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1.
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   

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In this paper, we consider the problem (Pε)(Pε) : Δ2u=un+4/n-4+εu,u>0Δ2u=un+4/n-4+εu,u>0 in Ω,u=Δu=0Ω,u=Δu=0 on ∂ΩΩ, where ΩΩ is a bounded and smooth domain in Rn,n>8Rn,n>8 and ε>0ε>0. We analyze the asymptotic behavior of solutions of (Pε)(Pε) which are minimizing for the Sobolev inequality as ε→0ε0 and we prove existence of solutions to (Pε)(Pε) which blow up and concentrate around a critical point of the Robin's function. Finally, we show that for εε small, (Pε)(Pε) has at least as many solutions as the Ljusternik–Schnirelman category of ΩΩ.  相似文献   

3.
Consider in a real Hilbert space H the Cauchy problem (P0P0): u(t)+Au(t)+Bu(t)=f(t)u(t)+Au(t)+Bu(t)=f(t), 0≤t≤T0tT; u(0)=u0u(0)=u0, where −A   is the infinitesimal generator of a C0C0-semigroup of contractions, B is a nonlinear monotone operator, and f is a given H-valued function. Inspired by the excellent book on singular perturbations by J.L. Lions, we associate with problem (P0P0) the following regularization (PεPε): −εu(t)+u(t)+Au(t)+Bu(t)=f(t)εu(t)+u(t)+Au(t)+Bu(t)=f(t), 0≤t≤T0tT; u(0)=u0u(0)=u0, u(T)=uTu(T)=uT, where ε>0ε>0 is a small parameter. We investigate existence, uniqueness and higher regularity for problem (PεPε). Then we establish asymptotic expansions of order zero, and of order one, for the solution of (PεPε). Problem (PεPε) turns out to be regularly perturbed of order zero, and singularly perturbed of order one, with respect to the norm of C([0,T];H)C([0,T];H). However, the boundary layer of order one is not visible through the norm of L2(0,T;H)L2(0,T;H).  相似文献   

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It is proven that the generalized Riemann problem for a class of quasilinear hyperbolic systems of balance laws admits a unique global piecewise C1C1 solution u=u(t,x)u=u(t,x) containing only nn shock waves with small amplitude on t?0t?0 and this solution possesses a global structure similar to that of the similarity solution u=U(x/t)u=U(x/t) of the corresponding homogeneous Riemann problem. As an application of our result, we prove the existence of global shock solutions, piecewise continuous and piecewise smooth solution with shock discontinuities, of the flow equations of a model class of fluids with viscosity induced by fading memory with a single jump initial data.  相似文献   

5.
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   

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We study the regularity up to the boundary of solutions to the Dirichlet problem for the fractional Laplacian. We prove that if u   is a solution of (−Δ)su=g(Δ)su=g in Ω  , u≡0u0 in RnRn\Ω, for some s∈(0,1)s(0,1) and g∈L(Ω)gL(Ω), then u   is Cs(Rn)Cs(Rn) and u/δs|Ωu/δs|Ω is CαCα up to the boundary ∂Ω   for some α∈(0,1)α(0,1), where δ(x)=dist(x,∂Ω)δ(x)=dist(x,Ω). For this, we develop a fractional analog of the Krylov boundary Harnack method.  相似文献   

11.
In this paper, we give a new proof of a result of R. Jones showing almost everywhere convergence of spherical means of actions of RdRd on Lp(X)Lp(X)-spaces are convergent for d?3d?3 and p>d/(d-1)p>d/(d-1).  相似文献   

12.
We consider the regularization of the backward in time problem for a nonlinear parabolic equation in the form ut+Au(t)=f(u(t),t)ut+Au(t)=f(u(t),t), u(1)=φu(1)=φ, where A is a positive self-adjoint unbounded operator and f is a local Lipschitz function. As known, it is ill-posed and occurs in applied mathematics, e.g. in neurophysiological modeling of large nerve cell systems with action potential f   in mathematical biology. A new version of quasi-reversibility method is described. We show that the regularized problem (with a regularization parameter β>0β>0) is well-posed and that its solution Uβ(t)Uβ(t) converges on [0,1][0,1] to the exact solution u(t)u(t) as β→0+β0+. These results extend some earlier works on the nonlinear backward problem.  相似文献   

