Stationary mean-field games Diogo A. Gomes
|
|
- Preston Rich
- 5 years ago
- Views:
Transcription
1 Stationary mean-field games Diogo A. Gomes
2 We consider is the periodic stationary MFG, { ɛ u + Du V (x) = g(m) + H ɛ m div(mdu) = 0, (1) where the unknowns are u : T d R, m : T d R, with m 0 and m = 1, and H R.
3 We suppose that V : T d R is C, g : R + R (or g : R + 0 R), C in the set m > 0, satisfying T d g(m) C T d mg(m). We say that (u, m, H) or (u, m) is a classical solution of, respectively, (1) if u and m are C, m > 0, (u, m) solves (1).
4 Maximum principle bounds Bounds for H Proposition Let u be a classical solution of (1). Suppose that g 0. Then, H sup T d V.
5 Maximum principle bounds Because u is periodic, it achieves a minimum at a point, x 0. At this point, Du(x 0 ) = 0 and u 0. Consequently, Hence, H sup V. V (x 0 ) H + g(m) H.
6 First-order estimates Proposition There exists a constant, C, such that, for any classical solution, (u, m, H), of (1), we have Td Du 2 2 (1 + m) + 1 g(m)mdx C. 2
7 First-order estimates Multiply the Hamilton-Jacobi equation by (m 1) and Fokker Planck equation by u, adding them, and integrating by parts gives Td Du 2 (1 + m) + mg(m)dx = V (m 1) + g(m)dx. 2 T d Using the assumption on g, we obtain the result.
8 First-order estimates Corollary Let (u, m, H) be a classical solution of (1). Suppose that g 0. Then, there exists a constant, C, not depending on the particular solution, such that H C.
9 First-order estimates We have: Du 2 2 L 1. From the assumptions and the preceding estimate, g(m) L 1. Therefore, integrating the Hamilton-Jacobi equation, we obtain the bound for H.
10 Integral Bernstein estimate Bernstein estimates Here, we examine the Hamilton-Jacobi equation, u(x) + Du(x) V (x) = H, with V L p. Our goal is to bound the norm of Du in L q for some q > 1.
11 Integral Bernstein estimate Lemma Let u C 3 and v = Du 2. Suppose that V C 1. Then, there exist, c, C > 0, which do not depend on u or V such that, for every p > 1, v p vdx T d 4pc ( (p + 1) 2 v T d ) d 2 (p+1)d ( d d 2 dx C v p+2 dx T d ) p+1 p+2 and 2 DV Du v p dx 1 D 2 u T d 2 T d 2 v p dx + C p T V d 2 v p dx.
12 Integral Bernstein estimate By integration by parts, we have the identity v p vdx = pv p 1 Dv 2 4p dx = T d T d (p + 1) 2 Dv p dx. T d Next, we use Sobolev s inequality to obtain Dv p dx+ T d T d v p+1 dx c moreover, from Young s inequality, v p+1 2 ( 2 = c 2 ( ) p+1 v p+1 dx v p+2 p+2 dx, T d T d v (p+1)d d 2 T d ) d 2 d dx,
13 Integral Bernstein estimate For the second inequality, we integrate again by parts to get DV Du v p dx = T d V u v p dx + p T d V v p 1 Dv Dudx. T d Next, we apply a weighted Cauchy inequality to each of the terms in the prior identity to get V u v p dx 1 D 2 u 2 v p dx + C V 2 v p dx. T d 8 T d T d Next, because v = Du 2, we have Dv = 2D 2 udu, hence p V v p 1 Dv Dudx 2p V v p D 2 u dx 1 D 2 u T d T d 8 T d 2 v p dx + C p T V d 2 v p dx. Using the two preceding bounds, we get the second estimate.
14 Integral Bernstein estimate Bernstein estimate Theorem Let u be C 3 and V C 1. Then, for any p > 1, there exists a constant, C p > 0, that depends only on H, such that ) Du 2d(p+1) C p (1 + V L d 2 (T d 2d(1+p). ) L d+2p (T d ) Note that γ p = 2d(1+p) d+2p d when p and that γ p is increasing when d > 2.
15 Integral Bernstein estimate We set v = Du 2. Differentiating the Hamilton-Jacobi equation Thus, v = 2 = 2 u xi = 1 2 v x i + V xi. d ( ) 2 d uxi x j 2 u xi u xi (2) i,j=1 i,j=1 i=1 d ( ) 2 d ( ) 1 uxi x j 2 u xi 2 v x i + V xi. i=1
16 Integral Bernstein estimate By multiplying (2) by v p and integrating over T d, we have v p vdx + 2 D 2 u 2 v p dx T d T d = Du Dv v p dx 2 DV Du v p dx. T d T d The Lemma provides bounds for the first term on the left-hand side and the last term on the right-hand side. For δ > 0, there exists a constant, C δ > 0, such that Du Dv v p dx δ D 2 u 2 v p dx + T d T d for every p > 1. C δ p + 1 T d v p+2 dx,
17 Integral Bernstein estimate Now, we claim that for any large enough p > 1, there exists C p > 0 that does not depend on u, such that ( ) (d 2) ( ) 1 v d(p+1) d(p+1) d 2 dx Cp V 2βp βp dx + Cp, (3) T d T d where β p is the conjugate exponent of d(p+1) (d 2)p. Further, β p d 2 when p.
