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1.
We give a Sobolev inequality with the weight K(x)K(x) belonging to the class A2GnA2Gn for the function |u|t|u|t and the weight K(x)−1K(x)1 for |∇u|2|u|2. The constant in the relevant inequality is seen to depend on the GnGn and A2A2 constants of the weight.  相似文献   

2.
In this note we derive a maximum principle for an appropriate functional combination of u(x)u(x) and |∇u|2|u|2, where u(x)u(x) is a strictly convex classical solution to a general class of Monge–Ampère equations. This maximum principle is then employed to establish some isoperimetric inequalities of interest in the theory of surfaces of constant Gauss curvature in RN+1RN+1.  相似文献   

3.
Let GG be an arbitrary finite group and let SS and TT be two subsets such that |S|≥2|S|2, |T|≥2|T|2, and |TS|≤|T|+|S|−1≤|G|−2|TS||T|+|S|1|G|2. We show that if |S|≤|G|−4|G|1/2|S||G|4|G|1/2 then either SS is a geometric progression or there exists a non-trivial subgroup HH such that either |HS|≤|S|+|H|−1|HS||S|+|H|1 or |SH|≤|S|+|H|−1|SH||S|+|H|1. This extends to the nonabelian case classical results for abelian groups. When we remove the hypothesis |S|≤|G|−4|G|1/2|S||G|4|G|1/2 we show the existence of counterexamples to the above characterization whose structure is described precisely.  相似文献   

4.
5.
We consider the simplest possible heat equation for director fields, ut=Δu+|∇u|2uut=Δu+|u|2u (|u|=1|u|=1), and construct axially symmetric traveling wave solutions defined in an infinitely long cylinder. The traveling waves have a point singularity of topological degree 0 or 1.  相似文献   

6.
The author deals with the quasilinear parabolic equation ut=[uα+g(u)]Δu+buα+1+f(u,∇u)ut=[uα+g(u)]Δu+buα+1+f(u,u) with Dirichlet boundary conditions in a bounded domain ΩΩ, where ff and gg are lower-order terms. He shows that, under suitable conditions on ff and gg, whether the solution is bounded or blows up in a finite time depends only on the first eigenvalue of −ΔΔ in ΩΩ with Dirichlet boundary condition. For some special cases, the result is sharp.  相似文献   

7.
In this paper we establish the boundedness of the extremal solution uu in dimension N=4N=4 of the semilinear elliptic equation −Δu=λf(u)Δu=λf(u), in a general smooth bounded domain Ω⊂RNΩRN, with Dirichlet data u|Ω=0u|Ω=0, where ff is a C1C1 positive, nondecreasing and convex function in [0,∞)[0,) such that f(s)/s→∞f(s)/s as s→∞s.  相似文献   

8.
For ΩΩ, an open bounded subset of RNRN with smooth boundary and 1<p<∞1<p<, we establish W1,p(Ω)W1,p(Ω)a priori bounds and prove the compactness of solution sets to differential inequalities of the form
|divA(x,∇u)|≤F(x,u,∇u),|divA(x,u)|F(x,u,u),
which are bounded in L(Ω)L(Ω). The main point in this work is that the nonlinear term FF may depend on ∇uu and may grow as fast as a power of order pp in this variable. Such growth conditions have been used extensively in the study of boundary value problems for nonlinear ordinary differential equations and are known as Bernstein–Nagumo growth conditions. In addition, we use these results to establish a sub-supersolution theorem.  相似文献   

9.
We study some properties of the solutions of (E) −Δpu+|∇u|q=0Δpu+|u|q=0 in a domain Ω⊂RNΩRN, mostly when p≥q>p−1pq>p1. We give a universal a priori estimate of the gradient of the solutions with respect to the distance to the boundary. We give a full classification of the isolated singularities of the nonnegative solutions of (E), a partial classification of isolated singularities of the negative solutions. We prove a general removability result expressed in terms of some Bessel capacity of the removable set. We extend our estimates to equations on complete noncompact manifolds satisfying a lower bound estimate on the Ricci curvature, and derive some Liouville type theorems.  相似文献   

10.
We consider two-dimensional mixed problems in an exterior domain for a semilinear strongly damped wave equation with a power-type nonlinearity |u|p|u|p. If the initial data have a small weighted energy, we shall derive a global existence and energy decay results in the case when the power pp of the nonlinear term satisfies p>6p>6.  相似文献   

