Endpoint Strichartz estimates for the magnetic Schrödinger equation
|
|
- Gerard Stephens
- 6 years ago
- Views:
Transcription
1 Journal of Functional Analysis 58 (010) Endpoint Strichartz estimates for the magnetic Schrödinger equation Piero D Ancona a, Luca Fanelli b,,luisvega b, Nicola Visciglia c a SAPIENZA Università di Roma, Dipartimento di Matematica, Piazzale A. Moro, I Roma, Italy b Universidad del Pais Vasco, Departamento de Matemáticas, Apartado 644, 48080, Bilbao, Spain c Universitá di Pisa, Dipartimento di Matematica, Largo B. Pontecorvo 5, Pisa, Italy Received 4 January 009; accepted 4 February 010 Available online 19 February 010 Communicated by I. Rodnianski Abstract We prove Strichartz estimates for the Schrödinger equation with an electromagnetic potential, in dimension n 3. The decay and regularity assumptions on the potentials are almost critical, i.e., close to the Coulomb case. In addition, we require repulsivity and a nontrapping condition, which are expressed as smallness of suitable components of the potentials, while the potentials themselves can be large. The proof is based on smoothing estimates and new Sobolev embeddings for spaces associated to magnetic potentials. 010 Elsevier Inc. All rights reserved. Keywords: Strichartz estimates; Dispersive equations; Schrödinger equation; Magnetic potential 1. Introduction Recent research on linear and nonlinear dispersive equations is largely focused on measuring precisely the rate of decay of solutions. Indeed, decay and Strichartz estimates are one of the central tools of the theory, with immediate applications to local and global well posedness, existence of low regularity solutions, and scattering. This point of view includes most fundamental equations of physics like the Schrödinger, Klein Gordon, wave and Dirac equations. Strichartz estimates appeared in [9]; the basic framework for this study was laid out in the two papers * Corresponding author. addresses: dancona@mat.uniroma1.it (P. D Ancona), luca.fanelli@ehu.es (L. Fanelli), luis.vega@ehu.es (L. Vega), viscigli@dm.unipi.it (N. Visciglia) /$ see front matter 010 Elsevier Inc. All rights reserved. doi: /j.jfa
2 38 P. D Ancona et al. / Journal of Functional Analysis 58 (010) [1,18], which examined in an exhaustive way the case of constant coefficient, unperturbed equations. This leads naturally to the possible extensions to equations perturbed with electromagnetic potentials or with variable coefficients; a general theory of dispersive properties for such equations is still under construction and very actively researched. In the present paper we shall focus on the time dependent Schrödinger equation i t u(t, x) = H u(t, x), u(0,x)= ϕ(x), x R n,n 3, (1.1) associated with the electromagnetic Schrödinger operator H := A + V(x), A := ia(x) (1.) where A = (A 1,...,A n ) : R n R n, V : R n R. We recall that in the unperturbed case A 0, V 0, dispersive properties are best expressed in terms of the mixed norms on R 1+n L p L q := L p( R t ; L q( R n )) x as follows: for every n 3, e it ϕ L p t L q x c n ϕ L, provided the couple (p, q) satisfies the admissibility condition p = n n, p. (1.3) q These estimates are usually referred to as Strichartz estimates. Our main goal is to find sufficient conditions on the potentials A,V such that Strichartz estimates are true for the perturbed equation (1.1). In the purely electric case A 0 the literature is extensive and almost complete; we may cite among many others the papers [1,,13,14,3]. It is now clear that in this case the decay V(x) 1/ x is critical for the validity of Strichartz estimates; suitable counterexamples were constructed in [15]. In the magnetic case A 0, the Coulomb decay A 1/ x is likely to be critical, however no explicit counterexamples are available at the time. An intense research is ongoing concerning Strichartz estimates for the magnetic Schrödinger equation, see e.g. [5,7,8, 11,1]; see also [] for a more general class of first order perturbations. Due to the perturbative techniques used in the above mentioned papers, an assumption concerning absence of zero-energy resonances for the perturbed operator H is typically required in order to preserve the dispersion. In the case A 0 it was shown in [] how this abstract condition can be dispensed with, by directly proving some weak dispersive estimates (also called Morawetz or smoothing estimates) via multipliers methods. Here we shall give a very short proof of Strichartz estimates for the magnetic Schrödinger equation with potentials of almost Coulomb decay, based uniquely on the weak dispersive estimates proved in [10]. The leading theme is that direct multiplier techniques allow to avoid, under suitable repulsivity conditions on V and nontrapping conditions on A (see also [9]), the presence of nondispersive components, and to preserve Strichartz estimates.