18 Integral Bernstein estimate To prove the previous claim, we use the lemma to get c p ( v d(p+1) d 2 T d ) d 2 d dx + 2 D 2 u 2 v p dx T d ) p+1 c p v (T p+2 p+2 dx Du Dv v p dx 2 DV Du v p dx, d T d T d where c p := 4p C (p+1) 2, for some constant C.
19 Integral Bernstein estimate From the second estimate in the Lemma, and Young s inequality, (z p+1 p+2 ɛz + C p,ɛ ), we have c p ( v d(p+1) d 2 T d ) d 2 d dx C p T d V 2 v p dx + [ ( ( Cδ p δ )] 2 + δ D 2 u 2 v p dx T ) d v p+2 dx + C p,δ. T d
20 Integral Bernstein estimate The key idea is now to use the Hamilton-Jacobi equation again: D 2 u 2 v p dx 1 u 2 v p dx = 1 v T d d T d d T d 2 + V H 2 v p dx 1 v 2 v p dx 1 V 2 v p dx 1 3d T d d T d d C c v p+2 dx C V 2 v p dx C p, T d T d where the second inequality follows from (a b c) a2 b 2 c 2. v p dx
21 Integral Bernstein estimate For a small δ and a large enough p, the preceding inequalities give Hence, ( ) d 2 v d(p+1) d d 2 dx C p V T T 2 v p dx + C p d d ( ) (d 2)p ( ) 1 d(p+1) C p dx V 2βp βp dx + Cp. Td v d(p+1) d 2 T d ( ) (d 2) ( ) 1 v d(p+1) d(p+1) d 2 dx Cp V 2βp βp dx + Cp. T d T d This last estimate gives (3), and the theorem follows.
22 The Bernstein method for MFGs We fix a C 2 potential, V : T d R, and look for a solution, (u, m, H), of the MFG u(x) + Du(x) V (x) = H + m α, m div (mdu) = 0, (4) udx = 0, mdx = 1, with u, m : T d R and H R.
23 The Bernstein method for MFGs Theorem Let (u, m, H) solve (4) and 0 < α 1 d 1. Suppose that u, m C 2 (T d ). Then, for every q > 1, there exists a constant, C q > 0, that depends only on V L 1+ α 1, such that (T d ) Du L q (T d ) C q.
24 The Bernstein method for MFGs We use Bernstein s estimate with V replaced by V (x) m α. By the previous results, we have H C, m α L 1+ 1 α (T d ) C. Because d α, γ p α, Bernstein estimate gives Du L q (T d ) C q for every q > 1.
25 Bootstrapping regularity Proposition Let (u, m, H) solve (4) with u and m in C (T d ), and let m > 0. Then, there exists a constant, C > 0, such that Hence, m, 1 m L. ln m W 1,q (T d ) C.
26 Bootstrapping regularity Standard elliptic regularity theory applied to Hamilton-Jacobi equation yields u W 2,q (T d ) C q, for every q > 1. Therefore, Morrey s Embedding Theorem implies that u C 1,β (T d ), for some β (0, 1).
27 Bootstrapping regularity Next, set w = 2 ln m. Straightforward computations show that w satisfies w Dw 2 Du Dw + 2 div(du) = 0. Integrating, we conclude that Dw L 2.
28 Bootstrapping regularity The Bernstein estimate gives Dw L 2d(p+1) d 2 (T d ) C p C + Du Dw L 2d(1+p) + 2d(1+p) d+2p (T d ) div(du) L d+2p (T d ). Since 2d(p+1) d 2 > 2d(1+p) d+2p, we get Dw Lq for any q 1.
29 Bootstrapping regularity Hence, ln m is a Hölder continuous function. Because T d mdx = 1, m is bounded from above and from below. Consequently, ln m L q (T d ) is a priori bounded by some universal constant that depends only on q.
30 Bootstrapping regularity Proposition Let (u, m, H) solve (4) with u and m in C (T d ), and let m > 0. For any k 1 and q > 1, there exists a constant, C k,q > 0, such that D u k D, k m C k,q. L q (T d ) L q (T d )
31 Bootstrapping regularity The preceding results give u W 2,a (T d ) C a for every 1 < a <. Also, Proposition 7 gives ln m W 1,a (T d ) C for any 1 < a <. By differentiating the first equation in (4), we obtain D x u = D x g(m) D 2 udu. (5) Finally, we observe that the right-hand side of (5) is bounded in L a (T d ). Thus, u W 3,a (T d ) C 3,a, which leads to m W 2,a (T d ) C 2,a. The proof proceeds by iterating this procedure up to order k.
32 Continuation method stationary problems Here, we illustrate the continuation method by proving the existence of smooth solutions of u + Du V (x) = H + g(m), m div (Dum) = 0, T u = 0, d T m = 1. d (6)
33 Continuation method stationary problems First, for 0 λ 1, we consider the family of problems m λ div(du λ m λ ) = 0, u λ Du λ 2 2 λv + H λ + g(m λ ) = 0, T u d λ = 0, T m d λ = 1. (7)
34 Continuation method stationary problems Set { } H k (T d, R) = f H k (T d, R) : f dx = 0 T d and consider F k = H k (T d, R) H k (T d, R) R,, which is a Hilber space with norm w 2 F k = ψ 2 Ḣ k (T d,r) + f 2 H k (T d,r) + h 2 for w = (ψ, f, h) F k. H+(T k d, R), for k > d 2 is the set of (everywhere) positive functions in H k (T d, R). For any k > d 2, let F+ k = H k (T d, R) H+(T k d, R) R. A classical solution is a tuple, (u λ, m λ, H λ ) F+. k k 0
35 Continuation method stationary problems Theorem Assume that g, V C (T d ) with g (z) > 0 for z (0, + ) and that we have the a priori estimate for any solution of (7): H u λ W k,p (T d ) + m λ W k,p (T d ) C k,p. L (T d ) m λ Then, there exists a classical solution to (6).
