spaces 4.1 Inner Product Spaces Inner Product Space. A complex vector space E is called an inner product space (or a pre-hilbert space, or a unitary space) if there is a mapping (, )

### GRAND SOBOLEV SPACES AND THEIR APPLICATIONS TO VARIATIONAL PROBLEMS

LE MATEMATICHE Vol. LI (1996) Fasc. II, pp. 335347 GRAND SOBOLEV SPACES AND THEIR APPLICATIONS TO VARIATIONAL PROBLEMS CARLO SBORDONE Dedicated to Professor Francesco Guglielmino on his 7th birthday W

### Regularity Theory a Fourth Order PDE with Delta Right Hand Side

Regularity Theory a Fourth Order PDE with Delta Right Hand Side Graham Hobbs Applied PDEs Seminar, 29th October 2013 Contents Problem and Weak Formulation Example - The Biharmonic Problem Regularity Theory

### LECTURE 1: SOURCES OF ERRORS MATHEMATICAL TOOLS A PRIORI ERROR ESTIMATES. Sergey Korotov,

LECTURE 1: SOURCES OF ERRORS MATHEMATICAL TOOLS A PRIORI ERROR ESTIMATES Sergey Korotov, Institute of Mathematics Helsinki University of Technology, Finland Academy of Finland 1 Main Problem in Mathematical

### Nonlinear Systems and Control Lecture # 12 Converse Lyapunov Functions & Time Varying Systems. p. 1/1

Nonlinear Systems and Control Lecture # 12 Converse Lyapunov Functions & Time Varying Systems p. 1/1 p. 2/1 Converse Lyapunov Theorem Exponential Stability Let x = 0 be an exponentially stable equilibrium

### The Dirichlet boundary problems for second order parabolic operators satisfying a Carleson condition

The Dirichlet boundary problems for second order parabolic operators satisfying a Martin Dindos Sukjung Hwang University of Edinburgh Satellite Conference in Harmonic Analysis Chosun University, Gwangju,

### Regularity of the p-poisson equation in the plane

Regularity of the p-poisson equation in the plane Erik Lindgren Peter Lindqvist Department of Mathematical Sciences Norwegian University of Science and Technology NO-7491 Trondheim, Norway Abstract We

### Second Order Elliptic PDE

Second Order Elliptic PDE T. Muthukumar tmk@iitk.ac.in December 16, 2014 Contents 1 A Quick Introduction to PDE 1 2 Classification of Second Order PDE 3 3 Linear Second Order Elliptic Operators 4 4 Periodic

### Introduction. Christophe Prange. February 9, This set of lectures is motivated by the following kind of phenomena:

Christophe Prange February 9, 206 This set of lectures is motivated by the following kind of phenomena: sin(x/ε) 0, while sin 2 (x/ε) /2. Therefore the weak limit of the product is in general different

### Notions such as convergent sequence and Cauchy sequence make sense for any metric space. Convergent Sequences are Cauchy

Banach Spaces These notes provide an introduction to Banach spaces, which are complete normed vector spaces. For the purposes of these notes, all vector spaces are assumed to be over the real numbers.

### Oblique derivative problems for elliptic and parabolic equations, Lecture II

of the for elliptic and parabolic equations, Lecture II Iowa State University July 22, 2011 of the 1 2 of the of the As a preliminary step in our further we now look at a special situation for elliptic.

### In this chapter we study elliptical PDEs. That is, PDEs of the form. 2 u = lots,

Chapter 8 Elliptic PDEs In this chapter we study elliptical PDEs. That is, PDEs of the form 2 u = lots, where lots means lower-order terms (u x, u y,..., u, f). Here are some ways to think about the physical

### NOTES ON SCHAUDER ESTIMATES. r 2 x y 2

NOTES ON SCHAUDER ESTIMATES CRISTIAN E GUTIÉRREZ JULY 26, 2005 Lemma 1 If u f in B r y), then ux) u + r2 x y 2 B r y) B r y) f, x B r y) Proof Let gx) = ux) Br y) u r2 x y 2 Br y) f We have g = u + Br

### Weak Convergence Methods for Energy Minimization

Weak Convergence Methods for Energy Minimization Bo Li Department of Mathematics University of California, San Diego E-mail: bli@math.ucsd.edu June 3, 2007 Introduction This compact set of notes present