3 P. D Ancona et al. / Journal of Functional Analysis 58 (010) We begin by introducing some notations. Regarding as usual the potential A as a 1-form, we define the corresponding magnetic field as the -form B = da, which can be identified with the anti-symmetric gradient of A: B M n n, B = DA (DA) t, (1.4) where (DA) ij = i A j, (DA) t ij = (DA) ji. In dimension 3, B is uniquely determined by the vector field curl A via the vector product We define the trapping component of B as when n = 3 this reduces to Bv = curl A v, v R 3. (1.5) B τ (x) = x B(x); (1.6) x B τ (x) = x curl A(x), n = 3, (1.7) x thus we see that B τ is a tangential vector. The trapping component may be interpreted as an obstruction to the dispersion of solutions; some explicit examples of potentials A with B τ = 0in dimension 3 are given in [9,10]. Moreover, by r V = V x x, we denote the radial derivative of V, and we decompose it into its positive and negative part r V = ( r V) + ( r V). The positive part ( r V) + also represents an obstruction to dispersion, and indeed we shall require it to be small in a suitable sense. To ensure good spectral properties of the operator we shall also assume that the negative part V is not too large in the sense of the Kato norm: Definition 1.1. Let n 3. A measurable function V(x)is said to be in the Kato class K n provided lim r 0 sup x R n x y r V(y) dy = 0. x y n We shall usually omit the reference to the space dimension and write simply K instead of K n. The Kato norm is defined as V K = sup x R n V(y) dy. x y n
4 330 P. D Ancona et al. / Journal of Functional Analysis 58 (010) A last notation we shall need is the radial-tangential norm f p L p r L (S r ) := sup f p dr. x =r In our results we always assume that the operators H and A := ( ia) are self-adjoint and positive on L, in order to ensure the existence of the propagator e ith and of the powers H s via the spectral theorem. There are several sufficient conditions for self-adjointness and positivity, which can be expressed in terms of the local integrability properties of the coefficients (see the standard references [4,19]); here we prefer to leave this as an abstract assumption. Our main result is the following: Theorem 1.1. Let n 3. Given A,V C 1 loc (Rn \{0}), assume the operators A = ( ia) and H = A + V are self-adjoint and positive on L. Moreover assume that 0 V K < π n Γ( n 1) (1.8) and j Z j sup A + j sup V <, (1.9) x C j j Z x C j where C j ={x: j x j+1 } and the Coulomb gauge condition Finally, when n = 3, we assume that for some M>0 div A = 0. (1.10) (M + 1 ) x 3 Bτ M L r L (S r ) + (M + 1) x ( r V) L + 1 r L (S r ) < 1, (1.11) while for n 4 we assume that x B τ (x) L + x 3 ( r V) + (x) L < (n 1)(n 3). (1.1) 3 Then, for any Schrödinger admissible couple (p, q), the following Strichartz estimates hold: e ith ϕ L p L q C ϕ L, p = n n, p, p if n = 3. (1.13) q In dimension n = 3, we have the endpoint estimate D 1 e ith ϕ L L 6 H 1 4 ϕ L. (1.14)
5 P. D Ancona et al. / Journal of Functional Analysis 58 (010) Remark 1.1. By optimizing condition (1.11) with respect to M, we can rephrase the condition as follows: writing for short α = x 3 Bτ L r L (S r ), β = x ( r V) + L 1 r L (S r ), we can rewrite it in the following equivalent form: ( ) α + α + β α + α + β α + β < 1 α + β. Notice that when B τ 0 the condition reduces to β<1/, and when ( r V) + 0 the condition reduces to α<1/4. Remark 1.. Let us remark that the regularity assumption A,V C 1 loc (Rn \{0}) is actually stronger than what we really require. For the validity of the theorem, we just need to give meaning to inequalities (1.11), (1.1). Remark 1.3. Assumptions (1.10), (1.11), and (1.1) imply the weak dispersion of the propagator e ith (see Theorems 1.9, 1.10, assumptions (1.4), and (1.7) in [10]). Actually assumption (1.4) in [10] seems to be stronger than (1.11), but reading carefully the proof of Theorem 1.9 in [10] it is clear that the real assumption is our (1.11) (see inequality (3.14) in [10]). The strict inequality in (1.11), (1.1) is essential, in order to dispose of the weighted L -estimateinthe above mentioned Theorems by [10] (see also inequality (3.5) below). Remark 1.4. We recall that, usually, suitable spectral conditions must be required for the dispersion to hold, see e.g. [7,8] where resonances at zero are excluded. In our case, such conditions are implied by the smallness assumptions (1.11), (1.1) which can be checked easily in concrete examples. The derivation of Strichartz estimates from the weak dispersive ones turns out to be remarkably simple if working on the half derivative D 1/ u, see Section 3 for details. As a drawback, the final estimates are expressed in terms of fractional Sobolev spaces generated by the perturbed magnetic operator A. Thus, in order to revert to standard Strichartz norms as in (1.13), we need suitable bounds for the perturbed Sobolev norms in terms of the standard ones. This is provided by the following theorem, which we think is of independent interest. Theorem 1.. Let n 3. Given A L loc (Rn ; R n ), V : R n R, assume the operators A = ( ia) and H = A + V are self-adjoint and positive on L. Moreover, assume that V + is of Kato class, V satisfies V K < and Γ( n π n 1), (1.15) A i A + V L n,, A L n,. (1.16)
6 33 P. D Ancona et al. / Journal of Functional Analysis 58 (010) Then the following estimate holds: In addition we have the reverse estimate H 1 4 f L q C q D 1 f L q, 1 <q<n, n 3. (1.17) H 1 4 f L q c q D 1 f L q, 4 <q<4, 3 n 3. (1.18). Proof of Theorem 1. We start with the proof of Theorem 1., divided into several steps. First we need to prove that the heat kernel associated with the operator H is well behaved under quite general assumptions: Proposition.1. Consider the self-adjoint operator H = ( ia(x)) + V(x) on L (R n ), n 3. Assume that A L loc (Rn, R n ), moreover the positive and negative parts V ± of V satisfy V + is of Kato class, (.1) V K <c n = π n/ /Γ (n/ 1). (.) Then e th is an integral operator and its heat kernel p t (x, y) satisfies the pointwise estimate p t (x, y) (πt) n/ e x y /(8t). (.3) 1 V K /c n Proof. We recall Simon s diamagnetic pointwise inequality (see e.g. Theorem B.13. in [7]), which holds under weaker assumptions than ours: for any test function g(x), e t[( ia(x)) V ] g e t( V) g. Notice that by choosing a delta sequence g ɛ of (positive) test functions, this implies an analogous pointwise inequality for the corresponding heat kernels. Now we can apply the second part of Proposition 5.1 in [6] which gives precisely estimate (.3) for the heat kernel of e t( V) under (.1), (.). The second tool we shall use is a weak type estimate for imaginary powers of self-adjoint operators, defined in the sense of spectral theory. This follows easily from the previous heat kernel bound and the techniques of Sikora and Wright (see [6]): Proposition.. Let H be as in Proposition.1, and assume in addition that H 0. Then for all y R the imaginary powers H iy satisfy the (1, 1) weak type estimate H iy L 1 L 1, C(1 + y ) n/. (.4)
