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1.
In this paper we study the existence theorem for solution of the nonlinear degenerate oblique boundary value problems for second order fully nonlinear elliptic equations F(x, u, Du, D²u) = 0 \quad x ∈ Ω, G(x, u, D, u) = 0, \qquad x ∈ ∂Ω where F (x, z, p, r) satisfies the natural structure conditions, G (x, z, q) satisfies G_q ≥ 0, G_x ≤ - G_0 < 0 and some structure conditions, vector τ is nowhere tangential to ∂Ω. This result extends the works of Lieberman G. M., Trudinger N. S. [2], Zhu Rujln [1] and Wang Feng [6].  相似文献   

2.
We consider the problem of existence for viscosity solutions of seoond order fully nonlinear elliptic partial differential equations F(D²u, Du, u, z) = 0. We prove existence results for viscosity solutions in W^{1,∞} under assumptions that function F satisfies the natural structure conditions. We do not assume F is convex.  相似文献   

3.
ln this paper we discuss the Dirichlet problems for a class of fully nonliucar elliptic equations F(D² u) = ψ(x, u)(ψ(x, u, ∇u)) \quad in Q u = φ(x) \quad on ∂Ω where F is represented by a symmetric function f(λ_1, …, λ_n) of the eigenvalues (λ_1, …,λ_n) of the Hessian D²u. This result extends the works of Caffarelli L., Nirenberg L., Spruck L. [2] to more general cases.  相似文献   

4.
Recently R. Jensen [1] has proved the uniqueness of viscosity solutions in W^{1,∞} of second order fully nonlinear elliptic equation F (D², Du, u) = 0. He does not assume F to be convex. In this paper we extend his result [1] to the case that F can be dependent on x, i. e. prove that the viscosity solutions in W^{1,∞} of the second order fully nonlinear elliptic equation F (D²u, Du, u, x) = 0 are unlique. We do not assume F to be convex either.  相似文献   

5.
In this paper we prove the existence of a positive solution to the following superlinear elliptic Dirichlet problem, - Σ^n_{i,j=1}aij(x, u, Du)D_{ij}u = f(x, u, Du) in Ω, \quad u = 0 on ∂Ω where f satisfies certain growth conditions.  相似文献   

6.
Let Ω be a bounded domain in R^4(n ≥ 4) with smooth boundary ∂Ω. We discuss the existence of nontrivial solutions of the Dirichlet problem {- Δu = a(x) |u|^{4/(a-2)}u + λu + g(x, u), \quad x ∈ Ω u = 0, \quad x ∈ ∂Ω where a(x) is a smooth function which is nonnegative on \overline{Ω} and positive somewhere, λ> 0 and λ ∉ σ(-Δ). We weaken the conditions on a(x) that are generally assumed in other papers dealing with this problem.  相似文献   

7.
ln this paper we are devoted to the free boundary problem {u_t = ΔA(u) \quad (x,t) ∈ G_{r,r} u(x, 0) = φ(x) \quad ∈ G_0 u|_r = 0 (\frac{∂A(u)}{∂x_i}v_i + ψ(x)v_1)|_r = 0, where A'(u) ≥ 0. Under suitable assumptions we obtain the existence and uniqueness of global radial solutions for n =2 and local radial solutions for n ≥ 3.  相似文献   

8.
In this paper, the existence of positive solutions for the mixed boundary problem of quasilinear elliptic equation {-div (|∇u|^{p-2}∇u) = |u|^{p^∗-2}u + f(x, u), \quad u > 0, \quad x ∈ Ω u|_Γ_0 = 0, \frac{∂u}{∂\overrightarrow{n}}|_Γ_1 = 0 is obtained, where Ω is a bounded smooth domain in R^N, ∂Ω = \overrightarrow{Γ}_0 ∪ \overrightarrow{Γ}_1, 2 ≤ p < N, p^∗ = \frac{Np}{N-p}, Γ_0 and Γ_1 are disjoint open subsets of ∂Ω.  相似文献   

