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1.
Ovidiu Savin 《Calculus of Variations and Partial Differential Equations》2005,22(3):303-320
We consider the following obstacle problem for Monge-Ampere equation
and discuss the regularity of the free boundary
. We prove that
is
if f is bounded away from 0 and
, and it is C
1,1 if
.Received: 4 February 2003, Accepted: 3 March 2004, Published online: 16 July 2004 相似文献
2.
Yavdat Il’yasov Thomas Runst 《Calculus of Variations and Partial Differential Equations》2005,22(1):101-127
This paper concerns with a family of inhomogeneous Neumann boundary value problems having indefinite nonlinearities which depend on a real parameter
. We discuss the existence and the multiplicity of positive solutions with respect to
. Developing the fibering method further, we can introduce a constructive concept of the calculation of certain nonlocal intervals
, the so-called sufficient intervals of the existence. Then we are able to prove some new results on the existence and the multiplicity of positive solutions
for
.Received: 22 December 2003, Accepted: 29 January 2004, Published online: 16 July 2004Mathematics Subject Classification (2000):
35J70, 35J65, 47H17 相似文献
3.
Manuel del Pino Michal Kowalczyk Monica Musso 《Calculus of Variations and Partial Differential Equations》2005,24(1):47-81
We consider the boundary value problem
in a bounded, smooth domain
in
with homogeneous Dirichlet boundary conditions. Here
0,k(x)
$$
" align="middle" border="0">
is a non-negative, not identically zero function. We find conditions under which there exists a solution
which blows up at exactly m points as
and satisfies
. In particular, we find that if
,
0
$" align="middle" border="0">
and
is not simply connected then such a solution exists for any given
Received: 11 February 2004, Accepted: 17 August 2004, Published online: 22 December 2004 相似文献
4.
Daniele Castorina Filomena Pacella 《Calculus of Variations and Partial Differential Equations》2005,23(2):125-138
We consider the subcritical problem
& 0 & \qquad\textrm{in} \; A\\
u & = & 0 & \qquad\textrm{on} \; \delta A\\
\end{array}
\right.$$" align="middle" border="0">
where A is an annulus in
,
,
is the critical Sobolev exponent and
0$" align="middle" border="0">
is a small parameter. We prove that solutions of (I) which concentrate at one or two points are axially symmetric.Received: 7 July 2003, Accepted: 10 May 2004, Published online: 16 July 2004Filomena Pacella: Research supported by MIUR, project Variational Methods and Nonlinear Differential Equations. 相似文献
5.
Given an almost complex structure J in a cylinder of
(p > 1) together with a compatible symplectic form
and given an arbitrary J-holomorphic curve
without boundary in that cylinder, we construct an holomorphic perturbation of
, for the canonical complex structure J
0 of
, such that the distance between these two curves in W
1,2 and
norms, in a sub-cylinder, are controled by quantities depending on J,
and by the area of
only. These estimates depend neither on the topology nor on the conformal class of
. They are key tools in the recent proof of the regularity of 1-1 integral currents in [RT].Received: 2 October 2003, Accepted: 18 November 2003, Published online: 25 February 2004 相似文献
6.
This is a follow-up of a paper of Bourgain, Brezis and Mironescu [2]. We study how the existence of the limit
for
continuous and
converging to
, is related to the weak regularity of
. This approach gives an alternative way of defining the Sobolev spaces W
1,p
. We also briefly discuss the
-convergence of (1) with respect to the
-topology.Received: 12 November 2002, Accepted: 7 January 2003, Published online: 22 September 2003Mathematics Subject Classification (2000):
46E35, 49J45Augusto C. Ponce: ponce@ann.jussieu.fr 相似文献
7.
We present a characterization of ideal knots, i.e., of closed knotted curves of prescribed thickness with minimal length, where we use the notion of global curvature for the definition of thickness. We show with variational methods that for an ideal knot
, the normal vector
at a curve point
is given by the integral over all vectors
against a Radon measure, where
realizes the given thickness. As geometric consequences we obtain in particular, that points without contact lie on straight segments of
, and for points
with exactly one contact point
we have that
points exactly into the direction of
Moreover, isolated contact points lie on straight segments of
, and curved arcs of
consist of contact points only, all realizing the prescribed thickness with constant (maximal) global curvature.Received: 1 January 2003, Accepted: 12 March 2003, Published online: 1 July 2003Mathematics Subject Classification (2000):
53A04, 57M25, 74K05, 74M15, 92C40 相似文献
8.
