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1.
New variational principles based on the concept of anti-selfdual (ASD) Lagrangians were recently introduced in “AIHP-Analyse
non linéaire, 2006”. We continue here the program of using such Lagrangians to provide variational formulations and resolutions
to various basic equations and evolutions which do not normally fit in the Euler-Lagrange framework. In particular, we consider
stationary boundary value problems of the form as well ass dissipative initial value evolutions of the form where is a convex potential on an infinite dimensional space, A is a linear operator and is any scalar. The framework developed in the above mentioned paper reformulates these problems as and respectively, where is an “ASD” vector field derived from a suitable Lagrangian L. In this paper, we extend the domain of application of this approach by establishing existence and regularity results under
much less restrictive boundedness conditions on the anti-selfdual Lagrangian L so as to cover equations involving unbounded operators. Our main applications deal with various nonlinear boundary value
problems and parabolic initial value equations governed by transport operators with or without a diffusion term.
Nassif Ghoussoub research was partially supported by a grant from the Natural Sciences and Engineering Research Council of
Canada. The author gratefully acknowledges the hospitality and support of the Centre de Recherches Mathématiques in Montréal
where this work was initiated.
Leo Tzou’s research was partially supported by a doctoral postgraduate scholarship from the Natural Science and Engineering
Research Council of Canada. 相似文献
2.
We consider a class of elliptic operators
with unbounded coefficients in a smooth exterior domain Ω and we prove that the Cauchy-Neumann problem associated with
admits, for any bounded and continuous initial datum, a unique bounded classical solution. We also provide pointwise gradient
estimates for such a solution.
Received: 5 July 2005; Revised: 20 December 2005 相似文献
3.
Maurizio Grasselli Giulio Schimperna Antonio Segatti Sergey Zelik 《Journal of Evolution Equations》2009,9(2):371-404
We study the modified Cahn–Hilliard equation proposed by Galenko et al. in order to account for rapid spinodal decomposition
in certain glasses. This equation contains, as additional term, the second-order time derivative of the (relative) concentration
multiplied by a (small) positive coefficient . Thus, in absence of viscosity effects, we are in presence of a Petrovsky type equation and the solutions do not regularize
in finite time. Many results are known in one spatial dimension. However, even in two spatial dimensions, the problem of finding
a unique solution satisfying given initial and boundary conditions is far from being trivial. A fairly complete analysis of
the 2D case has been recently carried out by Grasselli, Schimperna and Zelik. The 3D case is still rather poorly understood
but for the existence of energy bounded solutions. Taking advantage of this fact, Segatti has investigated the asymptotic
behavior of a generalized dynamical system which can be associated with the equation. Here we take a step further by establishing
the existence and uniqueness of a global weak solution, provided that is small enough. More precisely, we show that there exists such that well-posedness holds if (suitable) norms of the initial data are bounded by a positive function of which goes to + ∞ as tends to 0. This result allows us to construct a semigroup on an appropriate (bounded) phase space and, besides, to prove the existence of a global attractor. Finally, we show a regularity
result for the attractor by using a decomposition method and we discuss the existence of an exponential attractor.
相似文献
4.
We study the self-adjoint and dissipative realization A of a second order elliptic differential operator
with unbounded regular coefficients in
, where μ(dx) = ρ (x)dx is the associated invariant measure. We prove a maximal regularity result under suitable assumptions, that generalize the
well known conditions in the case of constant diffusion part.
Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday 相似文献
5.
We establish existence and pointwise estimates of fundamental solutions and Green’s matrices for divergence form, second order strongly elliptic systems in a domain $\Omega \subseteq {\mathbb{R}}^n, n \geq 3We establish existence and pointwise estimates of fundamental solutions and Green’s matrices for divergence form, second order
strongly elliptic systems in a domain , under the assumption that solutions of the system satisfy De Giorgi-Nash type local H?lder continuity estimates. In particular,
our results apply to perturbations of diagonal systems, and thus especially to complex perturbations of a single real equation. 相似文献
6.
Francesca Alessio Piero Montecchiari 《Calculus of Variations and Partial Differential Equations》2007,30(1):51-83
We consider a class of semilinear elliptic equations of the form
where is a periodic, positive function and is modeled on the classical two well Ginzburg-Landau potential . We show, via variational methods, that if the set of solutions to the one dimensional heteroclinic problem
has a discrete structure, then (0.1) has infinitely many solutions periodic in the variable y and verifying the asymptotic conditions as uniformly with respect to .