36 Continuation method stationary problems For large enough k, define E : R F k + F k 2 by E(λ, u, m, H) = m div(dum) u Du 2 2 λv + H + g(m) T d m + 1 Our system equivalent to E(λ, v λ ) = 0, where v λ = (u λ, m λ, H λ )..
37 Continuation method stationary problems The partial derivative of E in the second variable at v λ = (u λ, m λ, H λ ), L λ = D 2 E(λ, v λ ): F k F k 2, is L λ (w)(x) = f (x) div(du λ f (x) + m λ Dψ) ψ(x) Du λ Dψ + g (m λ (x))f (x) + h T d f, where w = (ψ, f, h) F k. In principle, L λ is a linear map on F k for a large enough k. However, it is easy to see that it admits a unique extension to F k for any k > 1.
38 Continuation method stationary problems Let Λ := { λ 0 λ 1, (7) has a classical solution (u λ, m λ, H λ ) }. Note that 0 Λ as (u 0, m 0, H 0 ) (0, 1, g(1)) is a solution to (7) for λ = 0. Our goal is to prove Λ = [0, 1]. The a priori bounds in the statement mean that Λ is a closed set. To prove that Λ is open, we s apply the implicit function theorem.
39 Continuation method stationary problems Let F = F 1. For w 1, w 2 F with smooth components, set B λ [w 1, w 2 ] = w 2 L λ (w 1 ). T d For smooth w 1, w 2, B λ [w 1, w 2 ] = [m λ Dψ 1 Dψ 2 + f 1 Du λ Dψ 2 f 2 Du λ Dψ 1 T d + g (m λ )f 1 f 2 + Df 1 Dψ 2 Df 2 Dψ 1 + h 1 f 2 h 2 f 1 ]. This last expression defines a bilinear form B λ : F F R.
40 Continuation method stationary problems Claim B λ is bounded, i.e., B λ [w 1, w 2 ] C w 1 F w 2 F. To prove the claim, we use Holder s inequality on each summand.
41 Continuation method stationary problems Claim There exists a linear bounded mapping, A: F F, such that B λ [w 1, w 2 ] = (Aw 1, w 2 ) F. This claim follows from Claim 10 and the Riesz Representation Theorem.
42 Continuation method stationary problems Claim There exists a positive constant, c, such that Aw F c w F for all w F. If the previous claim were false, then there would exist a sequence, w n F, with w n F = 1 such that Aw n 0.
43 Continuation method stationary problems Let w n = (ψ n, f n, h n ). Then, T d m λ Dψ n 2 + g (m λ )f 2 n = B λ [w n, w n ] 0. (8) By combining the a priori estimates on 1 m λ with the fact that g is strictly increasing and smooth, we have g (m λ ) > δ > 0.
44 Continuation method stationary problems Then, (8) implies that ψ n 0 in H 1 0 and f n 0 in L 2. Taking ˇw n = (f n f n, 0, 0) F, we get T d [ Df n 2 + m λ Dψ n Df n + f n Du λ Df n ] = B[w n, ˇw n ] = (Aw n, ˇw n ), Therefore, 1 ) 2 Df n 2 L 2 (T d ) ( Dψ C n 2 L 2 (T d ) + f n 2 L 2 (T d ) (Aw n, ˇw n ) 0, where the constant, C, depends only on u λ. Because Dψ n, f n 0 in L 2, we have f n 0 in H 1 (T d ).
45 Continuation method stationary problems Finally, we take w = (0, 1, 0). Accordingly, we get T d [ Du λ Dψ n + g (m λ )f n ] + h n = B[w n, w] = (Aw n, w) 0. Because Dψ n, f n 0 in L 2, we have h n 0. Hence, w n F 0, which contradicts w n F = 1.
46 Continuation method stationary problems Claim R(A) is closed in F. This claim follows from the preceding one.
47 Continuation method stationary problems Claim R(A) = F. By contradiction, suppose that R(A) F. Then, because R(A) is closed in F, there exists a vector, w 0, with w R(A). Let w = (ψ, f, h). Then, 0 = (Aw, w) = B λ [w, w] θ Dψ 2 + δ f 2. T d Therefore, ψ = 0 and f = 0. Next, we choose w = (0, 1, 0). Similarly, we have h = B λ [ w, w] = (A w, w) = 0. Thus, w = 0, and, consequently, R(A) = F.
48 Continuation method stationary problems I Claim For any w 0 F 0, there exists a unique w F such that B λ [w, w] = (w 0, w) F 0 for all w F. Consequently, w is the unique weak solution of the equation L λ (w) = w 0. Moreover, w F 2 and L λ (w) = w 0 in the sense of F 2.
49 Continuation method stationary problems Consider the functional w (w 0, w) F 0 on F. By the Riesz Representation Theorem, there exists ω F such that (w 0, w) F 0 = (ω, w) F. Taking w = A 1 ω, we get B[w, w] = (Aw, w) F = (ω, w) F = (w 0, w) F 0. Therefore, f is a weak solution to and ψ is a weak solution to f div(m λ Dψ + fdu λ ) = ψ 0 ψ Du λ Dψ + g (m λ )f + h = f 0.
