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1.
《Nonlinear Analysis: Theory, Methods & Applications》2004,57(3):349-362
We show that in a smooth bounded domain , n⩾2, all global nonnegative solutions of ut−Δum=up with zero boundary data are uniformly bounded in by a constant depending on and τ but not on u0, provided that 1<m<p<[(n+1)/(n−1)]m. Furthermore, we prove an a priori bound in depending on under the optimal condition 1<m<p<[(n+2)/(n−2)]m. 相似文献
2.
《Nonlinear Analysis: Theory, Methods & Applications》2004,56(2):185-199
In this paper we are concerned with positive solutions of the doubly nonlinear parabolic equation ut=div(um−1|∇u|p−2∇u)+Vum+p−2 in a cylinder , with initial condition u(·,0)=u0(·)⩾0 and vanishing on the parabolic boundary . Here (resp. ) is a bounded domain with smooth boundary, , , 1<p<N and m+p−2>0. The critical exponents are found and the nonexistence results are proved for . 相似文献
3.
In this paper we study the existence, uniqueness, and regularity of the solutions for the Cauchy problem for the evolution equation is in (0, 1), 0 ? t ? T, T is an arbitrary positive real number,f(s)?C1, and g(x, t)?L∞(0, T; L2(0, 1)). We prove the existence and uniqueness of the weak solutions for (1) using the Galerkin method and a compactness argument such as that of J. L. Lions. We obtain regular solutions using eigenfunctions of the one-dimensional Laplace operator as a basis in the Galerkin method. 相似文献
4.
The main result is the following. Let be a bounded Lipschitz domain in , d?2. Then for every with ∫f=0, there exists a solution of the equation divu=f in , satisfying in addition u=0 on and the estimate where C depends only on . However one cannot choose u depending linearly on f. To cite this article: J. Bourgain, H. Brezis, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 973–976. 相似文献
5.
Tomas Schonbek 《Journal of Differential Equations》1985,56(2):290-296
New and more elementary proofs are given of two results due to W. Littman: (1) Let . The estimate cannot hold for all u?C0∞(Q), Q a cube in , some constant C. (2) Let n ? 2, p ≠ 2. The estimate cannot hold for all C∞ solutions of the wave equation □u = 0 in ; all t ?; some function C: → . 相似文献
6.
Hermann König 《Journal of Functional Analysis》1977,24(1):32-51
For an open set Ω ? N, 1 ? p ? ∞ and λ ∈ +, let denote the Sobolev-Slobodetzkij space obtained by completing in the usual Sobolev-Slobodetzkij norm (cf. A. Pietsch, “r-nukleare Sobol. Einbett. Oper., Ellipt. Dgln. II,” Akademie-Verlag, Berlin, 1971, pp. 203–215). Choose a Banach ideal of operators , 1 ? p, q ? ∞ and a quasibounded domain Ω ? N. Theorem 1 of the note gives sufficient conditions on λ such that the Sobolev-imbedding map exists and belongs to the given Banach ideal : Assume the quasibounded domain fulfills condition Ckl for some l > 0 and 1 ? k ? N. Roughly this means that the distance of any to the boundary ?Ω tends to zero as for , and that the boundary consists of sufficiently smooth ?(N ? k)-dimensional manifolds. Take, furthermore, 1 ? p, q ? ∞, p > k. Then, if μ, ν are real positive numbers with λ = μ + v ∈ , μ > λ S(; p,q:N) and v > N/l · λD(;p,q), one has that belongs to the Banach ideal . Here λD(;p,q;N)∈+ and λS(;p,q;N)∈+ are the D-limit order and S-limit order of the ideal , introduced by Pietsch in the above mentioned paper. These limit orders may be computed by estimating the ideal norms of the identity mappings lpn → lqn for n → ∞. Theorem 1 in this way generalizes results of R. A. Adams and C. Clark for the ideals of compact resp. Hilbert-Schmidt operators (p = q = 2) as well as results on imbeddings over bounded domains.Similar results over general unbounded domains are indicated for weighted Sobolev spaces.As an application, in Theorem 2 an estimate is given for the rate of growth of the eigenvalues of formally selfadjoint, uniformly strongly elliptic differential operators with Dirichlet boundary conditions in , where Ω fulfills condition C1l.For an open set Ω in N, let denote the Sobolev-Slobodetzkij space obtained by completing in the usual Sobolev-Slobodetzkij norm, see below. Taking a fixed Banach ideal of operators and 1 ? p, q ? ∞, we consider quasibounded domains Ω in N and give sufficient conditions on λ such that the Sobolev imbedding operator exists and belongs to the Banach ideal. This generalizes results of C. Clark and R. A. Adams for compact, respectively, Hilbert-Schmidt operators (p = q = 2) to general Banach ideals of operators, as well as results on imbeddings over bounded domains. Similar results over general unbounded domains may be proved for weighted Sobolev spaces. As an application, we give an estimate for the rate of growth of the eigenvalues of formally selfadjoint, uniformly strongly elliptic differential operators with Dirichlet boundary conditions in , where Ω is a quasibounded open set in N. 相似文献
7.