11.
12.
Let FFvFFv be the set of faulty nodes in an nn-dimensional folded hypercube FQnFQn with |FFv|≤n−2|FFv|n2. In this paper, we show that if n≥3n3, then every edge of FQn−FFvFQnFFv lies on a fault-free cycle of every even length from 44 to 2n−2|FFv|2n2|FFv|, and if n≥2n2 and nn is even, then every edge of FQn−FFvFQnFFv lies on a fault-free cycle of every odd length from n+1n+1 to 2n−2|FFv|−12n2|FFv|1.  相似文献   

13.
A finite Sturmian   word ww is a balanced word over the binary alphabet {a,b}{a,b}, that is, for all subwords uu and vv of ww of equal length, ||u|a|v|a|≤1||u|a|v|a|1, where |u|a|u|a and |v|a|v|a denote the number of occurrences of the letter aa in uu and vv, respectively. There are several other characterizations, some leading to efficient algorithms for testing whether a finite word is Sturmian. These algorithms find important applications in areas such as pattern recognition, image processing, and computer graphics. Recently, Blanchet-Sadri and Lensmire considered finite semi-Sturmian words of minimal length and provided an algorithm for generating all of them using techniques from graph theory. In this paper, we exploit their approach in order to count the number of minimal semi-Sturmian words. We also present some other results that come from applying this graph theoretical framework to subword complexity.  相似文献   

14.
We study a multi-dimensional nonlocal active scalar equation of the form ut+v⋅∇u=0ut+vu=0 in R+×RdR+×Rd, where v=Λ−2+α∇uv=Λ2+αu with Λ=(−Δ)1/2Λ=(Δ)1/2. We show that when α∈(0,2]α(0,2] certain radial solutions develop gradient blowup in finite time. In the case when α=0α=0, the equations are globally well-posed with arbitrary initial data in suitable Sobolev spaces.  相似文献   

15.
16.
Consider a graph GG with a minimal edge cut FF and let G1G1, G2G2 be the two (augmented) components of G−FGF. A long-open question asks under which conditions the crossing number of GG is (greater than or) equal to the sum of the crossing numbers of G1G1 and G2G2—which would allow us to consider those graphs separately. It is known that crossing number is additive for |F|∈{0,1,2}|F|{0,1,2} and that there exist graphs violating this property with |F|≥4|F|4. In this paper, we show that crossing number is additive for |F|=3|F|=3, thus closing the final gap in the question.  相似文献   

17.
18.
By a perturbation method and constructing comparison functions, we reveal how the inhomogeneous term hh affects the exact asymptotic behaviour of solutions near the boundary to the problem △u=b(x)g(u)+λh(x)u=b(x)g(u)+λh(x), u>0u>0 in ΩΩ, u|Ω=∞u|Ω=, where ΩΩ is a bounded domain with smooth boundary in RNRN, λ>0λ>0, g∈C1[0,∞)gC1[0,) is increasing on [0,∞)[0,), g(0)=0g(0)=0, gg is regularly varying at infinity with positive index ρρ, the weight bb, which is non-trivial and non-negative in ΩΩ, may be vanishing on the boundary, and the inhomogeneous term hh is non-negative in ΩΩ and may be singular on the boundary.  相似文献   

19.
This paper deals with the asymptotic behavior of strong solutions to the 3D Navier–Stokes equations with a nonlinear damping term |u|β−1u(β≥3)|u|β1u(β3). First, we establish an upper bound for the difference between the solution of our equation and the heat equation in L2L2 space. Then, we optimize the upper bound of decay for the solutions and obtain their algebraic lower bound by using Fourier Splitting method.  相似文献   

20.
We prove the Strong Maximum Principle (SMP) under suitable assumptions for a class of quasilinear parabolic problems with the p  -Laplacian, p>1p>1, on bounded cylindrical domains of RN+1RN+1,
tu−Δpu−λ|u|p−2u≥0,tuΔpuλ|u|p2u0,
with nonnegative initial–boundary conditions and λ≤0λ0, and we give some counterexamples to the SMP if some of our assumptions are violated. We show that the Hopf Maximum Principle holds for 1<p<21<p<2, and give a counterexample to it for p>2p>2. Also the Weak Maximum Principle for λ≤λ1λλ1 is established.  相似文献   

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