Sharp Sobolev Strichartz estimates for the free Schrödinger propagator

Size: px
Start display at page:

Download "Sharp Sobolev Strichartz estimates for the free Schrödinger propagator"

Transcription

1 Sharp Sobolev Strichartz estimates for the free Schrödinger propagator Neal Bez, Chris Jeavons and Nikolaos Pattakos Abstract. We consider gaussian extremisability of sharp linear Sobolev Strichartz estimates and closely related sharp bilinear Ozawa Tsutsumi estimates for the free Schrödinger equation. Mathematics Subject Classification (010). Primary 35B45; Secondary 35Q40. Keywords. Sobolev Strichartz estimates, extremisers, Schrödinger equation. 1. Introduction For d 1 and s [0, d ), it is well-known that the solution u : R Rd C of the free Schrödinger equation i t u + u = 0, u(0) = u 0 Ḣs (R d ) (1.1) satisfies the global space-time estimate u L p(d,s) S(d, s) u 0 Ḣs (1.) for some finite constant S(d, s), which we assume to be the optimal (i.e. smallest) such constant. Here, p(d, s) = (d+) and, as usual, Ḣs (R d ) denotes the homogeneous Sobolev space with norm f Ḣs = ( ) s/ f L. This article will be concerned with optimal constants and extremisers, and we note immediately that S(d, s) and the shape of corresponding extremisers are only known in the rather special cases (d, s) {(1, 0), (, 0)} (see Foschi [5] and also Hundertmark Zharnitsky [6]). In such cases, the isotropic gaussian initial data u 0 (x) = exp( x ) is an extremiser. Very closely related are the sharp bilinear estimates ( ) d 4 (uv) L OT(d) u 0 L v 0 L (1.3) due to Ozawa Tsutsumi [7], where d and OT(d) = Γ( d ) 1 d π d 4.

2 N. Bez, C. Jeavons and N. Pattakos In (1.3), u and v are solutions of (1.1) with square-integrable initial data u 0 and v 0. The constant OT(d) is optimal and (u 0, v 0 ) is an extremising pair when u 0 (x) = v 0 (x) = exp( x ). The sharp estimate (1.3) was motivated by the case of one spatial dimension, in which case (1.3) is an identity ( ) 1 4 (uv) L = OT(1) u 0 L v 0 L. (1.4) This basic tool was established and used in [7] to prove local-wellposedness for certain nonlinear Schrödinger equations with nonlinearities including ( u )u and initial data in H 1/. Thus, (1.4) gives control on the null gauge form (uv) for the one-dimensional Schrödinger equation, and (1.3) gives estimates for the null gauge form in higher dimensions. We remark that taking d = and u 0 = v 0 in (1.3) immediately yields the optimal constant S(, 0) and its gaussian extremisability (this was not explicitly observed in [7]). The approach in [7] is different to the approaches in [5] and [6], so this provides an alternative derivation of this optimal constant. As far as we know, for the Sobolev Strichartz estimate (1.1), no conjecture has been made on the shape of extremising initial data in the case where s is strictly positive. Extremising initial data certainly exist for all admissible d 1 and s [0, d ) (see, for example, [8]). In this direction, our first observation is the following. Theorem 1.1. Only if s = 0 are gaussians u 0 such that û 0 (ξ) = exp(a ξ + ib ξ + c) for some a, c C, b R d, Re(a) < 0, critical points for the functional u 0 eit u 0 L p(d,s) u 0 Ḣs. (1.5) Theorem 1.1 of course implies that gaussians are not amongst the class of extremisers for (1.) for any admissible s which are strictly positive; i.e. s (0, d ). We find this outcome particularly interesting when measured against the analogous sharp estimates for the wave propagator e it. Here, it is known that if d and s [0, d 1 ) then u L p(d 1,s) W(d, s) u 0 Ḣs+ 1 for all solutions of the (pseudo) wave equation i t u + u = 0, u(0) = u 0 Ḣs+ 1 (R d ). (1.6) Again, we take W(d, s) to be the optimal constant, which is finite for the given range of parameters (d, s). It is known that initial data u 0 for which û 0 (ξ) = ξ 1 exp( ξ ) (1.7) are extremisers for (1.6) when (d, s) {(, 0), (3, 0), (5, 1 )} (uniquely, up to certain transformations). The cases (, 0) and (3, 0) were established by Foschi [5] and the case (5, 1 ) was established in []. Thus, u 0 satisfying (1.7)

3 Sharp Sobolev Strichartz estimates 3 is an extremiser for W(d, s) for two distinct values of s. Theorem 1.1 shows that this phenomena does not occur for S(d, s) and gaussian u 0. The condition s = 0 is, in fact, necessary and sufficient for gaussians to be critical points for the functional in (1.5). The sufficiency part was demonstrated by Hundertmark Zharnitsky [6]. See also the work of Christ Quilodrán [3] where a closely related result to Theorem 1.1 was established in the context of adjoint Fourier restriction estimates for the paraboloid; in fact, we prove Theorem 1.1 by making small modifications to their argument. For the cases (d, s) {(1, 0), (, 0)}, it is known that the isotropic gaussian u 0 (x) = exp( x ) is, up to certain transformations, the only extremising initial data for (1.1); see [5], [6]. Furthermore, it is conjectured ([5], [6]) that gaussians are the only extremisers for S(d, 0) for all d 1. Providing a full characterisation of the set of extremisers often requires delicate arguments, and applications of sharp estimates frequently demand that such a characterisation is established. For example, recent work of Duyckaerts Merle Roudenko [4] considered extremisers for the global Strichartz norm for solutions of the mass-critical nonlinear Schrödinger equation i t u + u + γ u 4 d u = 0, u(0) = u0 L (R d ), where γ = 1 in the focusing case and γ = 1 in the defocusing case. In particular, it was shown in [4] that for δ > 0 sufficiently small, I(δ) := sup u L p(d,0) u 0 =δ is attained for some initial data u 0 (δ) L (R d ) with u 0 (δ) = δ. When d = 1, they prove significantly more; it is shown that, as δ 0, I(δ) = S(d, 0)δ + γλ(d)δ 1+ 4 d + O(δ 1+ 8 d ), where Λ(d) is some positive constant, and that any extremising initial data u 0 (δ) is, for δ sufficiently small and up to certain transformations, close to δg 0, where G 0 is the isotropic centred gaussian which has been L - normalised. For this additional information concerning I(δ) when d = 1,, it was vital to know a full characterisation of the extremisers for S(d, 0). Our next result establishes a full characterisation of extremisers for the bilinear Sobolev Strichartz estimate (1.3) of Ozawa Tsutsumi; this question was left open in [7] and the following theorem says that extremisers for (1.3) must be isotropic centred gaussians, up to certain transformations. Theorem 1.. Let d. We have equality in the estimate (1.3) if and only if there exist a, c, d C, b C d, with Re(a) < 0, so that u 0 (x) = exp(a x + b x + c), v 0 (x) = exp(a x + b x + d). (1.8). Further remarks and proofs For d 1, q (1, (d+1) d ) and p = (d+)q d, it was shown in [3] that gaussians are critical points for the L q (P d, dσ) L p (R d+1 ) adjoint Fourier restriction