Supported by MURST Project ‘Metodi Variazionali ed Equazioni Differenziali Non Lineari’. 相似文献
7.
Kin Ming Hui 《Mathematische Annalen》2007,339(2):395-443
We prove the existence of a unique solution of the following Neumann problem , u > 0, in (a, b) × (0, T), u(x, 0) = u
0(x) ≥ 0 in (a, b), and , where if m < 0, if m = 0, and
m≤ 0, , and the case −1 < m ≤ 0, , for some constant p > 1 − m. We also obtain a similar result in higher dimensions. As a corollary we will give a new proof of a result of A. Rodriguez
and J.L. Vazquez on the existence of infinitely many finite mass solutions of the above equation in for any −1 < m ≤ 0. We also obtain the exact decay rate of the solution at infinity. 相似文献
8.
Given asymptotics types P, Q, pseudodifferential operators
are constructed in such a way that if u(t) possesses conormal asymptotics
of type P as t +0, then Au(t) possesses conormal asymptotics of
type Q as t +0. This is achieved by choosing the operators A in Schulzes
cone algebra on the half-line
, controlling their complete Mellin symbols
{
},
and prescribing the mapping properties of the residual
Green operators. The constructions lead to a coordinate invariant calculus,
including trace and potential operators at t = 0, in which a parametrix construction
for the elliptic elements is possible. Boutet de Monvels calculus for
pseudodifferential boundary problems occurs as a special case when P = Q is
the type resulting from Taylor expansion at t = 0. 相似文献
9.
N. I. Karachalios N. B. Zographopoulos 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2005,2(3):11-30
We study a real Ginzburg-Landau equation, in a bounded domain of
\mathbbRN ,\mathbb{R}^N , with a variable, generally non-smooth diffusion coefficient having a finite number of zeroes. By using the compactness of the embeddings of the weighted Sobolev spaces involved in the functional formulation of the problem, and the associated energy equation, we show the existence of a global attractor. The extension of the main result in the case of an unbounded domain is also discussed, where in addition, the diffusion coefficient has to be unbounded. Some remarks for the case of a complex Ginzburg-Landau equation are given. 相似文献
10.
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
11.
We study the semiflow defined by a semilinear parabolic equation with a singular square potential . It is known that the Hardy-Poincaré inequality and its improved versions, have a prominent role on the definition of the
natural phase space. Our study concerns the case 0 < μ ≤ μ*, where μ* is the optimal constant for the Hardy-Poincaré inequality. On a bounded domain of , we justify the global bifurcation of nontrivial equilibrium solutions for a reaction term f(s) = λs − |s|2γ
s, with λ as a bifurcation parameter. We remark some qualitative differences of the branches in the subcritical case μ < μ* and the critical case μ = μ*. The global bifurcation result is used to show that any solution , initiating form initial data tends to the unique nonnegative equilibrium. 相似文献
12.
Yonggeun Cho Tohru Ozawa Yong-Sun Shim 《Calculus of Variations and Partial Differential Equations》2009,34(3):321-339
In this paper, we consider elliptic estimates for a system with smooth variable coefficients on a domain containing the origin. We first show the invariance of the estimates under a domain expansion defined by the scale that with parameter R > 1, provided that the coefficients are in a homogeneous Sobolev space. Then we apply these invariant estimates to the global
existence of unique strong solutions to a parabolic system defined on an unbounded domain.
This paper was supported in part by research funds of Chonbuk National University in 2007. 相似文献
13.
The purpose of the paper is to study properties of solutions of the Cauchy problem for the equation
under the assumption
.
General selfsimilar solutions are constructed. Moreover, for initial data with some decay at infinity, we determine
the leading term of the asymptotics of solutions in
which is described by either solutions of the linear heat equation or by particular selfsimilar solutions of the original equation. 相似文献
14.
Generalizing previous results of M. Comte and P. Mironescu, it
is shown that for degree d large enough
(such that
), there
is a bifurcation branch in the set of the solutions of the Ginzburg-Landau
equation, emanating from the branch of radial solutions at the critical value
d of the parameter. Moreover, the solutions on the bifurcation branch admit
exactly d zeroes, and the energy on the bifurcation branch is strictly smaller
than the energy on the radial branch. 相似文献
15.
