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1.
We consider non-local linear Schrödinger-type critical systems of the type(1) where Ω is antisymmetric potential in L2(R,so(m)), v is an Rm valued map and Ωv denotes the matrix multiplication. We show that every solution vL2(R,Rm) of (1) is in fact in , for every 2?p<+∞, in other words, we prove that the system (1) which is a-priori only critical in L2 happens to have a subcritical behavior for antisymmetric potentials. As an application we obtain the regularity of weak 1/2-harmonic maps into C2 compact sub-manifolds without boundary.  相似文献   

2.
We prove Liouville type theorems for weak solutions of the Navier–Stokes and the Euler equations. In particular, if the pressure satisfies pL1(0,T;L1(RN)) with , then the corresponding velocity should be trivial, namely v=0 on RN×(0,T). In particular, this is the case when pL1(0,T;Hq(RN)), where Hq(RN), q∈(0,1], the Hardy space. On the other hand, we have equipartition of energy over each component, if pL1(0,T;L1(RN)) with . Similar results hold also for the magnetohydrodynamic equations.  相似文献   

3.
In this paper, we prove existence of radially symmetric minimizersuA(x)=UA(|x|), having UA(⋅)AC monotone and increasing, for the convex scalar multiple integral(∗ ) among those u(⋅) in the Sobolev space. Here, |u(x)| is the Euclidean norm of the gradient vector and BR is the ball ; while A is the boundary data.Besides being e.g. superlinear (but no growth needed if (∗) is known to have minimum), our Lagrangian?∗∗:R×R→[0,] is just convex lsc and and ?∗∗(s,⋅) is even; while ρ1(⋅) and ρ2(⋅) are Borel bounded away from .Remarkably, (∗) may also be seen as the calculus of variations reformulation of a distributed-parameter scalar optimal control problem. Indeed, state and gradient pointwise constraints are, in a sense, built-in, since ?∗∗(s,v)= is freely allowed.  相似文献   

4.
In this note the following inequality is proved. For any nonnegative measure μH−1(R2), xR2 and 0<r<1, there holds
(1)
where C is a positive constant. Using (1) an estimate for the vorticity maximal function similar to the estimate in Majda [A. Majda, Remarks on weak solutions for vortex sheets with a distinguished sign, Indiana Univ. Math. J. 42 (1993) 921–939] is established without assuming the initial vorticity having compact support. From this a more simple proof of the Delort's celebrated theorem [J.M. Delort, Existence de mappes de fourbillon en dimension deux, J. Amer. Math. Soc. 4 (1991) 553–586] is presented.  相似文献   

5.
In this paper and in a forthcoming one, we study difference equations in of the types (2)(4)(6) which are linked to families of conics, cubics and quartics, respectively. These equations generalize Lyness' one un+2un=a+un+1 studied in several papers, whose behavior was completely elucidated in [G. Bastien, M. Rogalski, in press] through methods which are transposed in the present paper for the study of (1) and (2), and in the forthcoming one for (3). In particular we prove in the present paper a form of chaotic behavior for solutions of difference equations (1) and (2), and find all the possible periods for these solutions.  相似文献   

6.
We consider the fully nonlinear integral systems involving Wolff potentials:(1) whereThis system includes many known systems as special cases, in particular, when and γ=2, system (1) reduces to(2) The solutions (u,v) of (2) are critical points of the functional associated with the well-known Hardy–Littlewood–Sobolev inequality. We can show that (2) is equivalent to a system of semi-linear elliptic PDEs which comprises the well-known Lane–Emden system and Yamabe equation.We obtain integrability and regularity for the positive solutions to systems (1). A regularity lifting method by contracting operators is used in proving the integrability, and while deriving the Lipschitz continuity, a brand new idea – Lifting Regularity by Shrinking Operators is introduced. We hope to see many more applications of this new idea in lifting regularities of solutions for nonlinear problems.  相似文献   

7.
We consider a process given by the SDE , t∈[0,T), with initial condition , where T∈(0,∞], αR, (Bt)t∈[0,T) is a standard Wiener process, b:[0,T)→R?{0} and σ:[0,T)→(0,∞) are continuously differentiable functions. Assuming , t∈[0,T), with some KR, we derive an explicit formula for the joint Laplace transform of and for all t∈[0,T) and for all αR. Our motivation is that the maximum likelihood estimator (MLE) of α can be expressed in terms of these random variables. As an application, we show that in case of α=K, K≠0,
  相似文献   

8.
The following Dirichlet problem
(1.1)
is considered, where , N≥2, KC2[0,1] and K(r)>0 for 0≤r≤1, , sf(s)>0 for s≠0. Assume moreover that f satisfies the following sublinear condition: f(s)/s>f(s) for s≠0. A sufficient condition is derived for the uniqueness of radial solutions of (1.1) possessing exactly k−1 nodes, where . It is also shown that there exists KC[0,1] such that (1.1) has three radial solutions having exactly one node in the case N=3.  相似文献   

9.
Let be a strictly increasing sequence of real numbers satisfying(0.1)aj+1−aj?σ>0. For an open box I in [0,1d), we write It is shown that the Hausdorff dimension of is d−1 whenever The case d=1 is due to Boshernitzan. The proof builds on his approach.Now let S1,…,Sd be strictly increasing in N. Define to be the set of x in [0, 1) for which A sequence S is said to fulfill condition D(C) if it containsBr=[ur,vr]∩S for which vrur→∞ and1+vrur?C#(Br). Kaufman has shown that is countable whenever S1,…,Sd fulfill condition D(C). Here it is shown that is finite under this hypothesis. An upper bound for is provided.  相似文献   

10.
If P is a polynomial on Rm of degree at most n, given by , and Pn(Rm) is the space of such polynomials, then we define the polynomial |P| by . Now if is a convex set, we define the norm on Pn(Rm), and then we investigate the inequality providing sharp estimates on for some specific spaces of polynomials. These ’s happen to be the unconditional constants of the canonical bases of the considered spaces.  相似文献   

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