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1.
We introduce the concept of Lp-maximal regularity for second order Cauchy problems. We prove Lp-maximal regularity for an abstract model problem and we apply the abstract results to prove existence, uniqueness and regularity
of solutions for nonlinear wave equations.
The author acknowledges with thanks the support provided by the Department ofApplied Analysis, University of Ulm, and the
travel grants provided by NBMH India and MSF Delhi, India. 相似文献
2.
We investigate R-bounded representations
, where X is a Banach space and G is a lca group. Observing that Ψ induces a (strongly continuous) group homomorphism
, we are then able to analyze certain classical homomorphisms U (e.g. translations in Lp (G)) from the viewpoint of R-boundedness and the theory of scalar-type spectral operators.
Dedicated to the memory of H. H. Schaefer 相似文献
3.
4.
We study Lp-viscosity solutions of fully nonlinear, second-order, uniformly elliptic partial differential equations (PDE) with measurable terms and quadratic nonlinearity. We present a sufficient condition under which the maximum principle holds for Lp-viscosity solution. We also prove stability and existence results for the equations under consideration. 相似文献
5.
Given
, a compact abelian group G and a function
, we identify the maximal (i.e. optimal) domain of the convolution
operator
(as an operator from Lp(G) to itself). This is the
largest Banach function space (with order continuous norm) into which Lp(G)
is embedded and to which
has a continuous extension, still with values
in Lp(G). Of course, the optimal domain depends on p and g. Whereas
is compact, this is not always so for the extension of
to its optimal domain.
Several characterizations of precisely when this is the case are presented. 相似文献
6.
Let X be a Banach space and let
A be a closed linear operator on
X. It is shown that the abstract Cauchy problem
enjoys maximal regularity in weighted
L
p
-spaces with weights
, where
,
if and only if it has the property of maximal
L
p
-regularity.
Moreover, it is also shown that the derivation operator
admits an
-calculus in weighted
L
p
-spaces.
Received: 26 February 2003 相似文献
7.
We consider the mixed problem,
in a class of Lipschitz graph domains in two dimensions with Lipschitz constant at most 1. We suppose the Dirichlet data,
f
D
, has one derivative in L
p
(D) of the boundary and the Neumann data, f
N
, is in L
p
(N). We find a p
0 > 1 so that for p in an interval (1, p
0), we may find a unique solution to the mixed problem and the gradient of the solution lies in L
p
.
L. Lanzani, L. Capogna and R. M. Brown were supported, in part, by the U.S. National Science Foundation. 相似文献
8.
We prove that for a fixed integer s2 every K
s,s
-free graph of average degree at least r contains a K
p
minor where
. A well-known conjecture on the existence of dense K
s,s
-free graphs would imply that the value of the exponent is best possible. Our result implies Hadwigers conjecture for K
s,s
-free graphs whose chromatic number is sufficiently large compared with s. 相似文献
9.
We discuss the analytic properties of curves γ whose global curvature function ρ
G
[γ]−1 is p-integrable. It turns out that the L
p
-norm is an appropriate model for a self-avoidance energy interpolating between “soft” knot energies in form of singular repulsive
potentials and “hard” self-obstacles, such as a lower bound on the global radius of curvature introduced by Gonzalez and Maddocks.
We show in particular that for all p > 1 finite -energy is necessary and sufficient for W
2,p
-regularity and embeddedness of the curve. Moreover, compactness and lower-semicontinuity theorems lead to the existence of
-minimizing curves in given isotopy classes. There are obvious extensions to other variational problems for curves and nonlinearly
elastic rods, where one can introduce a bound on to preclude self-intersections. 相似文献
10.
A weighted norm inequality for the Marcinkiewicz integral operator
is proved when belongs to
. We also give the weighted
Lp-boundedness for a class of Marcinkiewicz integral operators with rough
kernels
and
related to the Littlewood-Paley
-function and the
area integral S, respectively. 相似文献
11.
We prove that for every graph H and for every s there exists d=d(H,s) such that every graph of average degree at least d contains either a Ks,s as a subgraph or an induced subdivision of H. 相似文献
12.
Haïm Brezis Jean Van Schaftingen 《Calculus of Variations and Partial Differential Equations》2007,30(3):369-388
We obtain boundary estimates for the gradient of solutions to elliptic systems with Dirichlet or Neumann boundary conditions
and L
1–data, under some condition on the divergence of the data. Similar boundary estimates are obtained for div–curl and Hodge
systems. 相似文献
13.
