首页 | 本学科首页   官方微博 | 高级检索  
相似文献
 共查询到20条相似文献,搜索用时 125 毫秒
1.
In this article, the authors obtain an integral representation for the relaxation of the functional F(x,u, Ω) := {∫Ωf(x, u(x),εu(x))dx if u ∈ W1,1(Ω,Rn), ∞ otherwis,in the space of functions of bounded deformation, with respect to L1-convergence. Here εu represents the absolutely continuous part of the symmetrized distributional derivative Eu.f(x, p, ε) satisfying weak convexity assumption.  相似文献   

2.
The existence of solutions is obtained for a class of the non-periodic Schrdinger equation -Δu + V (x)u = f (x, u), x ∈ R N , by the generalized mountain pass theorem, where V is large at infinity and f is superlinear as |u| →∞.  相似文献   

3.
We give an existence result of the obstacle parabolic equations(b(x,u))/(t)-div(a(x,t,u,▽u))+div(φ(x,t,u))=f in Q_T,where b(x,u) is bounded function of u,the term-div(a(x,t,u,▽u)) is a Leray-Lions type operator and the function φ is a nonlinear lower order and satisfy only the growth condition.The second term f belongs to L~1(Q_T).The proof of an existence solution is based on the penalization methods.  相似文献   

4.
In this paper,we mainly discuss a priori bounds of the following degenerate elliptic equation,a ij(x)■ij u+b i(x)■iu+f(x,u)=0,in ΩRn,(*)where aij■iφ■jφ=0 on■Ω,andφis the defining function of ■Ω.Imposing suitable conditions on the coefficients and f(x,u),one can get the L∞-estimates of(*)via blow up method.  相似文献   

5.
Let u = u(t, x, p) satisfy the transport equation ?u/?t+p/p0 ?u/?x= f, where f =f(t, x, p) belongs to L~p((0, T) × R~3× R~3) for 1 p ∞ and ?/?t+p/p0 ?/?x is the relativisticfree transport operator from the relativistic Boltzmann equation. We show the regularity of ∫_(R~3) u(t, x, p)d p using the same method as given by Golse, Lions, Perthame and Sentis. This average regularity is considered in terms of fractional Sobolev spaces and it is very useful for the study of the existence of the solution to the Cauchy problem on the relativistic Boltzmann equation.  相似文献   

6.
In this article,we study the initial boundary value problem of generalized Pochhammer-Chree equation u_(tt)-u_(xx)-u_(xxt)-u_(xxtt)=f(u) xx,x ∈Ω,t 0,u(x,0) = u0(x),u t(x,0)=u1(x),x ∈Ω,u(0,t) = u(1,t) = 0,t≥0,where Ω=(0,1).First,we obtain the existence of local W k,p solutions.Then,we prove that,if f(s) ∈ΩC k+1(R) is nondecreasing,f(0) = 0 and |f(u)|≤C1|u| u 0 f(s)ds+C2,u 0(x),u 1(x) ∈ΩW k,p(Ω) ∩ W 1,p 0(Ω),k ≥ 1,1 p ≤∞,then for any T 0 the problem admits a unique solution u(x,t) ∈ W 2,∞(0,T;W k,p(Ω) ∩ W 1,p 0(Ω)).Finally,the finite time blow-up of solutions and global W k,p solution of generalized IMBq equations are discussed.  相似文献   

7.
In this paper we consider the bifurcation problem -div A(x, u)=λa(x)|u|^p-2u+f(x,u,λ) in Ω with p 〉 1.Under some proper assumptions on A(x,ξ),a(x) and f(x,u,λ),we show that the existence of an unbounded branch of positive solutions bifurcating Irom the principal eigenvalue of the problem --div A(x, u)=λa(x)|u|^p-2u.  相似文献   

8.
This paper investigates a class of nonlinear elliptic equations on a fractal domain. We establish a strong Sobolev-type inequality which leads to the existence of multiple non-trivial solutions of △u+ c(x)u = f(x, u), with zero Dirichlet boundary conditions on the Sierpihski gasket. Our existence results do not require any growth conditions of f(x,t) in t, in contrast to the classical theory of elliptic equations on smooth domains.  相似文献   

9.
In this paper, the periodic boundary problem and the initial value problem for the nonlinear system of parabolic type u_1=-A(x, t)u_(x4)+B(x, t)u_(x2)+(g(u))_(x2)+(grad h(u))_x+f(u)are studied, where u(x, t)=(u_1(x, t).…, u_J(x, t) is a J-dimensional unknown vector valued function, f(u) and g(u) are the J-dimensional vector valued function of u(x, t), h(u) is a scalar function of u, A(x, t) and B(x, t) are J×J matrices of functions. The existent, uniqueness and regularities of the generalized global solution and classical global solution of the problems are proved. When J=1, h(u)=0, g(u)=au~3, A=a_1, B=a_2, where a_1, a_2 a are constants, the system is a generalized diffusion model equation in population problem.  相似文献   

10.
In the present paper, we consider elliptic equations with nonlinear and nonhomogeneous Robin boundary conditions of the type{-div(B(x, u)▽u) = f in ?,u = 0 on Γ_0,B(x, u)▽u·n→+γ(x)h(u) =g on Γ_1,where f and g are the element of L~1(?) and L~1(Γ_1), respectively. We define a notion of renormalized solution and we prove the existence of a solution. Under additional assumptions on the matrix field B we show that the renormalized solution is unique.  相似文献   

