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1.
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
2.
Wei-Ming Ni 《Journal of Differential Equations》1983,50(2):289-304
In this paper, we consider the uniqueness of radial solutions of the nonlinear Dirichlet problem satisfies some appropriate conditions and Ω is a bounded smooth domain in n which possesses radial symmetry. Our uniqueness results apply to, for instance, , or more generally λu + ∑i = 1kaiupi, λ ? 0, ai > 0 and pi > 1 with appropriate upper bounds, and Ω a ball or an annulus. 相似文献
3.
J.H Michael 《Journal of Mathematical Analysis and Applications》1981,79(1):203-217
We consider the mixed boundary value problem , where Ω is a bounded open subset of n whose boundary Γ is divided into disjoint open subsets Γ+ and Γ? by an (n ? 2)-dimensional manifold ω in Γ. We assume A is a properly elliptic second order partial differential operator on and Bj, for j = 0, 1, is a normal jth order boundary operator satisfying the complementing condition with respect to A on . The coefficients of the operators and Γ+, Γ? and ω are all assumed arbitrarily smooth. As announced in [Bull. Amer. Math. Soc.83 (1977), 391–393] we obtain necessary and sufficient conditions in terms of the coefficients of the operators for the mixed boundary value problem to be well posed in Sobolev spaces. In fact, we construct an open subset of the reals such that, if then for is a Fredholm operator if and only if s ∈ . Moreover, = ?xewx, where the sets x are determined algebraically by the coefficients of the operators at x. If n = 2, x is the set of all reals not congruent (modulo 1) to some exceptional value; if n = 3, x is either an open interval of length 1 or is empty; and finally, if n ? 4, x is an open interval of length 1. 相似文献
4.
This paper deals with asymptotic behavior for (weak) solutions of the equation , on + × Ω; u(t, x) = 0, on + × ?Ω. If and β is coercive, we prove that the solutions are bounded in the energy space, under weaker assumptions than those used by G. Prouse in a previous work. If in addition and ? is srongly almost-periodic, we prove for strongly monotone β that all solutions are asymptotically almost-periodic in the energy space. The assumptions made on β are much less restrictive than those made by G. Prouse: mainly, we allow β to be multivalued, and in the one-dimensional case β need not be defined everywhere. 相似文献
5.
6.
M.G. Crandall S.-O. Londen J.A. Nohel 《Journal of Mathematical Analysis and Applications》1978,64(3):701-735
We study the nonlinear Volterra equation , (1) as well as the corresponding problem with infinite delay . (7) Under various assumptions on the nonlinear operators A, B and on the given functions a, F, f, h existence theorems are obtained for (1) and (7, followed by results concerning boundedness and asymptotic behaviour of solutions on (0 ? < ∞); two applications of the theory to problems of nonlinear heat flow with “infinite memory” are also discussed. 相似文献
7.
The usual Sobolev inequality in n, n ? 3, asserts that , with Sn being the sharp constant. This paper is concerned, instead, with functions restricted to bounded domains Ω ? n. Two kinds of inequalities are established: (i) If ? = 0 on ?Ω, then with and with . (ii) If ? ≠ 0 on ?Ω, then with . Some further results and open problems in this area are also presented. 相似文献
8.
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
9.
Pedro M Girão 《Comptes Rendus Mathematique》2002,334(2):105-108
Let N?5, a>0, be a smooth bounded domain in , , and 6u62=|?u|22+a|u|22. We prove there exists an α0>0 such that, for all , This inequality implies Cherrier's inequality. To cite this article: P.M. Girão, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 105–108 相似文献
10.
Let be a Dirichlet form in , where Ω is an open subset of n, n ? 2, and m a Radon measure on Ω; for each integer k with 1 ? k < n, let k be a Dirichlet form on some k-dimensional submanifold of Ω. The paper is devoted to the study of the closability of the forms E with domain and defined by: ki where 1 ? kp < ? < n, and where , gki denote restrictions of ?, g in to . Conditions are given for E to be closable if, for each i = 1,…, p, one has ki = n ? i. Other conditions are given for E to be nonclosable if, for some i, ki < n ? i. 相似文献
11.