50 Continuation method stationary problems Standard results from the regularity theory for elliptic equations combined with bootstrapping arguments give w = (ψ, f, h) F 2. Thus, L λ (w) = w 0. Consequently, L λ is a bijective operator from F 2 to F 0. Then, L λ is injective as an operator from F k to F k 2 for any k 2. To prove that it is also surjective, take any w 0 F k 2. Then, there exists w F 2 such that L λ (w) = w 0. Finally elliptic regularity and bootstrapping imply that w F k. Hence, L λ : F k F k 2 is surjective and, therefore, also bijective.
51 Continuation method stationary problems Claim L λ is an isomorphism from F k to F k 2 for any k 2. Because L λ : F k F k 2 is bijective, we just need to check that it is also bounded. The boundedness follows directly from bounds on u λ and m λ and the smoothness of V and g.
52 Continuation method stationary problems Claim The set Λ is open.
53 Continuation method stationary problems We choose k > d/2 + 1 so that H k 1 (T d, R) is an algebra. For each λ 0 Λ, the partial derivative, L = D 2 E(λ 0, v λ0 ): F k F k 2, is an isomorphism. By the Implicit Function Theorem, there exists a unique solution v λ F k + to E(λ, v λ ) = 0, in some neighborhood, U, of λ 0. Finally, because H k 1 (T d, R) is an algebra, bootstrapping yields that u λ and m λ are smooth. Therefore, v λ is a classical solution to (6). Hence, U Λ, which proves that Λ is open.
54 Continuation method stationary problems We have proven that Λ is both open and closed; hence, Λ = [0, 1]. This argument ends the proof of the theorem.
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
More informationarxiv: v1 [math.ap] 31 Jul 2014
EXISTENCE FOR STATIONARY MEAN FIELD GAMES WITH QUADRATIC HAMILTONIANS WITH CONGESTION arxiv:1407.8267v1 [math.ap] 31 Jul 2014 DIOGO A. GOMES AND HIROYOSHI MITAKE Abstract. In this paper, we investigate
More informationSobolev 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
More informationTD 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
More informationSobolev 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
More informationLaplace 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 informationA PROPERTY OF SOBOLEV SPACES ON COMPLETE RIEMANNIAN MANIFOLDS
Electronic Journal of Differential Equations, Vol. 2005(2005), No.??, pp. 1 10. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) A PROPERTY
More informationTHE L 2 -HODGE THEORY AND REPRESENTATION ON R n
THE L 2 -HODGE THEORY AND REPRESENTATION ON R n BAISHENG YAN Abstract. We present an elementary L 2 -Hodge theory on whole R n based on the minimization principle of the calculus of variations and some
More informationIntegral Representation Formula, Boundary Integral Operators and Calderón projection
Integral Representation Formula, Boundary Integral Operators and Calderón projection Seminar BEM on Wave Scattering Franziska Weber ETH Zürich October 22, 2010 Outline Integral Representation Formula Newton
More informationA 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
More informationThe continuity method
The continuity method The method of continuity is used in conjunction with a priori estimates to prove the existence of suitably regular solutions to elliptic partial differential equations. One crucial
More informationPDE Methods for Mean Field Games with Non-Separable Hamiltonian: Data in Sobolev Spaces (Continued) David Ambrose June 29, 2018
PDE Methods for Mean Field Games with Non-Separable Hamiltonian: Data in Sobolev Spaces Continued David Ambrose June 29, 218 Steps of the energy method Introduce an approximate problem. Prove existence
More informationPrice formation models Diogo A. Gomes
models Diogo A. Gomes Here, we are interested in the price formation in electricity markets where: a large number of agents owns storage devices that can be charged and later supply the grid with electricity;
More informationFact Sheet Functional Analysis
Fact Sheet Functional Analysis Literature: Hackbusch, W.: Theorie und Numerik elliptischer Differentialgleichungen. Teubner, 986. Knabner, P., Angermann, L.: Numerik partieller Differentialgleichungen.
More informationFunctional Analysis. Franck Sueur Metric spaces Definitions Completeness Compactness Separability...
Functional Analysis Franck Sueur 2018-2019 Contents 1 Metric spaces 1 1.1 Definitions........................................ 1 1.2 Completeness...................................... 3 1.3 Compactness......................................
More informationMINIMAL 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
More informationChapter 1 Foundations of Elliptic Boundary Value Problems 1.1 Euler equations of variational problems
Chapter 1 Foundations of Elliptic Boundary Value Problems 1.1 Euler equations of variational problems Elliptic boundary value problems often occur as the Euler equations of variational problems the latter
More informationEXISTENCE OF SOLUTIONS TO SYSTEMS OF EQUATIONS MODELLING COMPRESSIBLE FLUID FLOW
Electronic Journal o Dierential Equations, Vol. 15 (15, No. 16, pp. 1 8. ISSN: 17-6691. URL: http://ejde.math.txstate.e or http://ejde.math.unt.e tp ejde.math.txstate.e EXISTENCE OF SOLUTIONS TO SYSTEMS
More informationScientific Computing WS 2018/2019. Lecture 15. Jürgen Fuhrmann Lecture 15 Slide 1
Scientific Computing WS 2018/2019 Lecture 15 Jürgen Fuhrmann juergen.fuhrmann@wias-berlin.de Lecture 15 Slide 1 Lecture 15 Slide 2 Problems with strong formulation Writing the PDE with divergence and gradient
More informationTheory of PDE Homework 2
Theory of PDE Homework 2 Adrienne Sands April 18, 2017 In the following exercises we assume the coefficients of the various PDE are smooth and satisfy the uniform ellipticity condition. R n is always an
More information08a. Operators on Hilbert spaces. 1. Boundedness, continuity, operator norms
(February 24, 2017) 08a. Operators on Hilbert spaces Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ [This document is http://www.math.umn.edu/ garrett/m/real/notes 2016-17/08a-ops
More informationOblique 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.