7 P. D Ancona et al. / Journal of Functional Analysis 58 (010) Proof. By Theorem 3 in [5] we obtain immediately that our heat kernel bound (.3) implies the finite speed of propagation for the wave kernel cos(t H), in the sense of [5,6], i.e., ( ) cos(t H)φ,ψ L = 0 for all φ,ψ L with support in B(ξ 1,x 1 ), B(ξ,x ) respectively, provided t < 1/ ( x 1 x ξ 1 ξ ). Then we are in a position to apply Theorem from [6] which gives the required bound. We are ready to prove the first part of Theorem 1.. Proof of (1.17). We shall use the Stein Weiss interpolation theorem applied to the analytic family of operators T z = H z ( ) z. Here H z is defined by spectral theory while ( ) z e.g. by the Fourier transform. Writing z = x + iy, we can decompose T z = H iy H x ( ) x ( ) iy, y R, x [0, 1]. The operators H iy and ( ) iy are obviously bounded on L. On the side Rz = 0 the operator reduces to the composition of pure imaginary powers T iy = H iy ( ) iy and by the weak type estimate (.4) we obtain immediately by interpolation that H iy, and hence T iy is bounded on L p for all 1 <p< : T z f L p C ( 1 + y ) n/ f L p for Rz = 0, 1 <p<. (.5) Next we consider the case Rz = 1. We start by proving the estimate For f C c (Rn ) we can write Hf L r C f L r, 1 <r< n. (.6) Hf = f ia f + ( A i A + V ) f. We have then by Hölder s inequality in Lorentz spaces and assumption (1.16) A f L r C A L n, f L nr n r,r 1 r<n, and using the precised Sobolev embedding g L nr n r,r C g L r,r = g L r
8 334 P. D Ancona et al. / Journal of Functional Analysis 58 (010) (and the boundedness of Riesz operators, which rules out the case r = 1) we obtain A f L r C f L r, 1 <r<n. In a similar way, ( A i A + V ) f L r C A i A + V L n/, f L nr n r,r, 1 r< n, and again by the Sobolev embedding we conclude that f L nr n r,r C f L r ( A i A + V ) f L r C f L r, 1 <r< n. Summing up we obtain (.6). Combining (.6) with the L r -boundedness of the purely imaginary powers H iy and iy, we get T 1+iy : L r L r, 1 <r< n. (.7) Interpolating (.7) with (.5) we obtain for T 1/4 f L p C f L p 1 q = 3 4p + 1 4r, 1 <p<, 1 <r<n 1 <q<n which concludes the proof. We pass now to the proof of the reverse estimate (1.18). We shall need the following lemma: Lemma.1. Assume that A L loc( R n ), V K < 4π n/ /Γ (n/ 1). (.8) Then for some constant a<1 the following inequality holds: V f dx a A f L. (.9)
9 P. D Ancona et al. / Journal of Functional Analysis 58 (010) Proof. The proof follows a standard argument. We begin by showing that V f dx a f L, (.10) for some a<1. This can be restated as ( V 1/ ( ) 1/ f,v 1/ ( ) 1/ f ) a f, a <1, i.e. we must prove that the operator T = V 1/ ( ) 1/ is bounded on L with norm smaller than one. Equivalently, we must prove that the operator satisfies TT = V 1/ ( ) 1 V 1/ V 1/ ( ) 1 V 1/ f b f, b<1. Writing explicitly the kernel of ( ) 1, we are reduced to prove I = V (x) V (y) 1/ f(y)dy x y n where k n = 4π n/ /Γ (n/ 1), b<1. Now by Cauchy Schwartz V I (x) ( )( V (y) dy x y n dx kn b f f(y) ) dy dx x y n which gives I V K V (x) x y n f(y) dy dx = V K f and this proves (.10) under the smallness assumption (.8). Applying the same computation to the function f instead of f, we deduce from (.10) that V f dx a f L. (.11) Since A L loc, we can apply the diamagnetic inequality f A f, a.e. in R n (see e.g. [0]) to obtain (.9).
10 336 P. D Ancona et al. / Journal of Functional Analysis 58 (010) Proof of (1.18). We begin by proving the L inequality ( ) 1/ f L f L C H 1/ f L. (.1) We can write, with the notation A = ia, H 1/ f = (Hf, f ) = ( A f,f ) + V f = A f L + V f and this implies H 1/ f A f L V f. Thus by Lemma.1 we have for some a<1 H 1/ f (1 a) A f L so that, in order to prove (.1), it is sufficient to prove the inequality f L C A f L. (.13) Now, using as in the first half of the proof the Hölder inequality and the Sobolev embedding in Lorentz spaces, we can write A f L + V f C A + V L n/, f L n n C f L, by assumption (1.16). Then, by the diamagnetic inequality f A f we obtain Af L + V f C A f L. (.14) Moreover we have ( ia)f f Af f A f + Af which implies and combining this with (.14) we get f L C A f L f L A f L + Af L V f C A f L + V f.
11 P. D Ancona et al. / Journal of Functional Analysis 58 (010) Using again Lemma.1 we finally arrive at (.13), so that the claimed estimate (.1) is proved. Now we can use again the Stein Weiss interpolation theorem, applied to the analytic family of operators T z = ( ) z H z with z in the range 0 Rz 1/. Writing z = x + iy we have T z = ( ) iy ( ) x H x H iy, y R, x [0, 1/]. The operators H iy and ( ) iy are bounded on L, while estimate (.1) proves that ( ) 1/ H 1/ is also bounded on L. This shows that T z f L C f L for Rz = 1/. (.15) On the side Rz = 0 the operator reduces to the composition of pure imaginary powers T iy = ( ) iy H iy and arguing as in the proof of (.5) we get that T iy is bounded on L p for all 1 <p< : T z f L p C ( 1 + y ) n/ f L p for Rz = 0, 1 <p<. (.16) Then by the Stein Weiss interpolation theorem we obtain as above with 3. Proof of Strichartz estimates T 1/4 f L q C f L q 1 q = p, 1 <p< 4 3 <q<4. Let us first recall some well-known facts about the free propagator. First of all, the free Strichartz estimates for T(t)= e it, its dual operator and the operator TT are e it ϕ L p L q C ϕ L, (3.1) e is F(s, )ds C F L p, (3.) L q L t 0 e i(t s) F(s, )ds C F L p, (3.3) L q L p L q