4 4 N. Bez, C. Jeavons and N. Pattakos estimates associated to the paraboloid P d = {(ξ, ξ ) : ξ R d } R d+1 if and only if q =. Here, dσ is the measure supported on P d given by P d F dσ := R d F (ξ, ξ ) dξ and dξ is Lebesgue measure on R d. A mixednorm generalisation of this is also established in [3] and we remark that Theorem 1.1 may also be extended by measuring the solution in appropriate L r t L p x(r d+1 ) norms. To prove Theorem 1.1 we make minor modifications to the argument in [3] associated with replacing L q (P d, dσ) by Ḣs (R d ). Proof of Theorem 1.1. We fix d 1, s (0, d ) and let p = p(d, s). If Ψ is the functional Ψ(u 0 ) = eit u 0 p L p u 0 p Ḣ s, defined for nonzero u 0 Ḣs (R d ), then u 0 is a critical point if 1 lim ε 0 ε (Ψ(u 0 + εv 0 ) Ψ(u 0 )) = 0 for any v 0 Ḣs (R d ), where ε is a complex parameter. For brevity, we write u(t, ) = e it u 0 and v(t, ) = e it v 0. Using Lemma.3 of [3], which gives an expansion of F + εg p L as p ε 0, ε C, we obtain some constant γ > 1 such that ( ) u + εv p L = p u p L R + p u(t, x) p v(t, x) Re ε dxdt + O( ε γ ) p u(t, x) d+1 and u 0 +εv 0 p Ḣ s ( ) = u 0 p Ḣ s+(π)d p u 0 p Re ε û Ḣ s 0 (ξ) v 0 (ξ) ξ s dξ +O( ε γ ) R d as ε 0. It then follows that u 0 is a critical point if and only if there exists λ > 0 such that R d+1 u(t, x) p u(t, x) exp( i(x ξ t ξ )) dxdt = λ ξ s û 0 (ξ) (.1) for almost all ξ R d. For Theorem 1.1, it suffices to show that u 0 is not a critical point, where u 0 (x) = exp( 1 4 x ). This reduction follows because u 0 such that û 0 (ξ) = exp(a ξ + ib ξ + c), with a, c C and b R d, can be generated from a centred isotropic gaussian under the action of the group generated by: (1) space-time translations: u(t, x) u(t + t 0, x + x 0 ) with (t 0, x 0 ) R d+1 ; () parabolic dilations: u(t, x) u(µ t, µx) with µ > 0; (3) change of scale: u(t, x) µu(t, x) with µ > 0; (4) phase shift: u(t, x) e iθ u(t, x) with θ R.

5 Sharp Sobolev Strichartz estimates 5 The Euler Lagrange equation (.1) is invariant under each of the above actions. For u 0 (x) = exp( 1 4 x ) we have û 0 (ξ) = C d exp( ξ ) and u(t, x) = C d 1 (1 + it) d exp ( x 4(1 + it) for some positive constants C d (which may differ). Thus, (.1) is equivalent to I(a) = C d,s a s (.) for all a [0, ), where C d,s is some positive constant, ( ) H(t) (p )(1 + it) I(a) := exp a dt R (1 + it) d 4 (p ) p 1 it and H(t) := (1 it) d 4 (p 4) (p 1 it) d. A power series expansion of the exponential term leads to (p ) j I j I(a) = a j, (.3) j! where and R j=0 I j := (1 + it) d(s+1) j Hj (t) dt H j (t) = (1 it) d 4 (p 4) (p 1 it) d j. Since H j is holomorphic in the upper half-plane and with d(s+1) we have (1 + it) d(s+1) j Hj (t) C t d(s+1), > 1, it follows (using Lemma 4.1 of [3]) that for j > d(s+1) 1 I j = sin(γ j π) 0 ) r γj H j (i + ir) dr, (.4) where γ j := j d(s+1). Since H j(i + ir) > 0 for all r 0, it is clear that I j = 0 if and only if γ j Z. In the case where s (0, d ) N, using (.), (.3) and a power series uniqueness argument, it follows that I j = 0 for all j s and I j 0 for j = s. If, additionally, d(s+1) / N, then (.4) implies I j 0 for any j > d(s+1) 1, s}, which gives a contradiction. If, instead, N, then max{ d(s+1) for j = d(s+1) 1 we may use Cauchy s residue theorem to obtain I j = (1 + it) 1 H j (t) dt = πh j (i) 0. R Since s > 0 we have j s and so this is also a contradiction.

6 6 N. Bez, C. Jeavons and N. Pattakos In the remaining case where s (0, d ) and s / N, one can see that (.) cannot hold for all a [0, ) since (.3) implies that I(a) is k times (right) differentiable at a = 0 for each k N, whereas a a s is not. Regarding Theorem 1., we begin with the observation that the proof of (1.3) in [7], involving several well-chosen changes of variables, leads to the representation ( ) d 4 (uv) L = C d û 0 ((r p)ω η) v 0 (η) dσ ω,r (η) dσ(ω)drdp, M R d where M = S d 1 R, dσ ω,r (η) = δ(r ω η) dη, δ is the Dirac measure on R supported at the origin, and dσ is the induced Lebesgue measure on S d 1. The constant C d is explicitly computable (and whose value depends on the chosen convention for the Fourier transform). An application of Cauchy Schwarz with respect to the measure dσ r,ω for each fixed (ω, r, p) M yields (1.3). Using the standard fact that equality holds in the Cauchy Schwarz inequality precisely when the constituent functions are linearly dependent, we see that if (u 0, v 0 ) is an extremising pair, then there exists a scalar function Λ such that û 0 ((r p)ω η) = Λ(ω, r, p) v 0 (η) (.5) for almost all η R d (with respect to dσ ω,r ) in the support of the measure dσ ω,r and almost all (ω, r, p) M (with respect to the induced Lebesgue measure). A complete justification that (u 0, v 0 ) satisfies the geometric functional equation in (.5) if and only if (u 0, v 0 ) have the gaussian form in (1.8) requires a multiple-stage argument. The strategy behind the characterisation is to first argue that u 0 and v 0 must be equal (up to non-zero constants), and then establish that û 0 must have a certain amount of regularity. In fact, a delicate geometric argument shows that û 0 must be at least continuous. Once equipped with this information, and furthermore, that û 0 never vanishes, it is possible to solve (.5) by decomposing û 0 = fg into a product of logarithmically even and odd functions, where ( ) 1 f(η) = (û 0 (η)û 0 ( η)) 1 û0 (η) and g(η) =. û 0 ( η) The functional equation inherited by f and g, from û 0, is the classical orthogonal Cauchy functional equation h(η 1 + η ) = h(η 1 )h(η ) whenever η 1 and η are orthogonal vectors in R d. If f and g are normalised so that f(0) = g(0) = 1, this forces f(η) = exp(a η ) and g(η) = exp(b η) for some a C and b C d, and hence û 0 has the desired form (1.8). Full details of this argument can be found in [1] as part of a substantial analysis of sharp bilinear estimates of Ozawa Tsutsumi type.

