共查询到20条相似文献,搜索用时 31 毫秒
1.
Pavol Quittner Frédérique Simondon 《Journal of Mathematical Analysis and Applications》2005,304(2):614-631
We study a priori estimates of positive solutions of the equation t∂u−Δu=λu+a(x)up, x∈Ω, t>0, satisfying the homogeneous Dirichlet boundary conditions. Here Ω is a bounded domain in Rn, λ∈R, p>1 is subcritical, changes sign and a,p satisfy some additional technical hypotheses. Assume that the solution u blows up in a finite time T and the set is connected. Using our a priori bounds, we show that u blows up completely in Ω+ at t=T and the blow-up time T depends continuously on the initial data. 相似文献
2.
Lihe Wang 《Journal of Mathematical Analysis and Applications》2011,380(1):10-16
We consider the following free boundary problem in an unbounded domain Ω in two dimensions: Δpu=0 in Ω, on J0, on J1, where ∂Ω=J0∪J1. We prove that if 0<u<1 in Ω, Ji is the graph of a function in and gi is a constant for each i=0,1, then the free boundary ∂Ω must be two parallel straight lines and the solution u must be a linear function. The proof is based on maximum principle. 相似文献
3.
Let X1,X2,…,Xq be a system of real smooth vector fields satisfying Hörmander's rank condition in a bounded domain Ω of Rn. Let be a symmetric, uniformly positive definite matrix of real functions defined in a domain U⊂R×Ω. For operators of kind
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In this paper, we consider the semilinear elliptic problem in Ω, u=0 on ∂Ω, where Ω is a smooth bounded domain in RN, N?4, , is the critical Sobolev exponent, K(x) is a continuous function. When Ω and K(x) are invariant under a group of orthogonal transformations, we prove the existence of nodal and positive solutions for 0<λ<λ1, where λ1 is the first Dirichlet eigenvalue of on Ω. 相似文献
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We study the existence and uniqueness of the mixed boundary value problem for Laplace equation in a bounded Lipschitz domain Ω⊂Rn, n?3. Let the boundary ∂Ω of Ω be decomposed by , Γ1∩Γ2=∅. We will show that if the Neumann data ψ is in and the Dirichlet data f is in , then the mixed boundary value problem has a unique solution and the solution is represented by potentials. 相似文献
8.
Song Liang 《Journal of Functional Analysis》2004,216(1):71-85
We consider uniformly elliptic diffusion processes X(t,x) on Euclidean spaces , with some conditions in terms of the drift term (see assumptions A2 and A3). By using interpolation theory, we show a bounded property which gives an estimate of involving |x| and but not ||∇f||∞, and a power of smaller than 1. 相似文献
9.
Giuseppe Da Prato 《Journal of Differential Equations》2004,198(1):35-52
We study the realization AN of the operator in L2(Ω,μ) with Neumann boundary condition, where Ω is a possibly unbounded convex open set in , U is a convex unbounded function, DU(x) is the element with minimal norm in the subdifferential of U at x, and is a probability measure, infinitesimally invariant for . We show that AN is a dissipative self-adjoint operator in L2(Ω,μ). Log-Sobolev and Poincaré inequalities allow then to study smoothing properties and asymptotic behavior of the semigroup generated by AN. 相似文献
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We consider the elliptic equation -Δu+u=0 in a bounded, smooth domain Ω in R2 subject to the nonlinear Neumann boundary condition . Here ?>0 is a small parameter. We prove that any family of solutions u? for which ?∫∂Ωeu is bounded, develops up to subsequences a finite number m of peaks ξi∈∂Ω, in the sense that as ?→0. Reciprocally, we establish that at least two such families indeed exist for any given m?1. 相似文献
12.