12 338 P. D Ancona et al. / Journal of Functional Analysis 58 (010) for all Schrödinger admissible couples (p, q), ( p, q) satisfying p = n n, p, q with p ifn = (see [1,18]). Moreover, we recall the following estimate: D 1 for any admissible couple (p, q) as above, where t 0 e i(t s) F(s, )ds j Fj L L, (3.4) L p L q j Z F = j Z F j, supp F j { j x j+1}, x [0,t]. Estimate (3.4) was proved in [4] first; actually it follows by mixing the free Strichartz estimates for T(t) with the dual of the local smoothing estimates which were proved independently by [3,17,8] and [30]. In the paper [4] the endpoint estimate for p = is not proved (and indeed it predates the Keel Tao paper [18]). The endpoint case p = in dimension n 3 is a consequence of Lemma 3 in [16]. Finally, we need to recall the local smoothing estimates for the magnetic propagator e ith proved in [10] under assumptions less restrictive than the ones of the present paper: we have 1 sup A e ith ϕ 1 dxdt + sup e ith R>0 R R>0 R ϕ dσ R dt ( A ) 1 4 ϕ L, (3.5) x R x =R where the constant in the inequality only depends on B τ and ( r V) +. We are now ready to prove Theorem 1.1. Since div A = 0, we can expand H as follows: As a consequence, by the Duhamel formula we can write H = + ia A A + V. (3.6) t e ith ϕ = e it ϕ + where the perturbative operator R(x,D) is given by By (3.1) and (3.4) we have 0 e i(t s) R(x,D)e ith ϕds, (3.7) R(x,D) = ia A A + V. (3.8) D 1 e ith ϕ L p L q C D 1 ϕl + j χ j R(x,D)e ith ϕ L L, (3.9) j Z
13 P. D Ancona et al. / Journal of Functional Analysis 58 (010) where χ j is the characteristic function of the ring j x j+1. For the first term at the RHS of (3.9), by (1.18) we have D 1 ϕl C H 1 4 ϕl. (3.10) On the other hand, we can split the second term as follows j χj R(x,D)e ith ϕ L L j Z j Z j ( χ j A A e ith ϕ L L + χ j ( V A ) e ith ϕ L L ) = I + II. (3.11) By Hölder inequality, assumption (1.9) and the smoothing estimates (3.5) we have ( ) ( I j 1 sup A sup A e ith ϕ ) 1 dxdt ( A ) 1 4 ϕl, (3.1) j Z x j R>0 R x R ( II j ( sup V + A ) ) x j j Z ( 1 sup e ith R>0 R ϕ ) 1 dσ R dt x =R ( A ) 1 4 ϕ L. (3.13) Now we remark that all the assumptions of Theorem 1. are satisfied. Indeed, we know that A L loc ; moreover, assumption (1.9) implies that A 1/ x and V 1/ x, hence (1.16) is satisfied. Thus by Theorem 1. (which holds also in the special case V 0) we get ( A ) 1 4 ϕl C ϕ H 1 H 1 4 ϕl. Collecting (3.11), (3.1) and (3.13) we obtain j Z j χ j R(x,D)e ith ϕ L L H 1 4 ϕl (3.14) and by (3.9), (3.10) and (3.14) we deduce 1 D e ith ϕ L p L q 1 H 4 ϕl, (3.15) for any admissible couple (p, q); notice that this includes also the 3D endpoint estimate (1.14). In order to conclude the proof, it is now sufficient to use estimate (1.17) which gives 1 H 4 e ith ϕ L p L q 1 H 4 ϕl, (3.16) and commuting H 1/4 with the flow e ith we obtain (1.13). However, in dimension 3 (1.17) does not cover the endpoint q = 6 and we are left with (3.15).
14 340 P. D Ancona et al. / Journal of Functional Analysis 58 (010) References [1] J.A. Barceló, A. Ruiz, L. Vega, Some dispersive estimates for Schrödinger equations with repulsive potentials, J. Funct. Anal. 36 (006) 1 4. [] N. Burq, F. Planchon, J. Stalker, S. Tahvildar-Zadeh, Strichartz estimates for the wave and Schrödinger equations with potentials of critical decay, Indiana Univ. Math. J. 53 (6) (004) [3] P. Constantin, J.-C. Saut, Local smoothing properties of dispersive equations, J. Amer. Math. Soc. (1988) [4] H.L. Cycon, R. Froese, W. Kirsch, B. Simon, Schrödinger Operators with Applications to Quantum Mechanics and Global Geometry, Texts Monogr. Phys., Springer Verlag, Berlin, Heidelberg, New York, [5] P. D Ancona, L. Fanelli, Strichartz and smoothing estimates for dispersive equations with magnetic potentials, Comm. Partial Differential Equations 33 (008) [6] P. D Ancona, V. Pierfelice, On the wave equation with a large rough potential, J. Funct. Anal. 7 (005) [7] M.B. Erdoğan, M. Goldberg, W. Schlag, Strichartz and smoothing estimates for Schrödinger operators with almost critical magnetic potentials in three and higher dimensions, Forum Math. 1 (009) [8] M.B. Erdoğan, M. Goldberg, W. Schlag, Strichartz and smoothing estimates for Schrödinger operators with large magnetic potentials in R 3, J. Eur. Math. Soc. (JEMS) 10 (008) [9] L. Fanelli, Non-trapping magnetic fields and Morrey Campanato estimates for Schrödinger operators, J. Math. Anal. Appl. 357 (009) [10] L. Fanelli, L. Vega, Magnetic virial identities, weak dispersion and Strichartz inequalities, Math. Ann. 344 (009) [11] V. Georgiev, A. Stefanov, M. Tarulli, Smoothing Strichartz estimates for the Schrödinger equation with small magnetic potential, Discrete Contin. Dyn. Syst. Ser. A 17 (007) [1] J. Ginibre, G. Velo, Generalized Strichartz inequalities for the wave equation, J. Funct. Anal. 133 (1) (1995) [13] M. Goldberg, Dispersive estimates for the three-dimensional Schrödinger equation with rough potential, Amer. J. Math. 18 (006) [14] M. Goldberg, W. Schlag, Dispersive estimates for Schrödinger operators in dimensions one and three, Comm. Math. Phys. 51 (004) [15] M. Goldberg, L. Vega, N. Visciglia, Counterexamples of Strichartz inequalities for Schrödinger equations with repulsive potentials, Int. Math. Res. Not. 006 (006), article ID [16] A.D. Ionescu, C. Kenig, Well-posedness and local smoothing of solutions of Schrödinger equations, Math. Res. Lett. 1 (005) [17] T. Kato, K. Yajima, Some examples of smooth operators and the associated smoothing effect, Rev. Math. Phys. 1 (1989) [18] M. Keel, T. Tao, Endpoint Strichartz estimates, Amer. J. Math. 10 (5) (1998) [19] H. Leinfelder, C. Simader, Schrödinger operators with singular magnetic vector potentials, Math. Z. 176 (1981) [0] E.H. Lieb, M. Loss, Analysis, Grad. Stud. Math., vol. 14, 001. [1] J. Marzuola, J. Metcalfe, D. Tataru, Strichartz estimates and local smoothing estimates for asymptotically flat Schrödinger equations, J. Funct. Anal. 55 (008) [] L. Robbiano, C. Zuily, Strichartz estimates for Schrödinger equations with variable coefficients, Mem. Soc. Math. Fr. (N.S.) (005) [3] I. Rodnianski, W. Schlag, Time decay for solutions of Schrödinger equations with rough and time-dependent potentials, Invent. Math. 155 (3) (004) [4] A. Ruiz, L. Vega, On local regularity of Schrödinger equations, Int. Math. Res. Not. IMRN 1 (1993) [5] A. Sikora, Sharp pointwise estimates on heat kernels, Quart. J. Math. Oxford Ser. () 47 (187) (1996) [6] A. Sikora, J. Wright, Imaginary powers of Laplace operators, Proc. Amer. Math. Soc. 19 (000) [7] B. Simon, Schrödinger semigroups, Bull. Amer. Math. Soc. 7 (198) [8] P. Sjölin, Regularity of solutions to the Schrödinger equations, Duke Math. J. 55 (1987) [9] R. Strichartz, Restriction of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J. 44 (1977) [30] L. Vega, The Schrödinger equation: pointwise convergence to the initial date, Proc. Amer. Math. Soc. 10 (1988)
