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1.
Luis F. Ragognette 《Mathematische Nachrichten》2019,292(2):409-427
In this work we present a new representation formula for ultradistributions using the so‐called ultradifferential operators. The main difference between our representation result and other works is that here we do not break the duality of Gevrey functions of other s and their ultradistributions, i.e., we locally represent an element of by an infinite order operator acting on a function of class . Our main application was in the local solvability of the differential complex associated to a locally integrable structure in a Gevrey environment. 相似文献
2.
José R. dos Santos Filho Maurício Fronza da Silva 《Journal of Differential Equations》2009,247(10):2688-2704
F. Treves, in [17], using a notion of convexity of sets with respect to operators due to B. Malgrange and a theorem of C. Harvey, characterized globally solvable linear partial differential operators on C∞(X), for an open subset X of Rn.Let P=L+c be a linear partial differential operator with real coefficients on a C∞ manifold X, where L is a vector field and c is a function. If L has no critical points, J. Duistermaat and L. Hörmander, in [2], proved five equivalent conditions for global solvability of P on C∞(X).Based on Harvey-Treves's result we prove sufficient conditions for the global solvability of P on C∞(X), in the spirit of geometrical Duistermaat-Hörmander's characterizations, when L is zero at precisely one point. For this case, additional non-resonance type conditions on the value of c at the equilibrium point are necessary. 相似文献
3.
4.
Dietmar Vogt 《Archiv der Mathematik》2006,87(2):163-171
It is shown that for open convex
, d > 1 and a nontrivial polynomial P the space
does not have property
. If P is elliptic or homogeneous, then this holds for every open Ω. For
even
cannot occur and if it occurs for some Ω, then P must be hypoelliptic.
Received: 18 July 2005 相似文献
5.
A.P. Bergamasco P.L. Dattori da Silva R.B. Gonzalez 《Journal of Differential Equations》2018,264(5):3500-3526
Let be a vector field defined on the torus , where , are real-valued functions and belonging to the Gevrey class , , for . We present a complete characterization for the s-global solvability and s-global hypoellipticity of L. Our results are linked to Diophantine properties of the coefficients and, also, connectedness of certain sublevel sets. 相似文献
6.
Adalberto P. Bergamasco Paulo L. Dattori da Silva Marcelo R. Ebert 《Journal of Differential Equations》2009,246(4):1673-1702
We study the Gevrey solvability of a class of complex vector fields, defined on Ω?=(−?,?)×S1, given by L=∂/∂t+(a(x)+ib(x))∂/∂x, b?0, near the characteristic set Σ={0}×S1. We show that the interplay between the order of vanishing of the functions a and b at x=0 plays a role in the Gevrey solvability. 相似文献
7.
Ognjen Milatovic 《Differential Geometry and its Applications》2004,21(3):361-377
We consider a family of Schrödinger-type differential expressions L(κ)=D2+V+κV(1), where κ∈C, and D is the Dirac operator associated with a Clifford bundle (E,∇E) of bounded geometry over a manifold of bounded geometry (M,g) with metric g, and V and V(1) are self-adjoint locally integrable sections of EndE. We also consider the family I(κ)=*(∇F)∇F+V+κV(1), where κ∈C, and ∇F is a Hermitian connection on a Hermitian vector bundle F of bonded geometry over a manifold of bounded geometry (M,g), and V and V(1) are self-adjoint locally integrable sections of EndF. We give sufficient conditions for L(κ) and I(κ) to have a realization in L2(E) and L2(F), respectively, as self-adjoint holomorphic families of type (B). In the proofs we use Kato's inequality for Bochner Laplacian operator and Weitzenböck formula. 相似文献
8.
We prove that a first-order linear differential operator G with unbounded operator coefficients is Fredholm on spaces of functions on with values in a reflexive Banach space if and only if the corresponding strongly continuous evolution family has exponential dichotomies on both and and a pair of the ranges of the dichotomy projections is Fredholm, and that the Fredholm index of G is equal to the Fredholm index of the pair. The operator G is the generator of the evolution semigroup associated with the evolution family. In the case when the evolution family is the propagator of a well-posed differential equation u′(t)=A(t)u(t) with, generally, unbounded operators , the operator G is a closure of the operator . Thus, this paper provides a complete infinite-dimensional generalization of well-known finite-dimensional results by Palmer, and by Ben-Artzi and Gohberg. 相似文献
9.
Inspired by the work of Suzuki [12] on the concept of real-analytic solvability for first-order analytic linear partial differential operators we extend his results for the differential complexes associated to analytic locally integrable structures of corank one. We prove that such notion of solvability is related to the smooth solvability condition introduced by F. Treves [13] in 1983. In our arguments the natural extension to closed forms of the well-known Baouendi–Treves approximation formula, the so-called “Approximate Poincaré Lemma” (cf. [1], [14]), plays a key role. 相似文献
10.