13.
We consider an insurance company in the case when the premium rate is a bounded non-negative random function ctct and the capital of the insurance company is invested in a risky asset whose price follows a geometric Brownian motion with mean return a   and volatility σ>0σ>0. If β?2a/σ2-1>0β?2a/σ2-1>0 we find exact the asymptotic upper and lower bounds for the ruin probability Ψ(u)Ψ(u) as the initial endowment u   tends to infinity, i.e. we show that C*u?Ψ(u)?C*uC*u-β?Ψ(u)?C*u-β for sufficiently large u  . Moreover if ct=c*eγtct=c*eγt with γ?0γ?0 we find the exact asymptotics of the ruin probability, namely Ψ(u)∼uΨ(u)u-β. If β?0β?0, we show that Ψ(u)=1Ψ(u)=1 for any u?0u?0.  相似文献   

14.
We consider the semilinear elliptic equation Δu+K(|x|)up=0Δu+K(|x|)up=0 in RNRN for N>2N>2 and p>1p>1, and study separation phenomena of positive radial solutions. With respect to intersection and separation, we establish a classification of the solution structures, and investigate the structures of intersection, partial separation and separation. As a consequence, we obtain the existence of positive solutions with slow decay when the oscillation of the function r−?K(r)r?K(r) with ?>−2?>2 around a positive constant is small near r=∞r= and p   is sufficiently large. Moreover, if the assumptions hold in the whole space, the equation has the structure of separation and possesses a singular solution as the upper limit of regular solutions. We also reveal that the equation changes its nature drastically across a critical exponent pcpc which is determined by N   and the order of the behavior of K(r)K(r) as r=|x|→0r=|x|0 and ∞. In order to understand how subtle the structure is on K   at p=pcp=pc, we explain the criticality in a similar way as done by Ding and Ni (1985) [6] for the critical Sobolev exponent p=(N+2)/(N−2)p=(N+2)/(N2).  相似文献   

15.
This paper treats some variational principles for solutions of inhomogeneous p  -Laplacian boundary value problems on exterior regions U?RNU?RN with dimension N?3N?3. Existence-uniqueness results when p∈(1,N)p(1,N) are provided in a space E1,p(U)E1,p(U) of functions that contains W1,p(U)W1,p(U). Functions in E1,p(U)E1,p(U) are required to decay at infinity in a measure theoretic sense. Various properties of this space are derived, including results about equivalent norms, traces and an LpLp-imbedding theorem. Also an existence result for a general variational problem of this type is obtained.  相似文献   

16.
In the well-known work of P.-L. Lions [The concentration–compactness principle in the calculus of variations, The locally compact case, part 1. Ann. Inst. H. Poincaré, Analyse Non Linéaire 1 (1984) 109–1453] existence of positive solutions to the equation -Δu+u=b(x)up-1-Δu+u=b(x)up-1, u>0u>0, u∈H1(RN)uH1(RN), p∈(2,2N/(N-2))p(2,2N/(N-2)) was proved under assumption b(x)?b?lim|x|b(x)b(x)?b?lim|x|b(x). In this paper we prove the existence for certain functions b   satisfying the reverse inequality b(x)<bb(x)<b. For any periodic lattice L   in RNRN and for any b∈C(RN)bC(RN) satisfying b(x)<bb(x)<b, b>0b>0, there is a finite set Y⊂LYL and a convex combination bYbY of b(·-y)b(·-y), y∈YyY, such that the problem -Δu+u=bY(x)up-1-Δu+u=bY(x)up-1 has a positive solution u∈H1(RN)uH1(RN).  相似文献   

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For a non-degenerate convex subset Y of the n  -dimensional Euclidean space RnRn, let F(Y)F(Y) be the family of all fuzzy sets of RnRn which are upper semicontinuous, fuzzy convex and normal with compact supports contained in Y  . We show that the space F(Y)F(Y) with the topology of sendograph metric is homeomorphic to the separable Hilbert space ?2?2 if Y   is compact; and the space F(Rn)F(Rn) is also homeomorphic to ?2?2.  相似文献   

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