arxiv: v2 [math.ap] 11 Aug 2017
arxiv:6.789v [math.ap] Aug 7 STRICHARTZ ESTIMATES FOR THE MAGNETIC SCHRÖDINGER EQUATION WITH POTENTIALS V OF CRITICAL DECAY SEONGHAK KIM AND YOUNGWOO KOH Abstract. We study the Strichartz estimates for
More informationScattering 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 informationSTRICHARTZ ESTIMATES FOR SCHRÖDINGER OPERATORS WITH A NON-SMOOTH MAGNETIC POTENTIAL. Michael Goldberg. (Communicated by the associate editor name)
STICHATZ ESTIMATES FO SCHÖDINGE OPEATOS WITH A NON-SMOOTH MAGNETIC POTENTIA Michael Goldberg Department of Mathematics Johns Hopkins University 3400 N. Charles St. Baltimore, MD 228, USA Communicated by
More informationANALYTIC 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 informationA 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 informationarxiv: v1 [math.ap] 18 May 2017
Littlewood-Paley-Stein functions for Schrödinger operators arxiv:175.6794v1 [math.ap] 18 May 217 El Maati Ouhabaz Dedicated to the memory of Abdelghani Bellouquid (2/2/1966 8/31/215) Abstract We study
More informationSCATTERING 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 informationarxiv:math/ v1 [math.ap] 28 Oct 2005
arxiv:math/050643v [math.ap] 28 Oct 2005 A remark on asymptotic completeness for the critical nonlinear Klein-Gordon equation Hans Lindblad and Avy Soffer University of California at San Diego and Rutgers
More informationA Limiting Absorption Principle for the three-dimensional Schrödinger equation with L p potentials
A Limiting Absorption Principle for the three-dimensional Schrödinger equation with L p potentials M. Goldberg, W. Schlag 1 Introduction Agmon s fundamental work [Agm] establishes the bound, known as the
More informationScattering 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 informationarxiv:math/ v2 [math.ap] 3 Oct 2006
THE TAYLOR SERIES OF THE GAUSSIAN KERNEL arxiv:math/0606035v2 [math.ap] 3 Oct 2006 L. ESCAURIAZA From some people one can learn more than mathematics Abstract. We describe a formula for the Taylor series
More informationCUTOFF 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 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 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 informationInégalités 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 informationarxiv: 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 informationBLOWUP 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 informationDISPERSIVE 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 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 informationNotes. 1 Fourier transform and L p spaces. March 9, For a function in f L 1 (R n ) define the Fourier transform. ˆf(ξ) = f(x)e 2πi x,ξ dx.
Notes March 9, 27 1 Fourier transform and L p spaces For a function in f L 1 (R n ) define the Fourier transform ˆf(ξ) = f(x)e 2πi x,ξ dx. Properties R n 1. f g = ˆfĝ 2. δλ (f)(ξ) = ˆf(λξ), where δ λ f(x)
More informationTitle: Localized self-adjointness of Schrödinger-type operators on Riemannian manifolds. Proposed running head: Schrödinger-type operators on
Title: Localized self-adjointness of Schrödinger-type operators on Riemannian manifolds. Proposed running head: Schrödinger-type operators on manifolds. Author: Ognjen Milatovic Department Address: Department
More informationSHARP L p WEIGHTED SOBOLEV INEQUALITIES
Annales de l Institut de Fourier (3) 45 (995), 6. SHARP L p WEIGHTED SOBOLEV INEUALITIES Carlos Pérez Departmento de Matemáticas Universidad Autónoma de Madrid 28049 Madrid, Spain e mail: cperezmo@ccuam3.sdi.uam.es
More informationEnergy 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 informationProceedings of the 5th International Conference on Inverse Problems in Engineering: Theory and Practice, Cambridge, UK, 11-15th July 2005
Proceedings of the 5th International Conference on Inverse Problems in Engineering: Theory and Practice, Cambridge, UK, 11-15th July 2005 SOME INVERSE SCATTERING PROBLEMS FOR TWO-DIMENSIONAL SCHRÖDINGER
More informationDispersive Equations and Hyperbolic Orbits
Dispersive Equations and Hyperbolic Orbits H. Christianson Department of Mathematics University of California, Berkeley 4/16/07 The Johns Hopkins University Outline 1 Introduction 3 Applications 2 Main
More informationWEAK TYPE ESTIMATES FOR SINGULAR INTEGRALS RELATED TO A DUAL PROBLEM OF MUCKENHOUPT-WHEEDEN
WEAK TYPE ESTIMATES FOR SINGULAR INTEGRALS RELATED TO A DUAL PROBLEM OF MUCKENHOUPT-WHEEDEN ANDREI K. LERNER, SHELDY OMBROSI, AND CARLOS PÉREZ Abstract. A ell knon open problem of Muckenhoupt-Wheeden says
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 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 informationPiecewise 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 informationSELF-ADJOINTNESS OF DIRAC OPERATORS VIA HARDY-DIRAC INEQUALITIES
SELF-ADJOINTNESS OF DIRAC OPERATORS VIA HARDY-DIRAC INEQUALITIES MARIA J. ESTEBAN 1 AND MICHAEL LOSS Abstract. Distinguished selfadjoint extension of Dirac operators are constructed for a class of potentials
More informationGlobal 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 informationHOMEOMORPHISMS OF BOUNDED VARIATION
HOMEOMORPHISMS OF BOUNDED VARIATION STANISLAV HENCL, PEKKA KOSKELA AND JANI ONNINEN Abstract. We show that the inverse of a planar homeomorphism of bounded variation is also of bounded variation. In higher
More informationGlobal regularity of a modified Navier-Stokes equation