9.
We prove in this paper theC regularity for a “very strict” local minimum of classC loc ρ , ρ>3, of functionals with genuine degenerate quasiconvex integrand depending on a vector-valued function u. Such a minimum satisfies the condition: for all x∈Ω, there exists a neighbourhoodK(x) ofx in Ω andC 1 (x)>0,C 2 (x)>0,1≥ε(x)>0, such that for all real ϕ∈c 0 (K). This work is supported by the National Natural Science Foundation of China and the Fok Ying Tung Education Foundation.  相似文献   

10.
ln this paper we consider the model problem for a second order quasilinear degenerate parabolic equation {D_xG(u) = t^{2N-1}D²_xK(u) + t^{N-1}D_x,F(u) \quad for \quad x ∈ R,t > 0 u(x,0) = A \quad for \quad x < 0, u(x,0) = B \quad for \quad x > 0 where A < B, and N > O are given constants; K(u) =^{def} ∫^u_Ak(s)ds, G(u)=^{def} ∫^u_Ag(s)ds, and F(u) =^{def} ∫^u_Af(s)ds are real-valued absolutely continuous functions defined on [A, B] such that K(u) is increasing, G(u) strictly increasing, and \frac{F(B)}{G(B)}G(u) - F(u) nonnegative on [A, B]. We show that the model problem has a unique discontinuous solution u_0 (x, t) when k(s) possesses at least one interval of degeneracy in [A, B] and that on each curve of discontinuity, x = z_j(t) =^{def} s_jt^N, where s_j= const., j=l,2, …, u_0(x, t) must satisfy the following jump conditions, 1°. u_0(z_j(t) - 0, t) = a_j, u_0 (z_j(t) + 0, t) = b_j, and u_0(z_j(t) - 0, t) = [a_j, b_j] where {[a_j, b_j]; j = 1, 2, …} is the collection of all intervals of degeneracy possessed by k (s) in [A, B], that is, k(s) = 0 a. e. on [a_j, b_j], j = 1, 2, …, and k(s) > 0 a. e. in [A, B] \U_j[a_j, b_j], and 2°. (z_j(t)G(u_0(x, t)) + t^{2N-1}D_xK(u_0(x, t)) + t^{N-1}F(u_0(x, t)))|\frac{s=s_j+0}{s=s_j-0} = 0  相似文献   

11.
In this paper we deal with the limit behaviour of the bounded solutions uε of quasi-linear equations of the form of Ω with Dirichlet boundary conditions on σΩ. The map a=a(x,ϕ) is periodic in x, monotone in ϕ, and satisfies suitable coerciveness and growth conditions. The function H=H(x,s,ϕ) is assumed to be periodic in x, continuous in [s,ϕ] and to grow at most like |ξ|p. Under these assumptions on a and H we prove that there exists a function H0=H0(s,ϕ) with the same behaviour of H, such that, up to a subsequence, (uε) converges to a solution u of the homogenized problem -div(b(Du)) + γ|u|p-2u = H0(u,Du) + h(x) on Ω, where b depends only on a and has analogous qualitative properties.  相似文献   

12.
We are concerned with the Dirichlet problem of {div A(x, Du) + B(z) = 0 \qquad in Ω u= u_0 \qquad \qquad on ∂ Ω Here Ω ⊂ R^N is a bounded domain, A(x, p) = (A¹ (x, p), ... >A^N (x, p}) satisfies min{|p|^{1+α}, |p|^{1+β}} ≤ A(x, p) ⋅ p ≤ α_0(|p|^{1+α}+|p|^{1+β}) with 0 < α ≤ β. We show that if A is Lipschitz, B and u_0 are bounded and β < max {\frac{N+2}{N}α + \frac{2}{N},α + 2}, then there exists a C¹-weak solution of (0.1).  相似文献   