7 References Sharp Sobolev Strichartz estimates 7 [1] N. Bez, C. Jeavons, N. Pattakos, On sharp bilinear Strichartz estimates of Ozawa Tsutsumi type, in preparation. [] N. Bez and K. M. Rogers, A sharp Strichartz estimate for the wave equation with data in the energy space, J. Eur. Math. Soc. 15 (013), [3] M. Christ, R. Quilodrán, Gaussians rarely extremize adjoint Fourier restriction inequalities for paraboloids, Proc. Amer. Math. Soc., to appear. [4] T. Duyckaerts, F. Merle and S. Roudenko, Maximizers for the Strichartz norm for small solutions of mass-critical NLS, Ann. Sc. Norm. Super. Pisa Cl. Sci., Ann. Scuola Norm. Sup. Pisa 10 (011), [5] D. Foschi, Maximizers for the Strichartz inequality, J. Eur. Math. Soc. 9 (007), [6] D. Hundertmark and V. Zharnitsky, On sharp Strichartz inequalities in low dimensions, Int. Math. Res. Not. (006), Art. ID 34080, 18 pp. [7] T. Ozawa and Y. Tsutsumi, Space-time estimates for null gauge forms and nonlinear Schrödinger equations, Differential Integral Equations 11 (1998), 01. [8] S. Shao, Maximizers for the Strichartz and the Sobolev Strichartz inequalities for the Schrödinger equation, Electron. J. Differential Equations (009), No. 3, 13 pp. Neal Bez School of Mathematics University of Birmingham Birmingham B15 TT United Kingdom Chris Jeavons School of Mathematics University of Birmingham Birmingham B15 TT United Kingdom Nikolaos Pattakos School of Mathematics University of Birmingham Birmingham B15 TT United Kingdom

On sharp bilinear Strichartz estimates of Ozawa Tsutsumi type

On sharp bilinear Strichartz estimates of Ozawa Tsutsumi type c 2017 The Mathematical Society of Japan J. Math. Soc. Japan Vol. 69, No. 2 (2017) pp. 459 476 doi: 10.2969/jmsj/06920459 On sharp bilinear Strichartz estimates of Ozawa Tsutsumi type By Jonathan Bennett,

More information

A COUNTEREXAMPLE TO AN ENDPOINT BILINEAR STRICHARTZ INEQUALITY TERENCE TAO. t L x (R R2 ) f L 2 x (R2 )

A 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 information

Bound-state solutions and well-posedness of the dispersion-managed nonlinear Schrödinger and related equations

Bound-state solutions and well-posedness of the dispersion-managed nonlinear Schrödinger and related equations Bound-state solutions and well-posedness of the dispersion-managed nonlinear Schrödinger and related equations J. Albert and E. Kahlil University of Oklahoma, Langston University 10th IMACS Conference,

More information

MAXIMIZERS FOR THE STRICHARTZ AND THE SOBOLEV-STRICHARTZ INEQUALITIES FOR THE SCHRÖDINGER EQUATION

MAXIMIZERS FOR THE STRICHARTZ AND THE SOBOLEV-STRICHARTZ INEQUALITIES FOR THE SCHRÖDINGER EQUATION Electronic Journal of Differential Euations, Vol. 9(9), No. 3, pp. 1 13. ISSN: 17-6691. UR: http://ede.math.tstate.edu or http://ede.math.unt.edu ftp ede.math.tstate.edu (login: ftp) MAXIMIZERS FOR THE

More information

A PHYSICAL SPACE PROOF OF THE BILINEAR STRICHARTZ AND LOCAL SMOOTHING ESTIMATES FOR THE SCHRÖDINGER EQUATION

A PHYSICAL SPACE PROOF OF THE BILINEAR STRICHARTZ AND LOCAL SMOOTHING ESTIMATES FOR THE SCHRÖDINGER EQUATION A PHYSICAL SPACE PROOF OF THE BILINEAR STRICHARTZ AND LOCAL SMOOTHING ESTIMATES FOR THE SCHRÖDINGER EQUATION TERENCE TAO Abstract. Let d 1, and let u, v : R R d C be Schwartz space solutions to the Schrödinger

More information

1D Quintic NLS with White Noise Dispersion

1D Quintic NLS with White Noise Dispersion 2011 年 8 月 8 日 1D Quintic NLS with White Noise Dispersion Yoshio TSUTSUMI, Kyoto University, Arnaud DEBUSSCHE, ENS de Cachan, Bretagne 1D quintic NLS with white noise dispersion idu + xu 2 dβ(t) =λ u 4

More information

1D Quintic NLS with White Noise Dispersion

1D Quintic NLS with White Noise Dispersion 2011 年 1 月 7 日 1D Quintic NLS with White Noise Dispersion Yoshio TSUTSUMI, Kyoto University, Arnaud DEBUSSCHE, ENS de Cachan, Bretagne 1D quintic NLS with white noise dispersion idu + xu 2 dβ(t) =λ u 4

More information

BLOWUP THEORY FOR THE CRITICAL NONLINEAR SCHRÖDINGER EQUATIONS REVISITED

BLOWUP THEORY FOR THE CRITICAL NONLINEAR SCHRÖDINGER EQUATIONS REVISITED BLOWUP THEORY FOR THE CRITICAL NONLINEAR SCHRÖDINGER EQUATIONS REVISITED TAOUFIK HMIDI AND SAHBI KERAANI Abstract. In this note we prove a refined version of compactness lemma adapted to the blowup analysis

More information

SCATTERING FOR THE TWO-DIMENSIONAL NLS WITH EXPONENTIAL NONLINEARITY

SCATTERING FOR THE TWO-DIMENSIONAL NLS WITH EXPONENTIAL NONLINEARITY SCATTERING FOR THE TWO-DIMENSIONAL NLS WITH EXPONENTIAL NONLINEARITY S. IBRAHIM, M. MAJDOUB, N. MASMOUDI, AND K. NAKANISHI Abstract. We investigate existence and asymptotic completeness of the wave operators

More information

Why do Solutions of the Maxwell Boltzmann Equation Tend to be Gaussians? A Simple Answer

Why do Solutions of the Maxwell Boltzmann Equation Tend to be Gaussians? A Simple Answer Documenta Math. 1267 Why do Solutions of the Maxwell Boltzmann Equation Tend to be Gaussians? A Simple Answer Dirk Hundertmark, Young-Ran Lee Received: December 20, 2016 Communicated by Heinz Siedentop

More information

CUTOFF RESOLVENT ESTIMATES AND THE SEMILINEAR SCHRÖDINGER EQUATION

CUTOFF RESOLVENT ESTIMATES AND THE SEMILINEAR SCHRÖDINGER EQUATION CUTOFF RESOLVENT ESTIMATES AND THE SEMILINEAR SCHRÖDINGER EQUATION HANS CHRISTIANSON Abstract. This paper shows how abstract resolvent estimates imply local smoothing for solutions to the Schrödinger equation.