Thierry Cazenave 《Journal of Functional Analysis》1985,60(1):36-55
Uniform estimates in of global solutions to nonlinear Klein-Gordon equations of the form , where Ω is an open subset of N, m > 0, and g satisfies some growth conditions are established. 相似文献
8.
Alan McIntosh 《Journal of Functional Analysis》1985,61(3):307-327
Consider an elliptic sesquilinear form defined on × by , where is a closed subspace of which contains , Ω is a bounded Lipschitz domain in n, for all ζ?n with ¦ζ¦ = 1. Let L be the operator with largest domain satisfying J[u, v] = (Lu, v) for all υ∈. Then L + λI is a maximal accretive operator in for λ a sufficiently large real number. It is proved that is a bounded operator from to provided mild regularity of the coefficients is assumed. In addition it is shown that if the coefficients depend differentiably on a parameter t in an appropriate sense, then the corresponding square root operators also depend differentiably on t. The latter result is new even when the forms J are hermitian. 相似文献
9.
We consider the problem of minimizing integral functionals of the form , where Ω ?p, u:ω → and ▽[k]u denotes the set of all partial derivatives of u with orders ?k. The method is based on a characterization of null Lagrangians L(▽ku) depending only on derivatives of order k. Applications to elasticity and other theories of mechanics are given. 相似文献
10.
Variational problems for the multiple integral , where and are studied. A new condition on g, called W1,p-quasiconvexity is introduced which generalizes in a natural way the quasiconvexity condition of C. B. Morrey, it being shown in particular to be necessary for sequential weak lower semicontinuity of in and for the existence of minimizers for certain related integrals. Counterexamples are given concerning the weak continuity properties of Jacobians in , p ? n = m. An existence theorem for nonlinear elastostatics is proved under optimal growth hypotheses. 相似文献
11.
We study the bifurcation problem ?Δu=g(u)+λ|?u|2+μ in on , where λ,μ?0 and is a smooth bounded domain in . The singular character of the problem is given by the nonlinearity g which is assumed to be decreasing and unbounded around the origin. In this Note we prove that the above problem has a positive classical solution (which is unique) if and only if λ(a+μ)<λ1, where a=limt→+∞g(t) and λ1 is the first eigenvalue of the Laplace operator in . We also describe the decay rate of this solution, as well as a blow-up result around the bifurcation parameter. To cite this article: M. Ghergu, V. R?dulescu, C. R. Acad. Sci. Paris, Ser. I 338 (2004). 相似文献
12.
Let (Wt) = (W1t,W2t,…,Wdt), d ? 2, be a d-dimensional standard Brownian motion and let A(t) be a bounded measurable function from + into the space of d × d skew-symmetric matrices and x(t) such a function into d. A class of stochastic processes (LtA,x), a particular example of which is Levy's “stochastic area” , is dealt with.The joint characteristic function of Wt and L1A,x is calculated and based on this result a formula for fundamental solutions for the hypoelliptic operators which generate the diffusions (Wt,LtA,x) is given. 相似文献
13.
Nonlinear partial differential operators having the form G(u) = g(u, D1u,…, DNu), with g?C(R × RN), are here shown to be precisely those operators which are local, (locally) uniformly continuous on, , and (roughly speaking) translation invariant. It is also shown that all such partial differential operators are necessarily bounded and continuous with respect to the norm topologies of . 相似文献
14.