In this paper we investigate the spectral exponent, i.e. logarithm of the spectral radius of operators having the form
and acting in spaces Lp(X, μ), where X is a compact topological space, φk∈C(X), φ = (φk)k=1N∈C(X)N, and
are linear positive operators (Ukf≥ 0 for f≥ 0). We consider the spectral exponent ln r(Aφ) as a functional depending on vector-function φ. We prove that ln r(Aφ) is continuous and on a certain subspace
of C(X)N is also convex. This yields that the spectral exponent is the Fenchel-Legendre transform of a convex functional
defined on a set
of continuous linear positive and normalized functionals on the subspace
of coefficients φ that is
相似文献
14.
Sun-Sig Byun 《Journal of Evolution Equations》2007,7(3):415-428
This work treats Lp regularity theory for weak solutions of parabolic equations in divergence form with discontinuous coefficients on nonsmooth
domains. We essentially obtain an optimal condition on the coefficients under which the global W1,p regularity theory holds.
This work was supported by SNU foundation in 2005. 相似文献
15.
Let be a positive C
0-semigroup on L
p
(Ω), with infinitesimal
generator A. In this paper, it is proved that if there exists a such that and ,
where A
*
is the adjoint of A, then the growth bound of T(t) is upper bounded
by b when p = 1, and by when 1 lt; p lt; α and c D(A), where .
This is an operator version of a classical stability result
on Z-matrix. As application examples, some new results on the asymptotic
behaviours of population system and neutron transport system are obtained.
Submitted: March 1, 2001?Revised: August 28, 2002 相似文献
16.
In this paper we prove Lp estimates (p≥2) for the uniform norm of the paths of solutions of quasilinear stochastic partial differential equations (SPDE) of parabolic
type. Our method is based on a version of Moser's iteration scheme developed by Aronson and Serrin in the context of non-linear
parabolic PDE. 相似文献
17.
Robert J. Taggart 《Mathematische Zeitschrift》2009,261(4):933-949
Suppose that {T
t
: t ≥ 0} is a symmetric diffusion semigroup on L
2(X) and denote by its tensor product extension to the Bochner space , where belongs to a certain broad class of UMD spaces. We prove a vector-valued version of the Hopf–Dunford–Schwartz ergodic theorem
and show that this extends to a maximal theorem for analytic continuations of on . As an application, we show that such continuations exhibit pointwise convergence. 相似文献
18.
We show that any pointwise multiplier for BMO(ℝn) generates a function p from the class (ℝn) of those functions for which the Hardy-Littlewood maximal operator is bounded on the variable Lp space. In particular, this gives a positive answer to Diening's conjecture saying that there are discontinuous functions
which nevertheless belong to (ℝn). 相似文献
19.
The C
*-algebra
generated by the n poly-Bergman and m antipoly-Bergman projections and by the operators of multiplication by piecewise continuous functions on the Lebesgue space
L
2(Π) over the upper half-plane is studied. Making use of a local principle, limit operators techniques, and the Plamenevsky
results on two-dimensional convolution operators with symbols admitting homogeneous discontinuities we reduce the study to
simpler C
*-algebras associated with points
and pairs
. Applying a symbol calculus for the abstract unital C
*-algebras generated by N orthogonal projections sum of which equals the unit and by M = n + m one-dimensional orthogonal projections and using relations for the Gauss hypergeometric function, we study local algebras
at points
being the discontinuity points of coefficients. A symbol calculus for the C
*-algebra
is constructed and a Fredholm criterion for the operators
is obtained. 相似文献
20.
Li Fenggao 《高校应用数学学报(英文版)》2003,18(2):223-229
Let AG(n, F
q) be the n-dimensional affine space over F
q, where F
q is a finite field with q elements. Denote by Γ
(m) the graph induced by m-flats of AG(n, F
q). For any two adjacent vertices E and F of
is studied. In particular, sizes of maximal cliques in Γ
(m) are determined and it is shown that Γ
(m) is not edge-regular when m<n−1.
Supported by the National Natural Science Foundation of China (19571024) and Hunan Provincial Department of Education (02C512). 相似文献