More information

arxiv: v1 [math.ap] 20 Nov 2007

arxiv: v1 [math.ap] 20 Nov 2007 Long range scattering for the Maxwell-Schrödinger system with arbitrarily large asymptotic data arxiv:0711.3100v1 [math.ap] 20 Nov 2007 J. Ginibre Laboratoire de Physique Théorique Université de Paris

More information

SCALE INVARIANT FOURIER RESTRICTION TO A HYPERBOLIC SURFACE

SCALE INVARIANT FOURIER RESTRICTION TO A HYPERBOLIC SURFACE SCALE INVARIANT FOURIER RESTRICTION TO A HYPERBOLIC SURFACE BETSY STOVALL Abstract. This result sharpens the bilinear to linear deduction of Lee and Vargas for extension estimates on the hyperbolic paraboloid

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

RANDOM PROPERTIES BENOIT PAUSADER

RANDOM PROPERTIES BENOIT PAUSADER RANDOM PROPERTIES BENOIT PAUSADER. Quasilinear problems In general, one consider the following trichotomy for nonlinear PDEs: A semilinear problem is a problem where the highest-order terms appears linearly

More information

TADAHIRO OH 0, 3 8 (T R), (1.5) The result in [2] is in fact stated for time-periodic functions: 0, 1 3 (T 2 ). (1.4)

TADAHIRO OH 0, 3 8 (T R), (1.5) The result in [2] is in fact stated for time-periodic functions: 0, 1 3 (T 2 ). (1.4) PERIODIC L 4 -STRICHARTZ ESTIMATE FOR KDV TADAHIRO OH 1. Introduction In [], Bourgain proved global well-posedness of the periodic KdV in L T): u t + u xxx + uu x 0, x, t) T R. 1.1) The key ingredient

More information

Sharp blow-up criteria for the Davey-Stewartson system in R 3

Sharp blow-up criteria for the Davey-Stewartson system in R 3 Dynamics of PDE, Vol.8, No., 9-60, 011 Sharp blow-up criteria for the Davey-Stewartson system in R Jian Zhang Shihui Zhu Communicated by Y. Charles Li, received October 7, 010. Abstract. In this paper,

More information

A REMARK ON AN EQUATION OF WAVE MAPS TYPE WITH VARIABLE COEFFICIENTS

A REMARK ON AN EQUATION OF WAVE MAPS TYPE WITH VARIABLE COEFFICIENTS A REMARK ON AN EQUATION OF WAVE MAPS TYPE WITH VARIABLE COEFFICIENTS DAN-ANDREI GEBA Abstract. We obtain a sharp local well-posedness result for an equation of wave maps type with variable coefficients.

More information

SCATTERING FOR THE NON-RADIAL 3D CUBIC NONLINEAR SCHRÖDINGER EQUATION

SCATTERING FOR THE NON-RADIAL 3D CUBIC NONLINEAR SCHRÖDINGER EQUATION SCATTERING FOR THE NON-RADIAL 3D CUBIC NONLINEAR SCHRÖDINGER EQUATION THOMAS DUYCKAERTS, JUSTIN HOLMER, AND SVETLANA ROUDENKO Abstract. Scattering of radial H 1 solutions to the 3D focusing cubic nonlinear

More information

Solutions to the Nonlinear Schrödinger Equation in Hyperbolic Space

Solutions to the Nonlinear Schrödinger Equation in Hyperbolic Space Solutions to the Nonlinear Schrödinger Equation in Hyperbolic Space SPUR Final Paper, Summer 2014 Peter Kleinhenz Mentor: Chenjie Fan Project suggested by Gigliola Staffilani July 30th, 2014 Abstract In

More information

M ath. Res. Lett. 15 (2008), no. 6, c International Press 2008 SCATTERING FOR THE NON-RADIAL 3D CUBIC NONLINEAR SCHRÖDINGER EQUATION

M ath. Res. Lett. 15 (2008), no. 6, c International Press 2008 SCATTERING FOR THE NON-RADIAL 3D CUBIC NONLINEAR SCHRÖDINGER EQUATION M ath. Res. Lett. 15 (2008), no. 6, 1233 1250 c International Press 2008 SCATTERING FOR THE NON-RADIAL 3D CUBIC NONLINEAR SCHRÖDINGER EQUATION Thomas Duyckaerts, Justin Holmer, and Svetlana Roudenko Abstract.

More information

A semilinear Schrödinger equation with magnetic field

A semilinear Schrödinger equation with magnetic field A semilinear Schrödinger equation with magnetic field Andrzej Szulkin Department of Mathematics, Stockholm University 106 91 Stockholm, Sweden 1 Introduction In this note we describe some recent results

More information

Scattering for cubic-quintic nonlinear Schrödinger equation on R 3

Scattering for cubic-quintic nonlinear Schrödinger equation on R 3 Scattering for cubic-quintic nonlinear Schrödinger equation on R 3 Oana Pocovnicu Princeton University March 9th 2013 Joint work with R. Killip (UCLA), T. Oh (Princeton), M. Vişan (UCLA) SCAPDE UCLA 1

More information

Energy transfer model and large periodic boundary value problem for the quintic NLS

Energy transfer model and large periodic boundary value problem for the quintic NLS Energy transfer model and large periodic boundary value problem for the quintic NS Hideo Takaoka Department of Mathematics, Kobe University 1 ntroduction This note is based on a talk given at the conference

More information

Nonlinear Schrödinger Equation BAOXIANG WANG. Talk at Tsinghua University 2012,3,16. School of Mathematical Sciences, Peking University.