### arxiv: v2 [math.ap] 28 Nov 2016

ONE-DIMENSIONAL SAIONARY MEAN-FIELD GAMES WIH LOCAL COUPLING DIOGO A. GOMES, LEVON NURBEKYAN, AND MARIANA PRAZERES arxiv:1611.8161v [math.ap] 8 Nov 16 Abstract. A standard assumption in mean-field game

### Boundary regularity of solutions of degenerate elliptic equations without boundary conditions

Boundary regularity of solutions of elliptic without boundary Iowa State University November 15, 2011 1 2 3 4 If the linear second order elliptic operator L is non with smooth coefficients, then, for any

### On the Intrinsic Differentiability Theorem of Gromov-Schoen

On the Intrinsic Differentiability Theorem of Gromov-Schoen Georgios Daskalopoulos Brown University daskal@math.brown.edu Chikako Mese 2 Johns Hopkins University cmese@math.jhu.edu Abstract In this note,

### 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

### A metric space X is a non-empty set endowed with a metric ρ (x, y):

Chapter 1 Preliminaries References: Troianiello, G.M., 1987, Elliptic differential equations and obstacle problems, Plenum Press, New York. Friedman, A., 1982, Variational principles and free-boundary

### Existence and Regularity of Stable Branched Minimal Hypersurfaces

Pure and Applied Mathematics Quarterly Volume 3, Number 2 (Special Issue: In honor of Leon Simon, Part 1 of 2 ) 569 594, 2007 Existence and Regularity of Stable Branched Minimal Hypersurfaces Neshan Wickramasekera

### ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS EXERCISES I (HARMONIC FUNCTIONS)

ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS EXERCISES I (HARMONIC FUNCTIONS) MATANIA BEN-ARTZI. BOOKS [CH] R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. Interscience Publ. 962. II, [E] L.

### 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

### Global minimization. Chapter Upper and lower bounds

Chapter 1 Global minimization The issues related to the behavior of global minimization problems along a sequence of functionals F are by now well understood, and mainly rely on the concept of -limit.

### REGULAR LAGRANGE MULTIPLIERS FOR CONTROL PROBLEMS WITH MIXED POINTWISE CONTROL-STATE CONSTRAINTS

REGULAR LAGRANGE MULTIPLIERS FOR CONTROL PROBLEMS WITH MIXED POINTWISE CONTROL-STATE CONSTRAINTS fredi tröltzsch 1 Abstract. A class of quadratic optimization problems in Hilbert spaces is considered,

### OPTIMAL POTENTIALS FOR SCHRÖDINGER OPERATORS. 1. Introduction In this paper we consider optimization problems of the form. min F (V ) : V V, (1.

OPTIMAL POTENTIALS FOR SCHRÖDINGER OPERATORS G. BUTTAZZO, A. GEROLIN, B. RUFFINI, AND B. VELICHKOV Abstract. We consider the Schrödinger operator + V (x) on H 0 (), where is a given domain of R d. Our

### AFFINE MAXIMAL HYPERSURFACES. Xu-Jia Wang. Centre for Mathematics and Its Applications The Australian National University

AFFINE MAXIMAL HYPERSURFACES Xu-Jia Wang Centre for Mathematics and Its Applications The Australian National University Abstract. This is a brief survey of recent works by Neil Trudinger and myself on

### Gradient Estimate of Mean Curvature Equations and Hessian Equations with Neumann Boundary Condition

of Mean Curvature Equations and Hessian Equations with Neumann Boundary Condition Xinan Ma NUS, Dec. 11, 2014 Four Kinds of Equations Laplace s equation: u = f(x); mean curvature equation: div( Du ) =

### Characterisation of Accumulation Points. Convergence in Metric Spaces. Characterisation of Closed Sets. Characterisation of Closed Sets

Convergence in Metric Spaces Functional Analysis Lecture 3: Convergence and Continuity in Metric Spaces Bengt Ove Turesson September 4, 2016 Suppose that (X, d) is a metric space. A sequence (x n ) X is

### ANALYTIC SEMIGROUPS AND APPLICATIONS. 1. Introduction

ANALYTIC SEMIGROUPS AND APPLICATIONS KELLER VANDEBOGERT. Introduction Consider a Banach space X and let f : D X and u : G X, where D and G are real intervals. A is a bounded or unbounded linear operator

### 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

### Some Remarks About the Density of Smooth Functions in Weighted Sobolev Spaces

Journal of Convex nalysis Volume 1 (1994), No. 2, 135 142 Some Remarks bout the Density of Smooth Functions in Weighted Sobolev Spaces Valeria Chiadò Piat Dipartimento di Matematica, Politecnico di Torino,