Global regularity of a modified Navier-Stokes equation Tobias Grafke, Rainer Grauer and Thomas C. Sideris Institut für Theoretische Physik I, Ruhr-Universität Bochum, Germany Department of Mathematics,
More informationA 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 informationTADAHIRO 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 informationSMOOTHING ESTIMATES OF THE RADIAL SCHRÖDINGER PROPAGATOR IN DIMENSIONS n 2
Acta Mathematica Scientia 1,3B(6):13 19 http://actams.wipm.ac.cn SMOOTHING ESTIMATES OF THE RADIAL SCHRÖDINGER PROPAGATOR IN DIMENSIONS n Li Dong ( ) Department of Mathematics, University of Iowa, 14 MacLean
More informationTHE FORM SUM AND THE FRIEDRICHS EXTENSION OF SCHRÖDINGER-TYPE OPERATORS ON RIEMANNIAN MANIFOLDS
THE FORM SUM AND THE FRIEDRICHS EXTENSION OF SCHRÖDINGER-TYPE OPERATORS ON RIEMANNIAN MANIFOLDS OGNJEN MILATOVIC Abstract. We consider H V = M +V, where (M, g) is a Riemannian manifold (not necessarily
More informationGlobal 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 informationOn the role of geometry in scattering theory for nonlinear Schrödinger equations
On the role of geometry in scattering theory for nonlinear Schrödinger equations Rémi Carles (CNRS & Université Montpellier 2) Orléans, April 9, 2008 Free Schrödinger equation on R n : i t u + 1 2 u =
More information1-D cubic NLS with several Diracs as initial data and consequences
1-D cubic NLS with several Diracs as initial data and consequences Valeria Banica (Univ. Pierre et Marie Curie) joint work with Luis Vega (BCAM) Roma, September 2017 1/20 Plan of the talk The 1-D cubic
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 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 informationand finally, any second order divergence form elliptic operator
Supporting Information: Mathematical proofs Preliminaries Let be an arbitrary bounded open set in R n and let L be any elliptic differential operator associated to a symmetric positive bilinear form B
More informationStrong uniqueness for second order elliptic operators with Gevrey coefficients
Strong uniqueness for second order elliptic operators with Gevrey coefficients Ferruccio Colombini, Cataldo Grammatico, Daniel Tataru Abstract We consider here the problem of strong unique continuation
More informationPotential Analysis meets Geometric Measure Theory
Potential Analysis meets Geometric Measure Theory T. Toro Abstract A central question in Potential Theory is the extend to which the geometry of a domain influences the boundary regularity of the solution
More informationEndpoint resolvent estimates for compact Riemannian manifolds
Endpoint resolvent estimates for compact Riemannian manifolds joint work with R. L. Frank to appear in J. Funct. Anal. (arxiv:6.00462) Lukas Schimmer California Institute of Technology 3 February 207 Schimmer
More informationFour-Fermion Interaction Approximation of the Intermediate Vector Boson Model
Four-Fermion Interaction Approximation of the Intermediate Vector Boson odel Yoshio Tsutsumi Department of athematics, Kyoto University, Kyoto 66-852, JAPAN 1 Introduction In this note, we consider the
More informationNONTRIVIAL SOLUTIONS TO INTEGRAL AND DIFFERENTIAL EQUATIONS
Fixed Point Theory, Volume 9, No. 1, 28, 3-16 http://www.math.ubbcluj.ro/ nodeacj/sfptcj.html NONTRIVIAL SOLUTIONS TO INTEGRAL AND DIFFERENTIAL EQUATIONS GIOVANNI ANELLO Department of Mathematics University
More informationAN EIGENVALUE PROBLEM FOR THE SCHRÖDINGER MAXWELL EQUATIONS. Vieri Benci Donato Fortunato. 1. Introduction
Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume, 998, 83 93 AN EIGENVALUE PROBLEM FOR THE SCHRÖDINGER MAXWELL EQUATIONS Vieri Benci Donato Fortunato Dedicated to
More informationNONLOCAL DIFFUSION EQUATIONS
NONLOCAL DIFFUSION EQUATIONS JULIO D. ROSSI (ALICANTE, SPAIN AND BUENOS AIRES, ARGENTINA) jrossi@dm.uba.ar http://mate.dm.uba.ar/ jrossi 2011 Non-local diffusion. The function J. Let J : R N R, nonnegative,
More informationProperties of the Scattering Transform on the Real Line
Journal of Mathematical Analysis and Applications 58, 3 43 (001 doi:10.1006/jmaa.000.7375, available online at http://www.idealibrary.com on Properties of the Scattering Transform on the Real Line Michael
More informationMath The Laplacian. 1 Green s Identities, Fundamental Solution
Math. 209 The Laplacian Green s Identities, Fundamental Solution Let be a bounded open set in R n, n 2, with smooth boundary. The fact that the boundary is smooth means that at each point x the external
More informationOn the Solvability Conditions for a Linearized Cahn-Hilliard Equation
Rend. Istit. Mat. Univ. Trieste Volume 43 (2011), 1 9 On the Solvability Conditions for a Linearized Cahn-Hilliard Equation Vitaly Volpert and Vitali Vougalter Abstract. We derive solvability conditions
More informationarxiv: v1 [math.ap] 12 Mar 2009
LIMITING FRACTIONAL AND LORENTZ SPACES ESTIMATES OF DIFFERENTIAL FORMS JEAN VAN SCHAFTINGEN arxiv:0903.282v [math.ap] 2 Mar 2009 Abstract. We obtain estimates in Besov, Lizorkin-Triebel and Lorentz spaces
More informationSINGULAR INTEGRALS WITH ANGULAR INTEGRABILITY
SINGULAR INTEGRALS WITH ANGULAR INTEGRABILITY FEDERICO CACCIAFESTA AND RENATO LUCÀ Abstract. In this note we rove a class of shar inequalities for singular integral oerators in weighted Lebesgue saces
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 informationJUHA KINNUNEN. Harmonic Analysis
JUHA KINNUNEN Harmonic Analysis Department of Mathematics and Systems Analysis, Aalto University 27 Contents Calderón-Zygmund decomposition. Dyadic subcubes of a cube.........................2 Dyadic cubes
More informationRemarks on the blow-up criterion of the 3D Euler equations
Remarks on the blow-up criterion of the 3D Euler equations Dongho Chae Department of Mathematics Sungkyunkwan University Suwon 44-746, Korea e-mail : chae@skku.edu Abstract In this note we prove that the
More informationA RELATIONSHIP BETWEEN THE DIRICHLET AND REGULARITY PROBLEMS FOR ELLIPTIC EQUATIONS. Zhongwei Shen
A RELATIONSHIP BETWEEN THE DIRICHLET AND REGULARITY PROBLEMS FOR ELLIPTIC EQUATIONS Zhongwei Shen Abstract. Let L = diva be a real, symmetric second order elliptic operator with bounded measurable coefficients.