Abdelhamid Meziani 《Journal of Differential Equations》2011,251(10):2896-2931
This paper deals with the global solvability of a complex vector field with real analytic coefficients in two real variables. The vector field is assumed to satisfy the Nirenberg-Treves condition (P) for local solvability. Normal forms for the vector field near the one-dimensional orbits are obtained and a generalization of the Riemann-Hilbert problem is considered. 相似文献
11.
12.
Gabriele Bonanno Giovanni Molica Bisci 《Nonlinear Analysis: Theory, Methods & Applications》2011,74(14):4785-4795
The aim of this paper is to establish a multiplicity result for an eigenvalue non-homogeneous Neumann problem which involves a nonlinearity fulfilling a nonstandard growth condition. Precisely, a recent critical points result for differentiable functionals is exploited in order to prove the existence of a determined open interval of positive eigenvalues for which the problem admits at least three weak solutions in an appropriate Orlicz-Sobolev space. 相似文献
13.
Zdzisław Brzeźniak 《Potential Analysis》1995,4(1):1-45
We study abstract stochastic evolution equations in M-type 2 Banach spaces. Applications to stochastic partial differential equations inL
p
spaces withp2 are given. For example, solutions of such equations are Hölder continuous in the space variables.The author is an Alexander von Humboldt Stiftung fellow 相似文献
14.
Results of Hörmander on evolution operators together with a characterization of the present authors [Ann. Inst. Fourier, Grenoble 40, 619–655 (1990)] are used to prove the following: Let P ∈ ?[z1,...,z n ] and denote by P m its principal part. If P ? Pm is dominated by P m then the following assertions for the partial differential operators P(D) and P m(D) are equivalent for N ∈ S n?1:
- P(D) and/or Pm D)admit a continuous linear right inverse on C ∞(H +(N)).
- P(D) admits a continuous linear right inverse on C ∞ (? n ) and a fundamental solution E ∈ C ∞(?n) satisfying Supp $E \subset \overline {H - (N)} $
15.
We investigate the splitting of short exact sequences of the form
0→X→Y→E→0, 相似文献
16.
M. Thamban Nair 《Integral Equations and Operator Theory》2002,44(1):79-92
A class of regularization methods using unbounded regularizing operators is considered for obtaining stable approximate solutions for ill-posed operator equations. With an a posteriori as well as an a priori parameter choice strategy, it is shown that the method yields the optimal order. Error estimates have also been obtained under stronger assumptions on the generalized solution. The results of the paper unify and simplify many of the results available in the literature. For example, the optimal results of the paper include, as particular cases for Tikhonov regularization, the main result of Mair (1994) with an a priori parameter choice, and a result of Nair (1999) with an a posteriori parameter choice. Thus the observations of Mair (1994) on Tikhonov regularization of ill-posed problems involving finitely and infinitely smoothing operators is applicable to various other regularization procedures as well. Subsequent results on error estimates include, as special cases, an optimal result of Vainikko (1987) and also some recent results of Tautenhahn (1996) in the setting of Hilbert scales. 相似文献
17.
Eduardo V. Teixeira 《Journal of Differential Equations》2005,214(1):65-91
Let (E,F) be a locally convex space. We denote the bounded elements of E by . In this paper, we prove that if BEb is relatively compact with respect to the F topology and f:I×Eb→Eb is a measurable family of F-continuous maps then for each x0∈Eb there exists a norm-differentiable, (i.e. differentiable with respect to the ∥·∥F norm) local solution to the initial valued problem ut(t)=f(t,u(t)), u(t0)=x0. All of this machinery is developed to study the Lipschitz stability of a nonlinear differential equation involving the Hardy-Littlewood maximal operator. 相似文献
18.
Wend Werner 《Integral Equations and Operator Theory》1992,15(3):496-502
We determine the smooth points of certain spaces of bounded operatorsL(X,Y), including the cases whereX andY arel
p
-orc
0-direct sums of finite dimensional Banach spaces or subspaces of the latter enjoying the metric compact approximation property. We also remark that the operators not attaining their norm are nowhere dense inL(X,Y) wheneverK(X,Y) is anM-ideal inL(X,Y). 相似文献
19.
We study the boundary value problem in Ω, u = 0 on ∂Ω, where Ω is a bounded domain in with smooth boundary, λ is a positive real number, and the continuous functions p
1, p
2, and q satisfy 1 < p
2(x) < q(x) < p
1(x) < N and for any . The main result of this paper establishes the existence of two positive constants λ0 and λ1 with λ0 ≤ λ1 such that any is an eigenvalue, while any is not an eigenvalue of the above problem. 相似文献
20.
Andreas Weber 《Archiv der Mathematik》2008,90(3):279-283
We prove that the L
p
spectrum of a Riemannian product M
1×M
2 coincides with the set theoretic sum of the L
p
spectra of M
1
and M
2
.
Received: 13 June 2007 相似文献