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
12.
Abstract connections between integral kernels of positivity preserving semigroups and suitable Lp contractivity properties are established. Then these questions are studied for the semigroups generated by , the Dirichlet Laplacian for an open, connected region Ω. As an application under a suitable hypothesis, Sobolev estimates are proved valid up to ?Ω, of the form , where ?0 is the unique positive L2 eigenfunction of . 相似文献
13.
Richard Askey Deborah Tepper Haimo 《Journal of Mathematical Analysis and Applications》1977,59(1):119-129
We study degeneration for ? → + 0 of the two-point boundary value problems , and convergence of the operators T?+ and T?? on 2(?1, 1) connected with them, T?±u := τ?±u for all for all . Here ? is a small positive parameter, λ a complex “spectral” parameter; a, b and c are real ∞-functions, a(x) ? γ > 0 for all x? [?1, 1] and h is a sufficiently smooth complex function. We prove that the limits of the eigenvalues of T?+ and of T?? are the negative and nonpositive integers respectively by comparison of the general case to the special case in which a 1 and b c 0 and in which we can compute the limits exactly. We show that (T?+ ? λ)?1 converges for ? → +0 strongly to (T0+ ? λ)?1 if . In an analogous way, we define the operator T?+, n (n ? in the Sobolev space H0?n(? 1, 1) as a restriction of τ?+ and prove strong convergence of (T+?,n ? λ)?1 for ? → +0 in this space of distributions if . With aid of the maximum principle we infer from this that, if h?1, the solution of τ?+u ? λu = h, u(±1) = A ± B converges for ? → +0 uniformly on [?1, ? ?] ∪ [?, 1] to the solution of xu′ ? λu = h, u(±1) = A ± B for each p > 0 and for each λ ? if ? ?.Finally we prove by duality that the solution of τ??u ? λu = h converges to a definite solution of the reduced equation uniformly on each compact subset of (?1, 0) ∪ (0, 1) if h is sufficiently smooth and if 1 ? ?. 相似文献
14.
Jeng-Eng Lin 《Journal of Functional Analysis》1979,31(3):321-332
Consider a smooth solution of and is C1, and 1 < p < 5. Assume that the initial data decay sufficiently rapidly at infinity, , and for simplicity, qr ? 0. Then the local energy decays faster than exponentially. 相似文献
15.
Abbas Bahri 《Journal of Functional Analysis》1981,41(3):397-427
We study in this paper problems of the type , Ω bounded ? N, , (I) where is given and where . Our main result is that (I) has an infinite number of solutions for a residual set of ? in . In particular, for many n ∈ there exists an open and dense subset of ? in such that (I) has n distinct solutions for such an ? This result is to be related to the conjecture developed in [1] of the existence of an infinite number of solutions to (I). The proof relies on a general characterization of level sets for a certain class of functionals, when there are no critical value in a large enough interval. In addition to the study of problem (I), we apply this characterization to give another proof (using, e.g., Brouwer's fixed point theorem) for some classical results about even functionals and saddle points. 相似文献
16.
Derek W Robinson 《Journal of Functional Analysis》1977,24(3):280-290
Let U, V be two strongly continuous one-parameter groups of bounded operators on a Banach space with corresponding infinitesimal generators S, T. We prove the following: ∥Ut, ? Vt ∥ = O(t), t → 0, if and only if U = V; ∥Ut ? Vt∥ = O(tα), t → 0; with 0 ? α ? 1, if and only if , where Ω, P, are bounded operators on such that if and only if has a bounded extension to 1. Further results of this nature are inferred for semigroups, reflexive spaces, Hilbert spaces, and von Neumann algebras. 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.
19.
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
20.
A continuity theorem for an operator of the form when ? is not a continuous function is proven. 相似文献