More informationSOLVABILITY RELATIONS FOR SOME NON FREDHOLM OPERATORS
SOLVABILITY RELATIONS FOR SOME NON FREDHOLM OPERATORS Vitali Vougalter 1, Vitaly Volpert 2 1 Department of Mathematics and Applied Mathematics, University of Cape Town Private Bag, Rondebosch 7701, South
More informationSELF-ADJOINTNESS OF SCHRÖDINGER-TYPE OPERATORS WITH SINGULAR POTENTIALS ON MANIFOLDS OF BOUNDED GEOMETRY
Electronic Journal of Differential Equations, Vol. 2003(2003), No.??, pp. 1 8. ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp) SELF-ADJOINTNESS
More information4 Riesz Kernels. Since the functions k i (ξ) = ξ i. are bounded functions it is clear that R
4 Riesz Kernels. A natural generalization of the Hilbert transform to higher dimension is mutiplication of the Fourier Transform by homogeneous functions of degree 0, the simplest ones being R i f(ξ) =
More informationSTRICHARTZ ESTIMATES ON EXTERIOR POLYGONAL DOMAINS
STRICHARTZ ESTIMATES ON EXTERIOR POLYGONAL DOMAINS DEAN BASKIN, JEREMY L. MARZUOLA, AND JARED WUNSCH Abstract. Using a new local smoothing estimate of the first and third authors, we prove local-in-time
More informationarxiv: v1 [math.sp] 17 Oct 2017
Absence of eigenvalues of two-dimensional magnetic Schrödinger operators Luca Fanelli, David Krejčiřík and Luis Vega arxiv:171.6176v1 [math.sp] 17 Oct 17 a) Dipartimento di Matematica, SAPIENZA Università
More informationSharp estimates for a class of hyperbolic pseudo-differential equations
Results in Math., 41 (2002), 361-368. Sharp estimates for a class of hyperbolic pseudo-differential equations Michael Ruzhansky Abstract In this paper we consider the Cauchy problem for a class of hyperbolic
More informationON THE BEHAVIOR OF THE SOLUTION OF THE WAVE EQUATION. 1. Introduction. = u. x 2 j
ON THE BEHAVIO OF THE SOLUTION OF THE WAVE EQUATION HENDA GUNAWAN AND WONO SETYA BUDHI Abstract. We shall here study some properties of the Laplace operator through its imaginary powers, and apply the
More informationRANDOM 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 informationLow frequency resolvent estimates for long range perturbations of the Euclidean Laplacian
Low frequency resolvent estimates for long range perturbations of the Euclidean Laplacian Jean-Francois Bony, Dietrich Häfner To cite this version: Jean-Francois Bony, Dietrich Häfner. Low frequency resolvent
More informationPartial Differential Equations
Part II Partial Differential Equations Year 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2015 Paper 4, Section II 29E Partial Differential Equations 72 (a) Show that the Cauchy problem for u(x,
More informationExponential Energy Decay for the Kadomtsev-Petviashvili (KP-II) equation
São Paulo Journal of Mathematical Sciences 5, (11), 135 148 Exponential Energy Decay for the Kadomtsev-Petviashvili (KP-II) equation Diogo A. Gomes Department of Mathematics, CAMGSD, IST 149 1 Av. Rovisco
More informationSYMMETRY RESULTS FOR PERTURBED PROBLEMS AND RELATED QUESTIONS. Massimo Grosi Filomena Pacella S. L. Yadava. 1. Introduction
Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 21, 2003, 211 226 SYMMETRY RESULTS FOR PERTURBED PROBLEMS AND RELATED QUESTIONS Massimo Grosi Filomena Pacella S.
More informationBrunn-Minkowski inequality for the 1-Riesz capacity and level set convexity for the 1/2-Laplacian
Brunn-Minkowski inequality for the 1-Riesz capacity and level set convexity for the 1/2-Laplacian M. Novaga, B. Ruffini January 13, 2014 Abstract We prove that that the 1-Riesz capacity satisfies a Brunn-Minkowski
More informationStrichartz Estimates for the Schrödinger Equation in Exterior Domains
Strichartz Estimates for the Schrödinger Equation in University of New Mexico May 14, 2010 Joint work with: Hart Smith (University of Washington) Christopher Sogge (Johns Hopkins University) The Schrödinger
More informationSharp Sobolev Strichartz estimates for the free Schrödinger propagator
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
More informationarxiv: 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 informationA GLOBAL COMPACT ATTRACTOR FOR HIGH-DIMENSIONAL DEFOCUSING NON-LINEAR SCHRÖDINGER EQUATIONS WITH POTENTIAL TERENCE TAO
A GLOBAL COMPACT ATTRACTOR FOR HIGH-DIMENSIONAL DEFOCUSING NON-LINEAR SCHRÖDINGER EQUATIONS WITH POTENTIAL TERENCE TAO arxiv:85.1544v2 [math.ap] 28 May 28 Abstract. We study the asymptotic behavior of
More informationNONLINEAR PROPAGATION OF WAVE PACKETS. Ritsumeikan University, and 22
NONLINEAR PROPAGATION OF WAVE PACKETS CLOTILDE FERMANIAN KAMMERER Ritsumeikan University, 21-1 - 21 and 22 Our aim in this lecture is to explain the proof of a recent Theorem obtained in collaboration
More informationPara el cumpleaños del egregio profesor Ireneo Peral
On two coupled nonlinear Schrödinger equations Para el cumpleaños del egregio profesor Ireneo Peral Dipartimento di Matematica Sapienza Università di Roma Salamanca 13.02.2007 Coauthors Luca Fanelli (Sapienza
More informationarxiv: v1 [math.ap] 28 Mar 2014
GROUNDSTATES OF NONLINEAR CHOQUARD EQUATIONS: HARDY-LITTLEWOOD-SOBOLEV CRITICAL EXPONENT VITALY MOROZ AND JEAN VAN SCHAFTINGEN arxiv:1403.7414v1 [math.ap] 28 Mar 2014 Abstract. We consider nonlinear Choquard
More informationBibliography. 1. A note on complex Tauberian theorems, Mitt. Math. Sem. Giessen 200, (1991).