13.
In this paper, we study the following Eigen-problem {-\frac{∂}{∂x_i}(a_{ij}(x, u)\frac{∂u}{∂x_j}) + \frac{1}{2}a_{iju}(x,u)\frac{∂u}{∂x_i}\frac{∂u}{∂x_j} + h(x)u = μμ\frac{n+2}{n-2} \quad in Ω \qquad (0.1) u = 0 \quad on ∂Ω u > 0 \quad in Ω ⊂ R^n under some assumptions. First. we minimize I(u) = \frac{1}{2}∫_Ωa_{ij}(x, u)\frac{∂u}{∂x_i}\frac{∂u}{∂x_j} + h(x)u² over E_α = {u ∈ H¹_0(Ω); ∫_Ωu^α = 1} ( 2 < α < N = \frac{2n}{n-2}) to give a H¹_0-solution U_α of the perturbation problems of (0.1). Since I is not differentiable in H¹_0(Ω), the key point is the estimate of U_α. Then, we derive local uniform bounds of (U_α) and give a 'bad' solution of (0.1). Last, we remove the singular points of the 'bad' solution to obtain a solution of (0.1), our result is a extension of that of Brezis & Nirenberg.  相似文献   

14.
The following regularity of weak solutions of a class of elliptic equations of the form are investigated.  相似文献   

15.
We have obtained in this paper the existence of weak solutions to the Cauchy problem for a special system of quasillnear equations with physical interest of the form {\frac{∂}{∂t}(u + qz) + \frac{∂}{∂x}f(u) = 0 \frac{∂z}{∂t} + kφ(u)z = 0 for the assumed smooth function φ(u) by employing the viscosity method and the theory of compensated compactness.  相似文献   

16.
Denoting byu a vector in R N defined on a bounded open set Ω ⊂ R n , we setH(u)={Dij u} and consider a basic differential operator of second ordera(H(u)) wherea(ξ) is a vector in R N , which is elliptic in the sense that it satisfies the condition (A). After a rapid comparison between this condition (A) and the classical definition of ellipticity, we shall prove that, if seu∈H 2 (Ω) is a solution of the elliptic systema(H(u))=0 in Ω thenH(u)∈H loc 2, q for someq>2. We then deduce from this the so called fundamental internal estimates for the matrixH(u) and for the vectorsDu andu. We shall then present a first risult on h?lder regularity for the solutions of the system withf h?lder continuous in Ω, and a partial h?lder continuity risult for solutionsu∈H 2 (Ω) of a differential systema (x, u, Du, H (u))=b(x, u, Du)  相似文献   

17.
We propose a solution strategy for fractional programming problems of the form max xx g(x)/ (u(x)), where the function satisfies certain convexity conditions. It is shown that subject to these conditions optimal solutions to this problem can be obtained from the solution of the problem max xx g(x) + u(x), where is an exogenous parameter. The proposed strategy combines fractional programming andc-programming techniques. A maximal mean-standard deviation ratio problem is solved to illustrate the strategy in action.  相似文献   

18.
In this paper, we study the hypoellipticity problems for fully nonlinear panial differential equations of order m. For a solution u ∈ C^p_{loc}(Ω), if the linearized operator for the nonlinear equation on u satisfies some subelliptic conditions, we can deduce u ∈ C^∞(Ω) by using the paradifferential operator theory of J. -M. Bony.  相似文献   

19.
We characterize the maps F=F(u,A) defined for u ∈ W1,∞ and A open, which can be written as supremal functionals of the form F(u,A)=ess supx ∈ A f(x,u(x),Du(x)).  相似文献   

20.
The general difference schemes for the first boundary problem of the fully nonlinear parabolic systems of second order f(x, t, u, u_x, u_{xx}, u_t) = 0 are considered in the rectangular domain Q_T = {0 ≤ x ≤ l, 0 ≤ t ≤ T}, where u(x, t) and f(x, t, u, p, r, q) are two m-dimensional vector functions with m ≥ 1 for (x, t) ∈ Q_T and u, p, r, q ∈ R^m. The existence and the estimates of solutions for the finite difference system are established by the fixed point technique. The absolute and relative stability and convergence of difference schemes are justified by means of a series of a priori estimates. In the present study, the existence of unique smooth solution of the original problem is assumed. The similar results for nonlinear and quasilinear parabolic systems are also obtained.  相似文献   

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