Nonlinear Schrödinger Equation BAOXIANG WANG. Talk at Tsinghua University 2012,3,16. School of Mathematical Sciences, Peking University. Talk at Tsinghua University 2012,3,16 Nonlinear Schrödinger Equation BAOXIANG WANG School of Mathematical Sciences, Peking University 1 1 33 1. Schrödinger E. Schrödinger (1887-1961) E. Schrödinger, (1887,

More information

arxiv: v2 [math.ca] 23 Jun 2017

arxiv: v2 [math.ca] 23 Jun 2017 A SHARP k-plane STRICHARTZ INEQUALITY FOR THE SCHRÖDINGER EQUATION arxiv:1611.0369v [math.ca] 3 Jun 017 J. BENNETT, N. BEZ, T. C. FLOCK, S. GUTIÉRREZ AND M. ILIOPOULOU Abstract. We prove that X( u ) L

More information

ON WEAKLY NONLINEAR BACKWARD PARABOLIC PROBLEM

ON WEAKLY NONLINEAR BACKWARD PARABOLIC PROBLEM ON WEAKLY NONLINEAR BACKWARD PARABOLIC PROBLEM OLEG ZUBELEVICH DEPARTMENT OF MATHEMATICS THE BUDGET AND TREASURY ACADEMY OF THE MINISTRY OF FINANCE OF THE RUSSIAN FEDERATION 7, ZLATOUSTINSKY MALIY PER.,

More information

SYMMETRY IN REARRANGEMENT OPTIMIZATION PROBLEMS

SYMMETRY IN REARRANGEMENT OPTIMIZATION PROBLEMS Electronic Journal of Differential Equations, Vol. 2009(2009), No. 149, pp. 1 10. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu SYMMETRY IN REARRANGEMENT

More information

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

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 (, )

More information

A 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. 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 information

Recent developments on the global behavior to critical nonlinear dispersive equations. Carlos E. Kenig

Recent developments on the global behavior to critical nonlinear dispersive equations. Carlos E. Kenig Recent developments on the global behavior to critical nonlinear dispersive equations Carlos E. Kenig In the last 25 years or so, there has been considerable interest in the study of non-linear partial

More information

Uniqueness of ground state solutions of non-local equations in R N

Uniqueness of ground state solutions of non-local equations in R N Uniqueness of ground state solutions of non-local equations in R N Rupert L. Frank Department of Mathematics Princeton University Joint work with Enno Lenzmann and Luis Silvestre Uniqueness and non-degeneracy

More information

Piecewise Smooth Solutions to the Burgers-Hilbert Equation

Piecewise Smooth Solutions to the Burgers-Hilbert Equation Piecewise Smooth Solutions to the Burgers-Hilbert Equation Alberto Bressan and Tianyou Zhang Department of Mathematics, Penn State University, University Park, Pa 68, USA e-mails: bressan@mathpsuedu, zhang

More information

Scattering for NLS with a potential on the line

Scattering for NLS with a potential on the line Asymptotic Analysis 1 (16) 1 39 1 DOI 1.333/ASY-161384 IOS Press Scattering for NLS with a potential on the line David Lafontaine Laboratoire de Mathématiques J.A. Dieudonné, UMR CNRS 7351, Université

More information

doi: /j.jde

doi: /j.jde doi: 10.1016/j.jde.016.08.019 On Second Order Hyperbolic Equations with Coefficients Degenerating at Infinity and the Loss of Derivatives and Decays Tamotu Kinoshita Institute of Mathematics, University

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

On the minimum of certain functional related to the Schrödinger equation

On the minimum of certain functional related to the Schrödinger equation Electronic Journal of Qualitative Theory of Differential Equations 2013, No. 8, 1-21; http://www.math.u-szeged.hu/ejqtde/ On the minimum of certain functional related to the Schrödinger equation Artūras

More information

LINEAR EQUATIONS WITH UNKNOWNS FROM A MULTIPLICATIVE GROUP IN A FUNCTION FIELD. To Professor Wolfgang Schmidt on his 75th birthday

LINEAR EQUATIONS WITH UNKNOWNS FROM A MULTIPLICATIVE GROUP IN A FUNCTION FIELD. To Professor Wolfgang Schmidt on his 75th birthday LINEAR EQUATIONS WITH UNKNOWNS FROM A MULTIPLICATIVE GROUP IN A FUNCTION FIELD JAN-HENDRIK EVERTSE AND UMBERTO ZANNIER To Professor Wolfgang Schmidt on his 75th birthday 1. Introduction Let K be a field

More information

Jae Gil Choi and Young Seo Park

Jae Gil Choi and Young Seo Park Kangweon-Kyungki Math. Jour. 11 (23), No. 1, pp. 17 3 TRANSLATION THEOREM ON FUNCTION SPACE Jae Gil Choi and Young Seo Park Abstract. In this paper, we use a generalized Brownian motion process to define

More information

arxiv: v3 [math.ap] 1 Sep 2017

arxiv: v3 [math.ap] 1 Sep 2017 arxiv:1603.0685v3 [math.ap] 1 Sep 017 UNIQUE CONTINUATION FOR THE SCHRÖDINGER EQUATION WITH GRADIENT TERM YOUNGWOO KOH AND IHYEOK SEO Abstract. We obtain a unique continuation result for the differential

More information

A survey on l 2 decoupling

A survey on l 2 decoupling A survey on l 2 decoupling Po-Lam Yung 1 The Chinese University of Hong Kong January 31, 2018 1 Research partially supported by HKRGC grant 14313716, and by CUHK direct grants 4053220, 4441563 Introduction

More information

Global well-posedness for semi-linear Wave and Schrödinger equations. Slim Ibrahim

Global well-posedness for semi-linear Wave and Schrödinger equations. Slim Ibrahim Global well-posedness for semi-linear Wave and Schrödinger equations Slim Ibrahim McMaster University, Hamilton ON University of Calgary, April 27th, 2006 1 1 Introduction Nonlinear Wave equation: ( 2

More information

Some notes on a second-order random boundary value problem

Some notes on a second-order random boundary value problem ISSN 1392-5113 Nonlinear Analysis: Modelling and Control, 217, Vol. 22, No. 6, 88 82 https://doi.org/1.15388/na.217.6.6 Some notes on a second-order random boundary value problem Fairouz Tchier a, Calogero

More information

ON A MEASURE THEORETIC AREA FORMULA

ON A MEASURE THEORETIC AREA FORMULA ON A MEASURE THEORETIC AREA FORMULA VALENTINO MAGNANI Abstract. We review some classical differentiation theorems for measures, showing how they can be turned into an integral representation of a Borel

More information

ANALYTIC SMOOTHING EFFECT FOR NONLI TitleSCHRÖDINGER EQUATION IN TWO SPACE DIMENSIONS. Citation Osaka Journal of Mathematics.

ANALYTIC SMOOTHING EFFECT FOR NONLI TitleSCHRÖDINGER EQUATION IN TWO SPACE DIMENSIONS. Citation Osaka Journal of Mathematics. ANALYTIC SMOOTHING EFFECT FOR NONLI TitleSCHRÖDINGER EQUATION IN TWO SPACE DIMENSIONS Author(s) Hoshino, Gaku; Ozawa, Tohru Citation Osaka Journal of Mathematics. 51(3) Issue 014-07 Date Text Version publisher

More information

MAXIMIZATION AND MINIMIZATION PROBLEMS RELATED TO A p-laplacian EQUATION ON A MULTIPLY CONNECTED DOMAIN. N. Amiri and M.