Bent Fuglede 《Journal of Functional Analysis》1974,16(1):101-121
In Rn let Ω denote a Nikodym region (= a connected open set on which every distribution of finite Dirichlet integral is itself in . The existence of n commuting self-adjoint operators such that each Hj is a restriction of (acting in the distribution sense) is shown to be equivalent to the existence of a set Λ ?Rn such that the restrictions to Ω of the functions exp i ∑ λjxj form a total orthogonal family in . If it is required, in addition, that the unitary groups generated by H1,…, Hn act multiplicatively on , then this is shown to correspond to the requirement that Λ can be chosen as a subgroup of the additive group Rn. The measurable sets Ω ?Rn (of finite Lebesgue measure) for which there exists a subgroup Λ ?Rn as stated are precisely those measurable sets which (after a correction by a null set) form a system of representatives for the quotient of Rn by some subgroup Γ (essentially the dual of Λ). 相似文献
15.
Norbert Wielens 《Journal of Functional Analysis》1985,61(1):98-115
A necessary and sufficient condition is given for the generalized Schrödinger operator to be essentially self-adjoint in , under general assumptions on ? and for arbitrary domains Ω in n. In particular, if ? is strictly positive and locally Lipschitz continuous on , then A is essentially. self-adjoint. Examples of non-essential self-adjointness and a complete discussion of the one-dimensional case are also given. These results have applications to the problem of the essential self-adjointness of quantum Hamiltonians and to the uniqueness problem of Markov processes. 相似文献
16.
Zeev Schuss 《Journal of Mathematical Analysis and Applications》1977,59(2):227-241
Let A and B be uniformly elliptic operators of orders 2m and 2n, respectively, m > n. We consider the Dirichlet problems for the equations (?2(m ? n)A + B + λ2nI)u? = f and (B + λ2nI)u = f in a bounded domain Ω in Rk with a smooth boundary ?Ω. The estimate is derived. This result extends the results of [7, 9, 10, 12, 14, 15, 18]by giving estimates up to the boundary, improving the rate of convergence in ?, using lower norms, and considering operators of higher order with variable coefficients. An application to a parabolic boundary value problem is given. 相似文献
17.
M.P Heble 《Journal of Mathematical Analysis and Applications》1983,93(2):363-384
Given a cocycle a(t) of a unitary group {U1}, ?∞ < t < ∞, on a Hilbert space , such that a(t) is of bounded variation on [O, T] for every T > O, a(t) is decomposed as a(t) = f;t0Usxds + β(t) for a unique x ? , β(t) yielding a vector measure singular with respect to Lebesgue measure. The variance is defined as if existing. For a stationary diffusion process on 1, with Ω1, the space of paths which are natural extensions backwards in time, of paths confined to one nonsingular interval J of positive recurrent type, an information function I(ω) is defined on , based on the paths restricted to the time interval [0, 1]. It is shown that is continuous and bounded on . The shift τt, defines a unitary representation {Ut}. Assuming , dm being the stationary measure defined by the transition probabilities and the invariant measure on J, has a C∞ spectral density function f;. It is then shown that σ2({Ut}, I) = f;(O). 相似文献
18.
Hans G Kaper 《Journal of Mathematical Analysis and Applications》1978,63(3):691-718
In this article we discuss the solution of boundary value problems which are described by the linear integrodifferential equation , where t∈J?, x∈. We interpret the equation in functional form as an ordinary differential equation for the mapping u:J→L2(R,μ), where L2(R,μ) is a weighted L2-space. Emphasis is on the constructive aspects of the solution and on finding representations of the relevant isomorphisms. 相似文献
19.
Chan-nan Chang 《Journal of Number Theory》1973,5(6):456-476
Let L be a lattice over the integers of a quaternion algebra with center K which is a -adic field. Then the unitary group U(L) equals its own commutator subgroup and is generated by the unitary transvections and quasitransvections contained in it. Let g be a tableau, U(g), U+(g), , T(g) be the corresponding congruence subgroups of order g. Then , and (the subgroup generated by the unitary transvections and quasitransvections with order ≤ g). Let G be a subgroup of U(L) with o(G) = g, then G is normal in U(L) if and only if U(g) ? G ? T(g). 相似文献
20.
C.O Horgan 《Journal of Mathematical Analysis and Applications》1979,69(1):231-242
We consider inequalities of the form (1) for sufficiently regular functions u(x) defined on a bounded domain Ω in Rn. The inequality (1) follows from the Trace Theorem in interpolation spaces and so is called a trace inequality. Information on the optimal constants C (which depend on the domain geometry) is obtained through consideration of associated eigenvalue problems. 相似文献