K. J. Brown 《Calculus of Variations and Partial Differential Equations》2004,22(4):483-494
The Nehari manifold for the equation
for
together with Dirichlet boundary conditions is investigated in the case where
. Exploiting the relationship between the Nehari manifold and fibrering maps (i.e., maps of the form
where J is the Euler functional associated with the equation), we discuss how the Nehari manifold changes as
changes and show how this is linked to results on bifurcation from infinity which are associated with the problem.Received: 8 December 2003, Accepted: 10 May 2004, Published online: 16 July 2004Mathematics Subject Classification (2000):
35J20, 35J25 相似文献
9.
Angela Pistoia Olivier Rey 《Calculus of Variations and Partial Differential Equations》2003,18(3):243-251
Let
,
, be a bounded domain as defined by Flucher, Garroni and Müller [6], which has a singular point
such that the Robins function achieves its infimum at
. Considering the elliptic problem
in
; u = 0 on
, with p = (N + 2)/(N-2),
, and
a minimizing solution of
,
concentrates at
as
goes to zero.Received: 15 September 2002, Accepted: 5 November 2002, Published online: 16 May 2003Mathematics Subject Classification:
35J65Angela Pistoia: The author is supported by M.U.R.S.T., project Metodi variazionali e topologici nello studio di fenomeni non lineari 相似文献
10.
P. Courilleau J. Mossino 《Calculus of Variations and Partial Differential Equations》2004,20(1):65-91
We study the limit behaviour of some nonlinear monotone equations, such as:
, in a domain
which is thin in some directions (e.g.
is a plate or a thin cylinder). After rescaling to a fixed domain
, the above equation is transformed into:
, with convenient operators
and
. Assuming that
and the inverse of
have particular forms and satisfy suitable compensated compactness assumptions, we prove a closure result, that is we prove that the limit problem has the same form. This applies in particular to the limit behaviour of nonlinear monotone equations in laminated plates.Received: 16 October 2002, Accepted: 12 June 2003, Published online: 22 September 2003Mathematics Subject Classification (2000):
35B27, 35B40, 74Q15 相似文献
11.
We study the regularizing effect of perimeter penalties for a problem of optimal compliance in two dimensions. In particular, we consider minimizers of
where
The sets
,
, and the force f are given. We show that if we consider only scalar valued u and constant
, or if we consider the elastic energy
, then
is
away from where
is pinned. In the scalar case, we also show that, for any
of class
,
is
. The proofs rely on a notion of weak outward curvature of
, which we can bound without considering properties of the minimizing fields, together with a bootstrap argument.Received: 5 March 2002, Accepted: 3 September 2002, Published online: 17 December 2002 相似文献
12.
Nicola Visciglia 《Calculus of Variations and Partial Differential Equations》2005,24(2):167-184
We present a generalized version of the Hardy-Sobolev inequality, in which the homogeneous potential
is replaced by any potential V belonging to the Lorentz space
. We show that the best constant in these inequalities is achieved provided that
where
. We also analyze the limit case
. Finally an application to a non-linear eigenvalues problem with rough potentials is presented.Received: 19 September 2004, Accepted: 15 November 2004, Published online: 22 December 2004 相似文献
13.
B. Abdellaoui E. Colorado I. Peral 《Calculus of Variations and Partial Differential Equations》2005,23(3):327-345
For 1 < p < N and
we obtain the following improved Hardy-Sobolev Inequalities
where 1 < q < p and
if
,
if
, for some positive constant
.
Also we give an alternative proof of the optimal improved inequality for p = 2 by Wang-Willem in [16].
Received: 2 February 2004, Accepted: 12 July 2004, Published online: 3 September 2004
Mathematics Subject Classification (2000):
35J20, 35P05, 35R05, 46E30, 46E35
Partially supported by Project BFM2001-0183 相似文献
14.
Relaxation problems for a functional of the type
are analyzed, where
is a bounded smooth open subset of
and g is a Carathéodory function. The admissible functions u are forced to satisfy a pointwise gradient constraint of the type
for a.e.
being, for every
, a bounded convex subset of
, in general varying with x not necessarily in a smooth way. The relaxed functionals
and
of G obtained letting u vary respectively in
, the set of the piecewise C
1-functions in
, and in
in the definition of G are considered. For both of them integral representation results are proved, with an explicit representation formula for the density of
. Examples are proposed showing that in general the two densities are different, and that the one of
is not obtained from g simply by convexification arguments. Eventually, the results are discussed in the framework of Lavrentieff phenomenon, showing by means of an example that deep differences occur between
and
. Results in more general settings are also obtained.Received: 18 December 2002, Accepted: 18 November 2003, Published online: 16 July 2004Mathematics Subject Classification (2000):
49J45, 49J10, 49J53This work is part of the European Research Training Network Homogenization and Multiple Scales (HMS 2000), under contract HPRN-2000-00109. It is also part of the 2003-G.N.A.M.P.A. Project Metodi Variazionali per Strutture Sottili, Frontiere Oscillanti ed Energie Vincolate. 相似文献
15.