MAXIMIZATION AND MINIMIZATION PROBLEMS RELATED TO A p-laplacian EQUATION ON A MULTIPLY CONNECTED DOMAIN. N. Amiri and M. TAIWANESE JOURNAL OF MATHEMATICS Vol. 19, No. 1, pp. 243-252, February 2015 DOI: 10.11650/tjm.19.2015.3873 This paper is available online at http://journal.taiwanmathsoc.org.tw MAXIMIZATION AND MINIMIZATION

More information

Inégalités de dispersion via le semi-groupe de la chaleur

Inégalités de dispersion via le semi-groupe de la chaleur Inégalités de dispersion via le semi-groupe de la chaleur Valentin Samoyeau, Advisor: Frédéric Bernicot. Laboratoire de Mathématiques Jean Leray, Université de Nantes January 28, 2016 1 Introduction Schrödinger

More information

On the relation between scaling properties of functionals and existence of constrained minimizers

On the relation between scaling properties of functionals and existence of constrained minimizers On the relation between scaling properties of functionals and existence of constrained minimizers Jacopo Bellazzini Dipartimento di Matematica Applicata U. Dini Università di Pisa January 11, 2011 J. Bellazzini

More information

DISPERSIVE EQUATIONS: A SURVEY

DISPERSIVE EQUATIONS: A SURVEY DISPERSIVE EQUATIONS: A SURVEY GIGLIOLA STAFFILANI 1. Introduction These notes were written as a guideline for a short talk; hence, the references and the statements of the theorems are often not given

More information

Analysis of the (CR) equation in higher dimensions

Analysis of the (CR) equation in higher dimensions Analysis of the (CR) equation in higher dimensions arxiv:60.05736v [math.ap] 8 Oct 06 T. Buckmaster, P. Germain, Z. Hani, J. Shatah October 9, 06 Abstract This paper is devoted to the analysis of the continuous

More information

Michael Lacey and Christoph Thiele. f(ξ)e 2πiξx dξ

Michael Lacey and Christoph Thiele. f(ξ)e 2πiξx dξ Mathematical Research Letters 7, 36 370 (2000) A PROOF OF BOUNDEDNESS OF THE CARLESON OPERATOR Michael Lacey and Christoph Thiele Abstract. We give a simplified proof that the Carleson operator is of weaktype

More information

A Minimal Uncertainty Product for One Dimensional Semiclassical Wave Packets

A Minimal Uncertainty Product for One Dimensional Semiclassical Wave Packets A Minimal Uncertainty Product for One Dimensional Semiclassical Wave Packets George A. Hagedorn Happy 60 th birthday, Mr. Fritz! Abstract. Although real, normalized Gaussian wave packets minimize the product

More information

Kernel Method: Data Analysis with Positive Definite Kernels

Kernel Method: Data Analysis with Positive Definite Kernels Kernel Method: Data Analysis with Positive Definite Kernels 2. Positive Definite Kernel and Reproducing Kernel Hilbert Space Kenji Fukumizu The Institute of Statistical Mathematics. Graduate University

More information

A. Iosevich and I. Laba January 9, Introduction

A. Iosevich and I. Laba January 9, Introduction K-DISTANCE SETS, FALCONER CONJECTURE, AND DISCRETE ANALOGS A. Iosevich and I. Laba January 9, 004 Abstract. In this paper we prove a series of results on the size of distance sets corresponding to sets

More information

Global well-posedness for KdV in Sobolev spaces of negative index

Global well-posedness for KdV in Sobolev spaces of negative index Electronic Journal of Differential Equations, Vol. (), No. 6, pp. 7. ISSN: 7-669. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp) Global well-posedness for

More information

Almost Sure Well-Posedness of Cubic NLS on the Torus Below L 2

Almost Sure Well-Posedness of Cubic NLS on the Torus Below L 2 Almost Sure Well-Posedness of Cubic NLS on the Torus Below L 2 J. Colliander University of Toronto Sapporo, 23 November 2009 joint work with Tadahiro Oh (U. Toronto) 1 Introduction: Background, Motivation,

More information

The Chern-Simons-Schrödinger equation

The Chern-Simons-Schrödinger equation The Chern-Simons-Schrödinger equation Low regularity local wellposedness Baoping Liu, Paul Smith, Daniel Tataru University of California, Berkeley July 16, 2012 Paul Smith (UC Berkeley) Chern-Simons-Schrödinger

More information

GLOBAL WELL-POSEDNESS FOR NONLINEAR NONLOCAL CAUCHY PROBLEMS ARISING IN ELASTICITY

GLOBAL WELL-POSEDNESS FOR NONLINEAR NONLOCAL CAUCHY PROBLEMS ARISING IN ELASTICITY Electronic Journal of Differential Equations, Vol. 2017 (2017), No. 55, pp. 1 7. ISSN: 1072-6691. UL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu GLOBAL WELL-POSEDNESS FO NONLINEA NONLOCAL

More information

LECTURE NOTES : INTRODUCTION TO DISPERSIVE PARTIAL DIFFERENTIAL EQUATIONS

LECTURE NOTES : INTRODUCTION TO DISPERSIVE PARTIAL DIFFERENTIAL EQUATIONS LECTURE NOTES : INTRODUCTION TO DISPERSIVE PARTIAL DIFFERENTIAL EQUATIONS NIKOLAOS TZIRAKIS Abstract. The aim of this manuscript is to provide a short and accessible introduction to the modern theory of

More information

Nonlinear elliptic systems with exponential nonlinearities

Nonlinear elliptic systems with exponential nonlinearities 22-Fez conference on Partial Differential Equations, Electronic Journal of Differential Equations, Conference 9, 22, pp 139 147. http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu

More information

Finite Element Clifford Algebra: A New Toolkit for Evolution Problems

Finite Element Clifford Algebra: A New Toolkit for Evolution Problems Finite Element Clifford Algebra: A New Toolkit for Evolution Problems Andrew Gillette joint work with Michael Holst Department of Mathematics University of California, San Diego http://ccom.ucsd.edu/ agillette/

More information

BIHARMONIC WAVE MAPS INTO SPHERES

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.