11.
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   

12.
We consider a multidimensional diffusion XX with drift coefficient b(α,Xt)b(α,Xt) and diffusion coefficient ?σ(β,Xt)?σ(β,Xt). The diffusion sample path is discretely observed at times tk=kΔtk=kΔ for k=1…nk=1n on a fixed interval [0,T][0,T]. We study minimum contrast estimators derived from the Gaussian process approximating XX for small ??. We obtain consistent and asymptotically normal estimators of αα for fixed ΔΔ and ?→0?0 and of (α,β)(α,β) for Δ→0Δ0 and ?→0?0 without any condition linking ?? and ΔΔ. We compare the estimators obtained with various methods and for various magnitudes of ΔΔ and ?? based on simulation studies. Finally, we investigate the interest of using such methods in an epidemiological framework.  相似文献   

13.
Let r,s∈]1,2[r,s]1,2[ and λ,μ∈]0,+∞[λ,μ]0,+[. In this paper, we deal with the existence and multiplicity of nonnegative and nonzero solutions of the Dirichlet problem with 00 boundary data for the semilinear elliptic equation −Δu=λus−1−ur−1Δu=λus1ur1 in Ω⊂RNΩRN, where N≥2N2. We prove that there exists a positive constant ΛΛ such that the above problem has at least two solutions, at least one solution or no solution according to whether λ>Λλ>Λ, λ=Λλ=Λ or λ<Λλ<Λ. In particular, a result by Hernandéz, Macebo and Vega is improved and, for the semilinear case, a result by Díaz and Hernandéz is partially extended to higher dimensions. Finally, an answer to a conjecture, recently stated by the author, is also given.  相似文献   

14.
15.
We study the problem (−Δ)su=λeu(Δ)su=λeu in a bounded domain Ω⊂RnΩRn, where λ   is a positive parameter. More precisely, we study the regularity of the extremal solution to this problem. Our main result yields the boundedness of the extremal solution in dimensions n≤7n7 for all s∈(0,1)s(0,1) whenever Ω   is, for every i=1,...,ni=1,...,n, convex in the xixi-direction and symmetric with respect to {xi=0}{xi=0}. The same holds if n=8n=8 and s?0.28206...s?0.28206..., or if n=9n=9 and s?0.63237...s?0.63237.... These results are new even in the unit ball Ω=B1Ω=B1.  相似文献   

16.
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   

17.
An acyclic edge coloring of a graph GG is a proper edge coloring such that no bichromatic cycles are produced. The acyclic chromatic index a(G)a(G) of GG is the smallest integer kk such that GG has an acyclic edge coloring using kk colors. It was conjectured that a(G)≤Δ+2a(G)Δ+2 for any simple graph GG with maximum degree ΔΔ. In this paper, we prove that if GG is a planar graph, then a(G)≤Δ+7a(G)Δ+7. This improves a result by Basavaraju et al. [M. Basavaraju, L.S. Chandran, N. Cohen, F. Havet, T. Müller, Acyclic edge-coloring of planar graphs, SIAM J. Discrete Math. 25 (2011) 463–478], which says that every planar graph GG satisfies a(G)≤Δ+12a(G)Δ+12.  相似文献   

18.
19.
We establish symmetrization results for the solutions of the linear fractional diffusion equation tu+(−Δ)σ/2u=ftu+(Δ)σ/2u=f and its elliptic counterpart hv+(−Δ)σ/2v=fhv+(Δ)σ/2v=f, h>0h>0, using the concept of comparison of concentrations. The results extend to the nonlinear version, tu+(−Δ)σ/2A(u)=ftu+(Δ)σ/2A(u)=f, but only when the nondecreasing function A:R+R+A:R+R+ is concave. In the elliptic case, complete symmetrization results are proved for B(v)+(−Δ)σ/2v=fB(v)+(Δ)σ/2v=f when B(v)B(v) is a convex nonnegative function for v>0v>0 with B(0)=0B(0)=0, and partial results hold when B is concave. Remarkable counterexamples are constructed for the parabolic equation when A is convex, resp. for the elliptic equation when B   is concave. Such counterexamples do not exist in the standard diffusion case σ=2σ=2.  相似文献   

20.
Given a point AA in the real Grassmannian, it is well-known that one can construct a soliton solution uA(x,y,t)uA(x,y,t) to the KP equation. The contour plot   of such a solution provides a tropical approximation to the solution when the variables xx, yy, and tt are considered on a large scale and the time tt is fixed. In this paper we use several decompositions of the Grassmannian in order to gain an understanding of the contour plots of the corresponding soliton solutions. First we use the positroid stratification   of the real Grassmannian in order to characterize the unbounded line-solitons in the contour plots at y?0y?0 and y?0y?0. Next we use the Deodhar decomposition   of the Grassmannian–a refinement of the positroid stratification–to study contour plots at t?0t?0. More specifically, we index the components of the Deodhar decomposition of the Grassmannian by certain tableaux which we call Go-diagrams  , and then use these Go-diagrams to characterize the contour plots of solitons solutions when t?0t?0. Finally we use these results to show that a soliton solution uA(x,y,t)uA(x,y,t) is regular for all times tt if and only if AA comes from the totally non-negative part of the Grassmannian.  相似文献   

设为首页 | 免责声明 | 关于勤云 | 加入收藏

Copyright©北京勤云科技发展有限公司  京ICP备09084417号