T. Godoy U. Kaufmann 《NoDEA : Nonlinear Differential Equations and Applications》2007,14(3-4):443-453
Let be a smooth bounded domain, let a, b be two functions that are possibly discontinuous and unbounded with a ≥ 0 in and b > 0 in a set of positive measure and let 0 < p < 1 < q. We prove that there exists some 0 < Λ < ∞ such that the nonlinear Dirichlet periodic parabolic problem in has a positive solution for all 0 < λ < Λ and that there is no positive solution if λ > Λ. In some cases we also show the
existence of a minimal solution for all 0 < λ < Λ and that the solution u
λ can be chosen such that λ → u
λ is differentiable and increasing. We also give some upper and lower estimates for such a Λ. All results remain true for the
analogous elliptic problems.
Partially supported by CONICET, Secyt-UNC, ANPCYT and Agencia Cordoba Ciencia 相似文献
16.
P. Quittner W. Reichel 《Calculus of Variations and Partial Differential Equations》2008,32(4):429-452
Consider the equation −Δu = 0 in a bounded smooth domain , complemented by the nonlinear Neumann boundary condition ∂ν
u = f(x, u) − u on ∂Ω. We show that any very weak solution of this problem belongs to L
∞(Ω) provided f satisfies the growth condition |f(x, s)| ≤ C(1 + |s|
p
) for some p ∈ (1, p*), where . If, in addition, f(x, s) ≥ −C + λs for some λ > 1, then all positive very weak solutions are uniformly a priori bounded. We also show by means of examples that
p* is a sharp critical exponent. In particular, using variational methods we prove the following multiplicity result: if N ∈ {3, 4} and f(x, s) = s
p
then there exists a domain Ω and such that our problem possesses at least two positive, unbounded, very weak solutions blowing up at a prescribed point of
∂Ω provided . Our regularity results and a priori bounds for positive very weak solutions remain true if the right-hand side in the differential
equation is of the form h(x, u) with h satisfying suitable growth conditions. 相似文献
17.
Pavel Drábek Peter Takáč 《Calculus of Variations and Partial Differential Equations》2007,29(1):31-58
An improved Poincaré inequality and validity of the Palais-Smale condition are investigated for the energy functional on , 1 < p < ∞, where Ω is a bounded domain in , is a spectral (control) parameter, and is a given function, in Ω. Analysis is focused on the case λ = λ1, where −λ1 is the first eigenvalue of the Dirichlet p-Laplacian Δ
p
on , λ1 > 0, and on the “quadratization” of within an arbitrarily small cone in around the axis spanned by , where stands for the first eigenfunction of Δ
p
associated with −λ1. 相似文献
18.
Marc Briane Juan Casado–Díaz 《Calculus of Variations and Partial Differential Equations》2008,33(4):463-492
In this paper we study the limit, in the sense of the Γ-convergence, of sequences of two-dimensional energies of the type
, where A
n
is a symmetric positive definite matrix-valued function and μ
n
is a nonnegative Borel measure (which can take infinite values on compact sets). Under the sole equicoerciveness of A
n
we prove that the limit energy belongs to the same class, i.e. its reads as , where is a diffusion independent of μ
n
and μ is a nonnegative Borel measure which does depend on . This compactness result extends in dimension two the ones of [11,23] in which A
n
is assumed to be uniformly bounded. It is also based on the compactness result of [7] obtained for sequences of two-dimensional
diffusions (without zero-order term). Our result does not hold in dimension three or greater, since nonlocal effects may appear.
However, restricting ourselves to three-dimensional diffusions with matrix-valued functions only depending on two coordinates,
the previous two-dimensional result provides a new approach of the nonlocal effects. So, in the periodic case we obtain an
explicit formula for the limit energy specifying the kernel of the nonlocal term. 相似文献
19.
Thomas Bartsch Shuangjie Peng Zhitao Zhang 《Calculus of Variations and Partial Differential Equations》2007,30(1):113-136
We investigate elliptic equations related to the Caffarelli–Kohn–Nirenberg inequalities: and such that . For various parameters α, β and various domains Ω, we establish some existence and non-existence results of solutions in
rather general, possibly degenerate or singular settings. 相似文献
20.
Adimurthi K. Sandeep 《NoDEA : Nonlinear Differential Equations and Applications》2007,13(5-6):585-603
Let Ω be a bounded domain in
, we prove the singular Moser-Trudinger embedding:
if and only if
where
and
. We will also study the corresponding critical exponent problem. 相似文献