More information

VANISHING-CONCENTRATION-COMPACTNESS ALTERNATIVE FOR THE TRUDINGER-MOSER INEQUALITY IN R N

VANISHING-CONCENTRATION-COMPACTNESS ALTERNATIVE FOR THE TRUDINGER-MOSER INEQUALITY IN R N VAISHIG-COCETRATIO-COMPACTESS ALTERATIVE FOR THE TRUDIGER-MOSER IEQUALITY I R Abstract. Let 2, a > 0 0 < b. Our aim is to clarify the influence of the constraint S a,b = { u W 1, (R ) u a + u b = 1 } on

More information

Scattering theory for nonlinear Schrödinger equation with inverse square potential

Scattering theory for nonlinear Schrödinger equation with inverse square potential Scattering theory for nonlinear Schrödinger equation with inverse square potential Université Nice Sophia-Antipolis Based on joint work with: Changxing Miao (IAPCM) and Junyong Zhang (BIT) February -6,

More information

GAUSSIAN MEASURE OF SECTIONS OF DILATES AND TRANSLATIONS OF CONVEX BODIES. 2π) n

GAUSSIAN MEASURE OF SECTIONS OF DILATES AND TRANSLATIONS OF CONVEX BODIES. 2π) n GAUSSIAN MEASURE OF SECTIONS OF DILATES AND TRANSLATIONS OF CONVEX BODIES. A. ZVAVITCH Abstract. In this paper we give a solution for the Gaussian version of the Busemann-Petty problem with additional

More information

Duality of multiparameter Hardy spaces H p on spaces of homogeneous type

Duality of multiparameter Hardy spaces H p on spaces of homogeneous type Duality of multiparameter Hardy spaces H p on spaces of homogeneous type Yongsheng Han, Ji Li, and Guozhen Lu Department of Mathematics Vanderbilt University Nashville, TN Internet Analysis Seminar 2012

More information

for all subintervals I J. If the same is true for the dyadic subintervals I D J only, we will write ϕ BMO d (J). In fact, the following is true

for all subintervals I J. If the same is true for the dyadic subintervals I D J only, we will write ϕ BMO d (J). In fact, the following is true 3 ohn Nirenberg inequality, Part I A function ϕ L () belongs to the space BMO() if sup ϕ(s) ϕ I I I < for all subintervals I If the same is true for the dyadic subintervals I D only, we will write ϕ BMO

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

Dynamics of energy-critical wave equation

Dynamics of energy-critical wave equation Dynamics of energy-critical wave equation Carlos Kenig Thomas Duyckaerts Franke Merle September 21th, 2014 Duyckaerts Kenig Merle Critical wave 2014 1 / 22 Plan 1 Introduction Duyckaerts Kenig Merle Critical

More information

FOURIER METHODS AND DISTRIBUTIONS: SOLUTIONS

FOURIER METHODS AND DISTRIBUTIONS: SOLUTIONS Centre for Mathematical Sciences Mathematics, Faculty of Science FOURIER METHODS AND DISTRIBUTIONS: SOLUTIONS. We make the Ansatz u(x, y) = ϕ(x)ψ(y) and look for a solution which satisfies the boundary

More information

2 A Model, Harmonic Map, Problem

2 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 information

Switching, sparse and averaged control

Switching, sparse and averaged control Switching, sparse and averaged control Enrique Zuazua Ikerbasque & BCAM Basque Center for Applied Mathematics Bilbao - Basque Country- Spain zuazua@bcamath.org http://www.bcamath.org/zuazua/ WG-BCAM, February

More information

STRONGLY SINGULAR INTEGRALS ALONG CURVES. 1. Introduction. It is standard and well known that the Hilbert transform along curves: f(x γ(t)) dt

STRONGLY SINGULAR INTEGRALS ALONG CURVES. 1. Introduction. It is standard and well known that the Hilbert transform along curves: f(x γ(t)) dt STRONGLY SINGULAR INTEGRALS ALONG CURVES NORBERTO LAGHI NEIL LYALL Abstract. In this article we obtain L 2 bounds for strongly singular integrals along curves in R d ; our results both generalise and extend

More information

Sharp Well-posedness Results for the BBM Equation

Sharp Well-posedness Results for the BBM Equation Sharp Well-posedness Results for the BBM Equation J.L. Bona and N. zvetkov Abstract he regularized long-wave or BBM equation u t + u x + uu x u xxt = was derived as a model for the unidirectional propagation

More information

Decouplings and applications

Decouplings and applications April 27, 2018 Let Ξ be a collection of frequency points ξ on some curved, compact manifold S of diameter 1 in R n (e.g. the unit sphere S n 1 ) Let B R = B(c, R) be a ball with radius R 1. Let also a

More information

Unimodular Bilinear multipliers on L p spaces

Unimodular Bilinear multipliers on L p spaces Jotsaroop Kaur (joint work with Saurabh Shrivastava) Department of Mathematics, IISER Bhopal December 18, 2017 Fourier Multiplier Let m L (R n ), we define the Fourier multiplier operator as follows :

More information

On the logarithm of the minimizing integrand for certain variational problems in two dimensions

On the logarithm of the minimizing integrand for certain variational problems in two dimensions On the logarithm of the minimizing integrand for certain variational problems in two dimensions University of Kentucky John L. Lewis University of Kentucky Andrew Vogel Syracuse University 2012 Spring

More information

ON THE PROBABILISTIC CAUCHY THEORY FOR NONLINEAR DISPERSIVE PDES

ON THE PROBABILISTIC CAUCHY THEORY FOR NONLINEAR DISPERSIVE PDES ON THE PROBABILISTIC CAUCHY THEORY FOR NONLINEAR DISPERSIVE PDES ÁRPÁD BÉNYI, TADAHIRO OH, AND OANA POCOVNICU Abstract. In this note, we review some of the recent developments in the well-posedness theory

More information

A Sharpened Hausdorff-Young Inequality

A Sharpened Hausdorff-Young Inequality A Sharpened Hausdorff-Young Inequality Michael Christ University of California, Berkeley IPAM Workshop Kakeya Problem, Restriction Problem, Sum-Product Theory and perhaps more May 5, 2014 Hausdorff-Young

More information

A NOTE ON ALMOST PERIODIC VARIATIONAL EQUATIONS

A NOTE ON ALMOST PERIODIC VARIATIONAL EQUATIONS A NOTE ON ALMOST PERIODIC VARIATIONAL EQUATIONS PETER GIESL AND MARTIN RASMUSSEN Abstract. The variational equation of a nonautonomous differential equation ẋ = F t, x) along a solution µ is given by ẋ

More information

Invariant measures and the soliton resolution conjecture

Invariant measures and the soliton resolution conjecture Invariant measures and the soliton resolution conjecture Stanford University The focusing nonlinear Schrödinger equation A complex-valued function u of two variables x and t, where x R d is the space variable

More information

On Asymptotic Variational Wave Equations

On Asymptotic Variational Wave Equations On Asymptotic Variational Wave Equations Alberto Bressan 1, Ping Zhang 2, and Yuxi Zheng 1 1 Department of Mathematics, Penn State University, PA 1682. E-mail: bressan@math.psu.edu; yzheng@math.psu.edu

More information

On non negative solutions of some quasilinear elliptic inequalities

On 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 information

Some asymptotic properties of solutions for Burgers equation in L p (R)

Some asymptotic properties of solutions for Burgers equation in L p (R) ARMA manuscript No. (will be inserted by the editor) Some asymptotic properties of solutions for Burgers equation in L p (R) PAULO R. ZINGANO Abstract We discuss time asymptotic properties of solutions