More informationOn non negative solutions of some quasilinear elliptic inequalities
On non negative solutions of some quasilinear elliptic inequalities Lorenzo D Ambrosio and Enzo Mitidieri September 28 2006 Abstract Let f : R R be a continuous function. We prove that under some additional
More information1. Bounded linear maps. A linear map T : E F of real Banach
DIFFERENTIABLE MAPS 1. Bounded linear maps. A linear map T : E F of real Banach spaces E, F is bounded if M > 0 so that for all v E: T v M v. If v r T v C for some positive constants r, C, then T is bounded:
More informationWeek 6 Notes, Math 865, Tanveer
Week 6 Notes, Math 865, Tanveer. Energy Methods for Euler and Navier-Stokes Equation We will consider this week basic energy estimates. These are estimates on the L 2 spatial norms of the solution u(x,
More information2 A Model, Harmonic Map, Problem
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
More informationINVERSE FUNCTION THEOREM and SURFACES IN R n
INVERSE FUNCTION THEOREM and SURFACES IN R n Let f C k (U; R n ), with U R n open. Assume df(a) GL(R n ), where a U. The Inverse Function Theorem says there is an open neighborhood V U of a in R n so that
More informationHeight Functions. Michael Tepper February 14, 2006
Height Functions Michael Tepper February 14, 2006 Definition 1. Let Y P n be a quasi-projective variety. A function f : Y k is regular at a point P Y if there is an open neighborhood U with P U Y, and
More informationDETERMINATION OF THE BLOW-UP RATE FOR THE SEMILINEAR WAVE EQUATION
DETERMINATION OF THE LOW-UP RATE FOR THE SEMILINEAR WAVE EQUATION y FRANK MERLE and HATEM ZAAG Abstract. In this paper, we find the optimal blow-up rate for the semilinear wave equation with a power nonlinearity.
More informationMath212a Lecture 5 Applications of the spectral theorem for compact self-adjoint operators, 2. Gårding s inequality and its conseque
Math212a Lecture 5 Applications of the spectral theorem for compact self-adjoint operators, 2. Gårding s inequality and its consequences. September 16, 2014 1 Review of Sobolev spaces. Distributions and
More information(d). Why does this imply that there is no bounded extension operator E : W 1,1 (U) W 1,1 (R n )? Proof. 2 k 1. a k 1 a k
Exercise For k 0,,... let k be the rectangle in the plane (2 k, 0 + ((0, (0, and for k, 2,... let [3, 2 k ] (0, ε k. Thus is a passage connecting the room k to the room k. Let ( k0 k (. We assume ε k
More informationRégularité des équations de Hamilton-Jacobi du premier ordre et applications aux jeux à champ moyen
Régularité des équations de Hamilton-Jacobi du premier ordre et applications aux jeux à champ moyen Daniela Tonon en collaboration avec P. Cardaliaguet et A. Porretta CEREMADE, Université Paris-Dauphine,
More informationPartial 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
More informationADJOINT METHODS FOR OBSTACLE PROBLEMS AND WEAKLY COUPLED SYSTEMS OF PDE. 1. Introduction
ADJOINT METHODS FOR OBSTACLE PROBLEMS AND WEAKLY COPLED SYSTEMS OF PDE F. CAGNETTI, D. GOMES, AND H.V. TRAN Abstract. The adjoint method, recently introduced by Evans, is used to study obstacle problems,
More informationExercise Solutions to Functional Analysis
Exercise Solutions to Functional Analysis Note: References refer to M. Schechter, Principles of Functional Analysis Exersize that. Let φ,..., φ n be an orthonormal set in a Hilbert space H. Show n f n
More informationSome 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 informationA LOWER BOUND ON BLOWUP RATES FOR THE 3D INCOMPRESSIBLE EULER EQUATION AND A SINGLE EXPONENTIAL BEALE-KATO-MAJDA ESTIMATE. 1.
A LOWER BOUND ON BLOWUP RATES FOR THE 3D INCOMPRESSIBLE EULER EQUATION AND A SINGLE EXPONENTIAL BEALE-KATO-MAJDA ESTIMATE THOMAS CHEN AND NATAŠA PAVLOVIĆ Abstract. We prove a Beale-Kato-Majda criterion
More information1 )(y 0) {1}. Thus, the total count of points in (F 1 (y)) is equal to deg y0
1. Classification of 1-manifolds Theorem 1.1. Let M be a connected 1 manifold. Then M is diffeomorphic either to [0, 1], [0, 1), (0, 1), or S 1. We know that none of these four manifolds are not diffeomorphic
More informationFUNCTIONAL ANALYSIS LECTURE NOTES: ADJOINTS IN HILBERT SPACES
FUNCTIONAL ANALYSIS LECTURE NOTES: ADJOINTS IN HILBERT SPACES CHRISTOPHER HEIL 1. Adjoints in Hilbert Spaces Recall that the dot product on R n is given by x y = x T y, while the dot product on C n is
More informationAnalysis in weighted spaces : preliminary version
Analysis in weighted spaces : preliminary version Frank Pacard To cite this version: Frank Pacard. Analysis in weighted spaces : preliminary version. 3rd cycle. Téhéran (Iran, 2006, pp.75.