Wilhelm Schlag Bibliography 1. A note on complex Tauberian theorems, Mitt. Math. Sem. Giessen 200, 13 14 (1991). 2. Schauder and L p estimates for parabolic systems via Campanato spaces, Comm. PDE 21,
More informationarxiv: v2 [math.ap] 4 Dec 2013
ON D NLS ON NON-TRAPPING EXTERIOR DOMAINS FARAH ABOU SHAKRA arxiv:04.768v [math.ap] 4 Dec 0 Abstract. Global existence and scattering for the nonlinear defocusing Schrödinger equation in dimensions are
More informationTHE HARDY LITTLEWOOD MAXIMAL FUNCTION OF A SOBOLEV FUNCTION. Juha Kinnunen. 1 f(y) dy, B(x, r) B(x,r)
Appeared in Israel J. Math. 00 (997), 7 24 THE HARDY LITTLEWOOD MAXIMAL FUNCTION OF A SOBOLEV FUNCTION Juha Kinnunen Abstract. We prove that the Hardy Littlewood maximal operator is bounded in the Sobolev
More informationESTIMATES FOR ELLIPTIC HOMOGENIZATION PROBLEMS IN NONSMOOTH DOMAINS. Zhongwei Shen
W,p ESTIMATES FOR ELLIPTIC HOMOGENIZATION PROBLEMS IN NONSMOOTH DOMAINS Zhongwei Shen Abstract. Let L = div`a` x, > be a family of second order elliptic operators with real, symmetric coefficients on a
More informationNon-radial solutions to a bi-harmonic equation with negative exponent
Non-radial solutions to a bi-harmonic equation with negative exponent Ali Hyder Department of Mathematics, University of British Columbia, Vancouver BC V6TZ2, Canada ali.hyder@math.ubc.ca Juncheng Wei
More informationStrauss conjecture for nontrapping obstacles
Chengbo Wang Joint work with: Hart Smith, Christopher Sogge Department of Mathematics Johns Hopkins University Baltimore, Maryland 21218 wangcbo@jhu.edu November 3, 2010 1 Problem and Background Problem
More informationComplex geometrical optics solutions for Lipschitz conductivities
Rev. Mat. Iberoamericana 19 (2003), 57 72 Complex geometrical optics solutions for Lipschitz conductivities Lassi Päivärinta, Alexander Panchenko and Gunther Uhlmann Abstract We prove the existence of
More informationNonlinear 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 informationASYMPTOTIC STRUCTURE FOR SOLUTIONS OF THE NAVIER STOKES EQUATIONS. Tian Ma. Shouhong Wang
DISCRETE AND CONTINUOUS Website: http://aimsciences.org DYNAMICAL SYSTEMS Volume 11, Number 1, July 004 pp. 189 04 ASYMPTOTIC STRUCTURE FOR SOLUTIONS OF THE NAVIER STOKES EQUATIONS Tian Ma Department of
More informationLORENTZ SPACE ESTIMATES FOR VECTOR FIELDS WITH DIVERGENCE AND CURL IN HARDY SPACES
- TAMKANG JOURNAL OF MATHEMATICS Volume 47, Number 2, 249-260, June 2016 doi:10.5556/j.tkjm.47.2016.1932 This paper is available online at http://journals.math.tku.edu.tw/index.php/tkjm/pages/view/onlinefirst
More informationNULL CONDITION FOR SEMILINEAR WAVE EQUATION WITH VARIABLE COEFFICIENTS. Fabio Catalano
Serdica Math J 25 (999), 32-34 NULL CONDITION FOR SEMILINEAR WAVE EQUATION WITH VARIABLE COEFFICIENTS Fabio Catalano Communicated by V Petkov Abstract In this work we analyse the nonlinear Cauchy problem
More informationWell-posedness for the Fourth-order Schrödinger Equations with Quadratic Nonlinearity
Well-posedness for the Fourth-order Schrödinger Equations with Quadratic Nonlinearity Jiqiang Zheng The Graduate School of China Academy of Engineering Physics P. O. Box 20, Beijing, China, 00088 (zhengjiqiang@gmail.com)
More informationNONHOMOGENEOUS ELLIPTIC EQUATIONS INVOLVING CRITICAL SOBOLEV EXPONENT AND WEIGHT
Electronic Journal of Differential Equations, Vol. 016 (016), No. 08, pp. 1 1. ISSN: 107-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu NONHOMOGENEOUS ELLIPTIC
More informationDECAY ESTIMATES FOR THE KLEIN-GORDON EQUATION IN CURVED SPACETIME
Electronic Journal of Differential Equations, Vol. 218 218), No. 17, pp. 1 9. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu DECAY ESTIMATES FOR THE KLEIN-GORDON EQUATION
More informationLIMITING ABSORPTION PRINCIPLE AND STRICHARTZ ESTIMATES FOR DIRAC OPERATORS IN TWO AND HIGHER DIMENSIONS
LIMITING ABSORPTION PRINCIPLE AND STRICHARTZ ESTIMATES FOR DIRAC OPERATORS IN TWO AND HIGHER DIMENSIONS M. BURAK ERDOĞAN, MICHAEL GOLDBERG, WILLIAM R. GREEN Abstract. In this paper we consider Dirac operators
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 informationEIGENFUNCTIONS OF DIRAC OPERATORS AT THE THRESHOLD ENERGIES
EIGENFUNCTIONS OF DIRAC OPERATORS AT THE THRESHOLD ENERGIES TOMIO UMEDA Abstract. We show that the eigenspaces of the Dirac operator H = α (D A(x)) + mβ at the threshold energies ±m are coincide with the
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 informationGRAND SOBOLEV SPACES AND THEIR APPLICATIONS TO VARIATIONAL PROBLEMS
LE MATEMATICHE Vol. LI (1996) Fasc. II, pp. 335347 GRAND SOBOLEV SPACES AND THEIR APPLICATIONS TO VARIATIONAL PROBLEMS CARLO SBORDONE Dedicated to Professor Francesco Guglielmino on his 7th birthday W
More informationScientiae Mathematicae Japonicae Online, Vol. 5, (2001), Ryo Ikehata Λ and Tokio Matsuyama y
Scientiae Mathematicae Japonicae Online, Vol. 5, (2), 7 26 7 L 2 -BEHAVIOUR OF SOLUTIONS TO THE LINEAR HEAT AND WAVE EQUATIONS IN EXTERIOR DOMAINS Ryo Ikehata Λ and Tokio Matsuyama y Received November
More information. A NOTE ON THE RESTRICTION THEOREM AND GEOMETRY OF HYPERSURFACES
. A NOTE ON THE RESTRICTION THEOREM AND GEOMETRY OF HYPERSURFACES FABIO NICOLA Abstract. A necessary condition is established for the optimal (L p, L 2 ) restriction theorem to hold on a hypersurface S,
More information