We establish existence of nodal solutions to the pure critical exponent problem
in
u = 0 on
where
a bounded smooth domain which is invariant under an orthogonal involution of
We extend previous results for positive solutions due to Coron, Dancer, Ding, and Passaseo to existence and multiplicity results for solutions which change sign exactly once.Received: 4 April 2003, Accepted: 26 August 2003, Published online: 24 November 2003Mathematics Subject Classification (2000):
35J65, 35J20Research partially supported by PAPIIT, UNAM, under grant IN110902-3. 相似文献
16.
G. Cerami G. Devillanova S. Solimini 《Calculus of Variations and Partial Differential Equations》2005,23(2):139-168
In this paper we consider the problem
in
, where p > 2 and
if N > 2. Assuming that the potential a(x) is a regular function such that
0$" align="middle" border="0">
and that verifies suitable decay assumptions, but not requiring any symmetry property on it, we prove that the problem has infinitely many solutions.Received: 3 December 2003, Accepted: 10 May 2004, Published online: 22 December 2004 相似文献
17.
Finn and Kosmodemyanskii, Jr. gave an example of a domain
containing a disk
, and of a family of domains
converging to
as
, such that the heights u
t
of capillary surfaces in vertical tubes with the sections
in a gravity field g satisfy
for every
, but for which u
1< u
0 over
for all g > 0. In subsequent work, Finn and Lee characterized the most general convex
that leads to such a discontinuous transition when
is a disk. It has been suggested that the cause for this curious behavior is related to the fact that in all cases considered the boundaries of the
have a discontinuity in their curvatures, that is bounded below in magnitude. In the present note we present an alternative form of the example, in which the domains
are disks concentric to
. Thus, the limited smoothness in the original example of the convergence to
of the approxim
ating domains cannot be viewed as the root cause of the anomaly. The procedure presented here leads to explicit bounds, which were not available in the earlier forms of the example.Received: 3 September 2002, Accepted: 17 February 2003, Published online: 1 July 2003Mathematics Subject Classification:
76B45, 53A10, 49Q10 相似文献
18.
Marc Oliver Rieger Paolo Tilli 《Calculus of Variations and Partial Differential Equations》2005,23(4):373-390
We study by means of
-convergence the asymptotics of the rescaled Mumford-Shah functional
when
and prove the existence of a
-limit. The limit functional is easy to handle and can be used as a simple approximation to the original Mumford-Shah functional. Moreover, its minimizers can be interpreted as a sort of asymptotic probability distribution of the sets
. Some examples illustrate the use of this method in image segmentation.Received: 12 June 2004, Accepted: 12 July 2004, Published online: 10 December 2004 相似文献
19.
Ulrich Dierkes Dirk Schwab 《Calculus of Variations and Partial Differential Equations》2005,22(2):173-184
We construct quadratic forms on
which are subharmonic on any n-dimensional minimal submanifold in
and, more generally, on submanifolds of bounded mean curvature. This leads to nonexistence results for connected n-dimensional minimal submanifolds in
as well as to necessary conditions for the existence of connected submanifolds of bounded mean curvature with arbitrary codimension. Furthermore we discuss a barrier principle for n-dimensional submanifolds in
of bounded mean curvature.Received: 11 November 2003, Accepted: 29 January 2004, Published online: 2 April 2004Mathematics Subject Classification (2000):
35J60, 49Q05, 53C42 相似文献
20.
A Sobolev inequality with remainder term and critical equations on domains with topology for the polyharmonic operator 总被引:1,自引:0,他引:1
Thomas Bartsch Tobias Weth Michel Willem 《Calculus of Variations and Partial Differential Equations》2003,18(3):253-268
We prove a Sobolev inequality with remainder term for the imbedding
,
arbitrary, generalizing a corresponding result of Bianchi and Egnell for the case m = 1. We also show that the manifold of least energy solutions
of the equation
is a nondegenerate critical manifold for the corresponding variational integral. Finally we generalize the results of J. M. Coron on the existence of solutions of equations with critical exponent on domains with nontrivial topology to the biharmonic operator.Received: 21 March 2002, Accepted: 5 November 2002, Published online: 16 May 2003 相似文献