More informationLECTURE 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
More informationThe heat equation in time dependent domains with Neumann boundary conditions
The heat equation in time dependent domains with Neumann boundary conditions Chris Burdzy Zhen-Qing Chen John Sylvester Abstract We study the heat equation in domains in R n with insulated fast moving
More informationMean field games and related models
Mean field games and related models Fabio Camilli SBAI-Dipartimento di Scienze di Base e Applicate per l Ingegneria Facoltà di Ingegneria Civile ed Industriale Email: Camilli@dmmm.uniroma1.it Web page:
More informationOn m-accretive Schrödinger operators in L p -spaces on manifolds of bounded geometry
On m-accretive Schrödinger operators in L p -spaces on manifolds of bounded geometry Ognjen Milatovic Department of Mathematics and Statistics University of North Florida Jacksonville, FL 32224 USA. Abstract
More informationLecture 9 Metric spaces. The contraction fixed point theorem. The implicit function theorem. The existence of solutions to differenti. equations.
Lecture 9 Metric spaces. The contraction fixed point theorem. The implicit function theorem. The existence of solutions to differential equations. 1 Metric spaces 2 Completeness and completion. 3 The contraction
More informationREGULARITY 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
More informationEilenberg-Steenrod properties. (Hatcher, 2.1, 2.3, 3.1; Conlon, 2.6, 8.1, )
II.3 : Eilenberg-Steenrod properties (Hatcher, 2.1, 2.3, 3.1; Conlon, 2.6, 8.1, 8.3 8.5 Definition. Let U be an open subset of R n for some n. The de Rham cohomology groups (U are the cohomology groups
More informationFriedrich 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
More informationAsymptotic behavior of infinity harmonic functions near an isolated singularity
Asymptotic behavior of infinity harmonic functions near an isolated singularity Ovidiu Savin, Changyou Wang, Yifeng Yu Abstract In this paper, we prove if n 2 x 0 is an isolated singularity of a nonegative
More informationHille-Yosida Theorem and some Applications
Hille-Yosida Theorem and some Applications Apratim De Supervisor: Professor Gheorghe Moroșanu Submitted to: Department of Mathematics and its Applications Central European University Budapest, Hungary
More informationRegularity and nonexistence results for anisotropic quasilinear elliptic equations in convex domains
Regularity and nonexistence results for anisotropic quasilinear elliptic equations in convex domains Ilaria FRAGALÀ Filippo GAZZOLA Dipartimento di Matematica del Politecnico - Piazza L. da Vinci - 20133
More informationMoscow, Russia; National Research University Higher School of Economics, Moscow, Russia
Differentiability of solutions of stationary Fokker Planck Kolmogorov equations with respect to a parameter Vladimir I. Bogachev 1, Stanislav V. Shaposhnikov 2, and Alexander Yu. Veretennikov 3 1,2 Department
More informationNumerical Solutions to Partial Differential Equations
Numerical Solutions to Partial Differential Equations Zhiping Li LMAM and School of Mathematical Sciences Peking University Sobolev Embedding Theorems Embedding Operators and the Sobolev Embedding Theorem
More informationStability of Feedback Solutions for Infinite Horizon Noncooperative Differential Games
Stability of Feedback Solutions for Infinite Horizon Noncooperative Differential Games Alberto Bressan ) and Khai T. Nguyen ) *) Department of Mathematics, Penn State University **) Department of Mathematics,
More informationFinite-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 (, )
More informationFunctional Analysis Exercise Class
Functional Analysis Exercise Class Week 9 November 13 November Deadline to hand in the homeworks: your exercise class on week 16 November 20 November Exercises (1) Show that if T B(X, Y ) and S B(Y, Z)
More informationLocal pointwise a posteriori gradient error bounds for the Stokes equations. Stig Larsson. Heraklion, September 19, 2011 Joint work with A.