More information

The uniform uncertainty principle and compressed sensing Harmonic analysis and related topics, Seville December 5, 2008

The uniform uncertainty principle and compressed sensing Harmonic analysis and related topics, Seville December 5, 2008 The uniform uncertainty principle and compressed sensing Harmonic analysis and related topics, Seville December 5, 2008 Emmanuel Candés (Caltech), Terence Tao (UCLA) 1 Uncertainty principles A basic principle

More information

Minimization problems on the Hardy-Sobolev inequality

Minimization problems on the Hardy-Sobolev inequality manuscript No. (will be inserted by the editor) Minimization problems on the Hardy-Sobolev inequality Masato Hashizume Received: date / Accepted: date Abstract We study minimization problems on Hardy-Sobolev

More information

NORM DEPENDENCE OF THE COEFFICIENT MAP ON THE WINDOW SIZE 1

NORM DEPENDENCE OF THE COEFFICIENT MAP ON THE WINDOW SIZE 1 NORM DEPENDENCE OF THE COEFFICIENT MAP ON THE WINDOW SIZE Thomas I. Seidman and M. Seetharama Gowda Department of Mathematics and Statistics University of Maryland Baltimore County Baltimore, MD 8, U.S.A

More information

LARGE DEVIATIONS OF TYPICAL LINEAR FUNCTIONALS ON A CONVEX BODY WITH UNCONDITIONAL BASIS. S. G. Bobkov and F. L. Nazarov. September 25, 2011

LARGE DEVIATIONS OF TYPICAL LINEAR FUNCTIONALS ON A CONVEX BODY WITH UNCONDITIONAL BASIS. S. G. Bobkov and F. L. Nazarov. September 25, 2011 LARGE DEVIATIONS OF TYPICAL LINEAR FUNCTIONALS ON A CONVEX BODY WITH UNCONDITIONAL BASIS S. G. Bobkov and F. L. Nazarov September 25, 20 Abstract We study large deviations of linear functionals on an isotropic

More information

Analytic families of multilinear operators

Analytic families of multilinear operators Analytic families of multilinear operators Mieczysław Mastyło Adam Mickiewicz University in Poznań Nonlinar Functional Analysis Valencia 17-20 October 2017 Based on a joint work with Loukas Grafakos M.

More information

TD 1: Hilbert Spaces and Applications

TD 1: Hilbert Spaces and Applications Université Paris-Dauphine Functional Analysis and PDEs Master MMD-MA 2017/2018 Generalities TD 1: Hilbert Spaces and Applications Exercise 1 (Generalized Parallelogram law). Let (H,, ) be a Hilbert space.

More information

ANALYSIS QUALIFYING EXAM FALL 2017: SOLUTIONS. 1 cos(nx) lim. n 2 x 2. g n (x) = 1 cos(nx) n 2 x 2. x 2.

ANALYSIS QUALIFYING EXAM FALL 2017: SOLUTIONS. 1 cos(nx) lim. n 2 x 2. g n (x) = 1 cos(nx) n 2 x 2. x 2. ANALYSIS QUALIFYING EXAM FALL 27: SOLUTIONS Problem. Determine, with justification, the it cos(nx) n 2 x 2 dx. Solution. For an integer n >, define g n : (, ) R by Also define g : (, ) R by g(x) = g n

More information

Long-term dynamics of nonlinear wave equations

Long-term dynamics of nonlinear wave equations Long-term dynamics of nonlinear wave equations W. Schlag (University of Chicago) Recent Developments & Future Directions, September 2014 Wave maps Let (M, g) be a Riemannian manifold, and u : R 1+d t,x

More information

5.1 Approximation of convex functions

5.1 Approximation of convex functions Chapter 5 Convexity Very rough draft 5.1 Approximation of convex functions Convexity::approximation bound Expand Convexity simplifies arguments from Chapter 3. Reasons: local minima are global minima and

More information

Generation of undular bores and solitary wave trains in fully nonlinear shallow water theory

Generation of undular bores and solitary wave trains in fully nonlinear shallow water theory Generation of undular bores and solitary wave trains in fully nonlinear shallow water theory Gennady El 1, Roger Grimshaw 1 and Noel Smyth 2 1 Loughborough University, UK, 2 University of Edinburgh, UK

More information

ON THE CAUCHY-PROBLEM FOR GENERALIZED KADOMTSEV-PETVIASHVILI-II EQUATIONS

ON THE CAUCHY-PROBLEM FOR GENERALIZED KADOMTSEV-PETVIASHVILI-II EQUATIONS Electronic Journal of Differential Equations, Vol. 009(009), No. 8, pp. 1 9. ISSN: 107-6691. URL: http://ejde.math.tstate.edu or http://ejde.math.unt.edu ftp ejde.math.tstate.edu ON THE CAUCHY-PROBLEM

More information

HARDY INEQUALITIES WITH BOUNDARY TERMS. x 2 dx u 2 dx. (1.2) u 2 = u 2 dx.

HARDY INEQUALITIES WITH BOUNDARY TERMS. x 2 dx u 2 dx. (1.2) u 2 = u 2 dx. Electronic Journal of Differential Equations, Vol. 003(003), No. 3, pp. 1 8. ISSN: 107-6691. UL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp) HADY INEQUALITIES

More information

Fourier analysis, measures, and distributions. Alan Haynes

Fourier analysis, measures, and distributions. Alan Haynes Fourier analysis, measures, and distributions Alan Haynes 1 Mathematics of diffraction Physical diffraction As a physical phenomenon, diffraction refers to interference of waves passing through some medium

More information

Introduction to Real Analysis Alternative Chapter 1

Introduction to Real Analysis Alternative Chapter 1 Christopher Heil Introduction to Real Analysis Alternative Chapter 1 A Primer on Norms and Banach Spaces Last Updated: March 10, 2018 c 2018 by Christopher Heil Chapter 1 A Primer on Norms and Banach Spaces

More information

Heat equations with singular potentials: Hardy & Carleman inequalities, well-posedness & control

Heat equations with singular potentials: Hardy & Carleman inequalities, well-posedness & control Outline Heat equations with singular potentials: Hardy & Carleman inequalities, well-posedness & control IMDEA-Matemáticas & Universidad Autónoma de Madrid Spain enrique.zuazua@uam.es Analysis and control

More information

ASYMPTOTICALLY NONEXPANSIVE MAPPINGS IN MODULAR FUNCTION SPACES ABSTRACT

ASYMPTOTICALLY NONEXPANSIVE MAPPINGS IN MODULAR FUNCTION SPACES ABSTRACT ASYMPTOTICALLY NONEXPANSIVE MAPPINGS IN MODULAR FUNCTION SPACES T. DOMINGUEZ-BENAVIDES, M.A. KHAMSI AND S. SAMADI ABSTRACT In this paper, we prove that if ρ is a convex, σ-finite modular function satisfying

More information