Zhijun Zhang 《Journal of Mathematical Analysis and Applications》2005,308(2):532-540
By constructing the comparison functions and the perturbed method, it is showed that any solution u∈C2(Ω) to the semilinear elliptic problems Δu=k(x)g(u), x∈Ω, u|∂Ω=+∞ satisfies , where Ω is a bounded domain with smooth boundary in RN; , −2<σ, c0>0, ; g∈C1[0,∞), g?0 and is increasing on (0,∞), there exists ρ>0 such that , ∀ξ>0, , . 相似文献
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Tomasz Piasecki 《Journal of Differential Equations》2010,248(8):2171-2198
We investigate a steady flow of a viscous compressible fluid with inflow boundary condition on the density and inhomogeneous slip boundary conditions on the velocity in a cylindrical domain Ω=Ω0×(0,L)∈R3. We show existence of a solution , p>3, where v is the velocity of the fluid and ρ is the density, that is a small perturbation of a constant flow (, ). We also show that this solution is unique in a class of small perturbations of . The term u⋅∇w in the continuity equation makes it impossible to show the existence applying directly a fixed point method. Thus in order to show existence of the solution we construct a sequence (vn,ρn) that is bounded in and satisfies the Cauchy condition in a larger space L∞(0,L;L2(Ω0)) what enables us to deduce that the weak limit of a subsequence of (vn,ρn) is in fact a strong solution to our problem. 相似文献
15.
Nicu?or Costea 《Nonlinear Analysis: Theory, Methods & Applications》2009,71(9):4271-4278
We study the problem in Ω, u=0 on ∂Ω, where Ω is a bounded domain in RN, is a continuous function and λ and ε are two positive constants. We prove that for any ε>0 each λ∈(0,λ1) is an eigenvalue of the above problem, where λ1 is the principal eigenvalue of the Laplace operator on Ω. Moreover, for each eigenvalue λ∈(0,λ1) it corresponds a unique eigenfunction. The proofs will be based on the Banach fixed point theorem combined with adequate variational techniques. 相似文献
16.
A. Zaghdani 《Applied mathematics and computation》2010,217(5):1791-1810
We present two new coupling models for the three dimensional magnetostatic problem. In the first model, we propose a new coupled formulation, prove that it is well posed and solves Maxwell’s equations in the whole space. In the second, we propose a new coupled formulation for the Local Discontinuous Galerkin method, the finite element method and the boundary element method. This formulation is obtained by coupling the LDG method inside a bounded domain Ω1 with the FEM method inside a layer where Ω is a bounded domain which is made up of material of permeability μ and such that , and with a boundary element method involving Calderon’s equations. We prove that our formulation is consistent and well posed and we present some a priori error estimates for the method. 相似文献
17.
We consider the semilinear elliptic problem in Ω, u=0 on ∂Ω, where 0∈Ω is a smooth bounded domain in RN, N?4, , is the critical Sobolev exponent, f(x,⋅) has subcritical growth at infinity, K(x)>0 is continuous. We prove the existence of sign-changing solutions under different assumptions when Ω is a usual domain and a symmetric domain, respectively. 相似文献
18.
Let Ω⊂RN be an open set and F a relatively closed subset of Ω. We show that if the (N−1)-dimensional Hausdorff measure of F is finite, then the spaces and coincide, that is, F is a removable singularity for . Here is the closure of in H1(Ω) and H1(Ω) denotes the first order Sobolev space. We also give a relative capacity criterium for this removability. The space is important for defining realizations of the Laplacian with Neumann and with Robin boundary conditions. For example, if the boundary of Ω has finite (N−1)-dimensional Hausdorff measure, then our results show that we may replace Ω by the better set (which is regular in topology), i.e., Neumann boundary conditions (respectively Robin boundary conditions) on Ω and on coincide. 相似文献
19.
Xianling Fan 《Journal of Mathematical Analysis and Applications》2009,349(2):436-442
Consider the eigenvalue problem : −Δu=λf(x,u) in Ω, u=0 on ∂Ω, where Ω is a bounded smooth domain in RN. Denote by the set of all Carathéodory functions f:Ω×R→R such that for a.e. x∈Ω, f(x,⋅) is Lipschitzian with Lipschitz constant L, f(x,0)=0 and , and denote by (resp. ) the set of λ>0 such that has at least one nonzero classical (resp. weak) solution. Let λ1 be the first eigenvalue for the Laplacian-Dirichlet problem. We prove that and . Our result is a positive answer to Ricceri's conjecture if use f(x,u) instead of f(u) in the conjecture. 相似文献