Local pointwise a posteriori gradient error bounds for the Stokes equations Stig Larsson Department of Mathematical Sciences Chalmers University of Technology and University of Gothenburg Heraklion, September
More informationIntegral Jensen inequality
Integral Jensen inequality Let us consider a convex set R d, and a convex function f : (, + ]. For any x,..., x n and λ,..., λ n with n λ i =, we have () f( n λ ix i ) n λ if(x i ). For a R d, let δ a
More informationBasic Calculus Review
Basic Calculus Review Lorenzo Rosasco ISML Mod. 2 - Machine Learning Vector Spaces Functionals and Operators (Matrices) Vector Space A vector space is a set V with binary operations +: V V V and : R V
More informationA COUNTEREXAMPLE TO AN ENDPOINT BILINEAR STRICHARTZ INEQUALITY TERENCE TAO. t L x (R R2 ) f L 2 x (R2 )
Electronic Journal of Differential Equations, Vol. 2006(2006), No. 5, pp. 6. ISSN: 072-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) A COUNTEREXAMPLE
More informationOn the Ladyzhenskaya Smagorinsky turbulence model of the Navier Stokes equations in smooth domains. The regularity problem
J. Eur. Math. Soc. 11, 127 167 c European Mathematical Society 2009 H. Beirão da Veiga On the Ladyzhenskaya Smagorinsky turbulence model of the Navier Stokes equations in smooth domains. The regularity
More informationBoundedness, 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
More informationSPECTRAL PROPERTIES OF THE LAPLACIAN ON BOUNDED DOMAINS
SPECTRAL PROPERTIES OF THE LAPLACIAN ON BOUNDED DOMAINS TSOGTGEREL GANTUMUR Abstract. After establishing discrete spectra for a large class of elliptic operators, we present some fundamental spectral properties
More informationThe 2D Magnetohydrodynamic Equations with Partial Dissipation. Oklahoma State University
The 2D Magnetohydrodynamic Equations with Partial Dissipation Jiahong Wu Oklahoma State University IPAM Workshop Mathematical Analysis of Turbulence IPAM, UCLA, September 29-October 3, 2014 1 / 112 Outline
More informationNonlinear Elliptic Systems and Mean-Field Games
Nonlinear Elliptic Systems and Mean-Field Games Martino Bardi and Ermal Feleqi Dipartimento di Matematica Università di Padova via Trieste, 63 I-35121 Padova, Italy e-mail: bardi@math.unipd.it and feleqi@math.unipd.it
More informationHilbert spaces. 1. Cauchy-Schwarz-Bunyakowsky inequality
(October 29, 2016) Hilbert 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 2016-17/03 hsp.pdf] Hilbert spaces are
More information1 Continuity Classes C m (Ω)
0.1 Norms 0.1 Norms A norm on a linear space X is a function : X R with the properties: Positive Definite: x 0 x X (nonnegative) x = 0 x = 0 (strictly positive) λx = λ x x X, λ C(homogeneous) x + y x +
More informationNew Helmholtz-Weyl decomposition in L r and its applications to the mathematical fluid mechanics
New Helmholtz-Weyl decomposition in L r and its applications to the mathematical fluid mechanics Hideo Kozono Mathematical Institute Tohoku University Sendai 980-8578 Japan Taku Yanagisawa Department of
More informationCollatz cycles with few descents
ACTA ARITHMETICA XCII.2 (2000) Collatz cycles with few descents by T. Brox (Stuttgart) 1. Introduction. Let T : Z Z be the function defined by T (x) = x/2 if x is even, T (x) = (3x + 1)/2 if x is odd.
More informationMath 5520 Homework 2 Solutions
Math 552 Homework 2 Solutions March, 26. Consider the function fx) = 2x ) 3 if x, 3x ) 2 if < x 2. Determine for which k there holds f H k, 2). Find D α f for α k. Solution. We show that k = 2. The formulas
More informationL19: Fredholm theory. where E u = u T X and J u = Formally, J-holomorphic curves are just 1
L19: Fredholm theory We want to understand how to make moduli spaces of J-holomorphic curves, and once we have them, how to make them smooth and compute their dimensions. Fix (X, ω, J) and, for definiteness,
More informationNonlinear 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
More informationSOLUTION OF THE DIRICHLET PROBLEM WITH A VARIATIONAL METHOD. 1. Dirichlet integral
SOLUTION OF THE DIRICHLET PROBLEM WITH A VARIATIONAL METHOD CRISTIAN E. GUTIÉRREZ FEBRUARY 3, 29. Dirichlet integral Let f C( ) with open and bounded. Let H = {u C ( ) : u = f on } and D(u) = Du(x) 2 dx.
More informationSHARP BOUNDARY TRACE INEQUALITIES. 1. Introduction
SHARP BOUNDARY TRACE INEQUALITIES GILES AUCHMUTY Abstract. This paper describes sharp inequalities for the trace of Sobolev functions on the boundary of a bounded region R N. The inequalities bound (semi-)norms
More informationON LIQUID CRYSTAL FLOWS WITH FREE-SLIP BOUNDARY CONDITIONS. Chun Liu and Jie Shen
DISCRETE AND CONTINUOUS Website: http://aimsciences.org DYNAMICAL SYSTEMS Volume 7, Number2, April2001 pp. 307 318 ON LIQUID CRYSTAL FLOWS WITH FREE-SLIP BOUNDARY CONDITIONS Chun Liu and Jie Shen Department
More informationis 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).
More informationPoisson 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
More informationto appear in the Journal of the European Mathematical Society THE WOLFF GRADIENT BOUND FOR DEGENERATE PARABOLIC EQUATIONS
to appear in the Journal of the European Mathematical Society THE WOLFF GRADIENT BOUND FOR DEGENERATE PARABOLIC EUATIONS TUOMO KUUSI AND GIUSEPPE MINGIONE Abstract. The spatial gradient of solutions to
More informationVariable Exponents Spaces and Their Applications to Fluid Dynamics
Variable Exponents Spaces and Their Applications to Fluid Dynamics Martin Rapp TU Darmstadt November 7, 213 Martin Rapp (TU Darmstadt) Variable Exponent Spaces November 7, 213 1 / 14 Overview 1 Variable
More informationSOLUTIONS TO HOMEWORK ASSIGNMENT 4
SOLUTIONS TO HOMEWOK ASSIGNMENT 4 Exercise. A criterion for the image under the Hilbert transform to belong to L Let φ S be given. Show that Hφ L if and only if φx dx = 0. Solution: Suppose first that
More informationPDE Constrained Optimization selected Proofs
PDE Constrained Optimization selected Proofs Jeff Snider jeff@snider.com George Mason University Department of Mathematical Sciences April, 214 Outline 1 Prelim 2 Thms 3.9 3.11 3 Thm 3.12 4 Thm 3.13 5
More informationLectures on. Sobolev Spaces. S. Kesavan The Institute of Mathematical Sciences, Chennai.
Lectures on Sobolev Spaces S. Kesavan The Institute of Mathematical Sciences, Chennai. e-mail: kesh@imsc.res.in 2 1 Distributions In this section we will, very briefly, recall concepts from the theory
More informationChapter One. The Calderón-Zygmund Theory I: Ellipticity
Chapter One The Calderón-Zygmund Theory I: Ellipticity Our story begins with a classical situation: convolution with homogeneous, Calderón- Zygmund ( kernels on R n. Let S n 1 R n denote the unit sphere
More information2 Two-Point Boundary Value Problems
2 Two-Point Boundary Value Problems Another fundamental equation, in addition to the heat eq. and the wave eq., is Poisson s equation: n j=1 2 u x 2 j The unknown is the function u = u(x 1, x 2,..., x
More informationInequalities of Babuška-Aziz and Friedrichs-Velte for differential forms
Inequalities of Babuška-Aziz and Friedrichs-Velte for differential forms Martin Costabel Abstract. For sufficiently smooth bounded plane domains, the equivalence between the inequalities of Babuška Aziz
More informationHahn-Banach theorems. 1. Continuous Linear Functionals
(April 22, 2014) Hahn-Banach theorems Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ [This document is http://www.math.umn.edu/ garrett/m/fun/notes 2012-13/07c hahn banach.pdf] 1.
More informationS 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 informationVariational Theory of Solitons for a Higher Order Generalized Camassa-Holm Equation
International Journal of Mathematical Analysis Vol. 11, 2017, no. 21, 1007-1018 HIKAI Ltd, www.m-hikari.com https://doi.org/10.12988/ijma.2017.710141 Variational Theory of Solitons for a Higher Order Generalized
More informationBanach space valued Cauchy-Riemann equations with totally real boundary conditions corrected
Banach space valued Cauchy-Riemann equations with totally real boundary conditions corrected Katrin Wehrheim August 18, 2010 Abstract The main purpose of this paper is to give a general regularity result
More informationThe 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 informationSome 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,
More informationEXISTENCE AND REGULARITY RESULTS FOR SOME NONLINEAR PARABOLIC EQUATIONS
EXISTECE AD REGULARITY RESULTS FOR SOME OLIEAR PARABOLIC EUATIOS Lucio BOCCARDO 1 Andrea DALL AGLIO 2 Thierry GALLOUËT3 Luigi ORSIA 1 Abstract We prove summability results for the solutions of nonlinear
More informationA Brief Introduction to Functional Analysis
A Brief Introduction to Functional Analysis Sungwook Lee Department of Mathematics University of Southern Mississippi sunglee@usm.edu July 5, 2007 Definition 1. An algebra A is a vector space over C with
More informationMath 350 Fall 2011 Notes about inner product spaces. In this notes we state and prove some important properties of inner product spaces.
Math 350 Fall 2011 Notes about inner product spaces In this notes we state and prove some important properties of inner product spaces. First, recall the dot product on R n : if x, y R n, say x = (x 1,...,
More informationLandesman-Lazer type results for second order Hamilton-Jacobi-Bellman equations
Author manuscript, published in "Journal of Functional Analysis 258, 12 (2010) 4154-4182" Landesman-Lazer type results for second order Hamilton-Jacobi-Bellman equations Patricio FELMER, Alexander QUAAS,
More information6 Classical dualities and reflexivity
6 Classical dualities and reflexivity 1. Classical dualities. Let (Ω, A, µ) be a measure space. We will describe the duals for the Banach spaces L p (Ω). First, notice that any f L p, 1 p, generates the
More informationComplex Analysis Qualifying Exam Solutions
Complex Analysis Qualifying Exam Solutions May, 04 Part.. Let log z be the principal branch of the logarithm defined on G = {z C z (, 0]}. Show that if t > 0, then the equation log z = t has exactly one
More informationFourier Transform & Sobolev Spaces
Fourier Transform & Sobolev Spaces Michael Reiter, Arthur Schuster Summer Term 2008 Abstract We introduce the concept of weak derivative that allows us to define new interesting Hilbert spaces the Sobolev
More informationREGULARITY RESULTS FOR THE EQUATION u 11 u 22 = Introduction
REGULARITY RESULTS FOR THE EQUATION u 11 u 22 = 1 CONNOR MOONEY AND OVIDIU SAVIN Abstract. We study the equation u 11 u 22 = 1 in R 2. Our results include an interior C 2 estimate, classical solvability
More informationATTRACTORS FOR SEMILINEAR PARABOLIC PROBLEMS WITH DIRICHLET BOUNDARY CONDITIONS IN VARYING DOMAINS. Emerson A. M. de Abreu Alexandre N.
ATTRACTORS FOR SEMILINEAR PARABOLIC PROBLEMS WITH DIRICHLET BOUNDARY CONDITIONS IN VARYING DOMAINS Emerson A. M. de Abreu Alexandre N. Carvalho Abstract Under fairly general conditions one can prove that
More information