共查询到20条相似文献，搜索用时 515 毫秒
1.
Abdelmajid Siai 《Potential Analysis》2006,24(1):15-45
Let Ω be an open bounded set in ℝN, N≥3, with connected Lipschitz boundary ∂Ω and let a(x,ξ) be an operator of Leray–Lions type (a(⋅,∇u) is of the same type as the operator |∇u|p−2∇u, 1<p<N). If τ is the trace operator on ∂Ω, [φ] the jump across ∂Ω of a function φ defined on both sides of ∂Ω, the normal derivative
∂/∂νa related to the operator a is defined in some sense as 〈a(⋅,∇u),ν〉, the inner product in ℝN, of the trace of a(⋅,∇u) on ∂Ω with the outward normal vector field ν on ∂Ω. If β and γ are two nondecreasing continuous real functions everywhere
defined in ℝ, with β(0)=γ(0)=0, f∈L1(ℝN), g∈L1(∂Ω), we prove the existence and the uniqueness of an entropy solution u for the following problem,
in the sense that, if Tk(r)=max {−k,min (r,k)}, k>0, r∈ℝ, ∇u is the gradient by means of truncation (∇u=DTku on the set {|u|<k}) and
, u measurable; DTk(u)∈Lp(ℝN), k>0}, then
and u satisfies,
for every k>0 and every
.
Mathematics Subject Classifications (2000) 35J65, 35J70, 47J05. 相似文献
2.
Givenμ, κ, c>0, we consider the functional
defined on allR
n
-valued functionsu on the open subset Ω ofR
n
which are smooth outside a free discontinuity setS
u, on which the tracesu
+,u
− on both sides have equal normal component (i.e.,u has a tangential jump alongS
u).E
Du=Eu − 1/3 (divu)I, withEu denoting the linearized strain tensor.
The functionalF is obtained from the usual strain energy of linearized elasticity by addition of a term (the second integral) which penalizes
the jump discontin uities of the displacement.
The lower semicontinuous envelope
is studied, with respect to theL
1 (Ω;R
n
)-topology, on the spaceP(Ω) of the functions of bounded deformation with distributional divergence inL
2(Ω) (F is extended with value +∞ on the wholeP(Ω)). The following integral representation is proved:
whereϕ is a convex function with linear growth at infinity. NowEu is a measure,ɛ
Du represents the density of the absolutely continuous part of the absolutely continuous part ofE
Du, whileE
s
D
u denotes the singular part and ϕ∞ the recession function ofϕ.
Finally, we show that
coincides with the functional which intervenes in the minimum problem for the displacement in the theory of Hencky’s plasticity
with Tresca’s yield conditions. 相似文献
3.
Laura DeMarco 《Mathematische Annalen》2003,326(1):43-73
Let L(f)=∫log∥Df∥dμ
f
denote the Lyapunov exponent of a rational map, f:P
1→P
1
. In this paper, we show that for any holomorphic family of rational maps {f
λ
:λX} of degree d>1, T(f)=dd
c
L(f
λ
) defines a natural, positive (1,1)-current on X supported exactly on the bifurcation locus of the family. The proof is based on the following potential-theoretic formula
for the Lyapunov exponent:
Here F:C
2
→C
2
is a homogeneous polynomial lift of f; ; G
F
is the escape rate function of F; and capK
F
is the homogeneous capacity of the filled Julia set of F. We show, in particular, that the capacity of K
F
is given explicitly by the formula
where Res(F) is the resultant of the polynomial coordinate functions of F.
We introduce the homogeneous capacity of compact, circled and pseudoconvex sets K⊂C
2 and show that the Levi measure (determined by the geometry of ∂K) is the unique equilibrium measure. Such K⊂C
2 correspond to metrics of non-negative curvature on P
1, and we obtain a variational characterization of curvature.
Received: 28 November 2001 / Revised version: 2 April 2002 /
Published online: 10 February 2003 相似文献
4.
Tetsutaro Shibata 《Journal d'Analyse Mathématique》2007,102(1):347-358
We study the nonlinear Sturm-Liouville problem
where λ > 0 is an eigenvalue parameter and f(u) is a rapidly increasing function. For better understanding of the global behavior of the bifurcation branch in R+ × L
2(I), we establish precise asymptotic formulas up to the third term for the eigenvalue λ(α) associated with the eigenfunction
u
α with ‖u
α‖2 = α, as α → ∞. We show that there exists a new type of asymptotic formula for λ (α) as α → ∞. 相似文献
5.
Abstract. Let Ω and Π be two simply connected domains in the complex plane C which are not equal to the whole plane C and let λ
Ω
and λ
Π
denote the densities of the Poincare metric in Ω and Π , respectively. For f: Ω → Π analytic in Ω , inequalities of the type
are considered where M
n
(z,Ω, Π) does not depend on f and represents the smallest value possible at this place. We prove that
if Δ is the unit disk and Π is a convex domain. This generalizes a result of St. Ruscheweyh.
Furthermore, we show that
holds for arbitrary simply connected domains whereas the inequality 2
n-1
≤ C
n
(Ω,Π) is proved only under some technical restrictions upon Ω and Π . 相似文献
6.
Francesca Prinari 《Applied Mathematics and Optimization》2008,58(1):111-145
We study the weak* lower semicontinuity properties of functionals of the form
where Ω is a bounded open set of R
N
and u∈W
1,∞(Ω). Without a continuity assumption on f(⋅,ξ) we show that the supremal functional F is weakly* lower semicontinuous if and only if it is a level convex functional (i.e. it has convex sub-levels). In particular if F is weakly* lower semicontinuous, then it can be represented through a level convex function. Finally a counterexample shows that in
general it is not possible to represent F through the level convex envelope of f. 相似文献
7.
Aleksandar Ivić 《Archiv der Mathematik》2008,90(5):412-419
If
denotes the error term in the classical Rankin-Selberg problem, then it is proved that
where Δ1(x) = ∫
x
0 Δ(u)du. The latter bound is, up to ‘ɛ’, best possible.
Received: 8 February 2007 相似文献
8.
Michela Eleuteri 《Applications of Mathematics》2007,52(2):137-170
We prove some optimal regularity results for minimizers of the integral functional ∫ f(x, u, Du) dx belonging to the class K ≔ {u ∈ W
1,p
(Ω): u ⩾ ψ, where ψ is a fixed function, under standard growth conditions of p-type, i.e.
.
This research has been supported by INdAM.
On leave from: Dipartimento di Matematica, Universitá di Trento, via Sommarive 14, 38050 Povo (Trento), Italy, e-mail: eleuteri@science.unitn.it. 相似文献
9.
LetA be the class of normalized analytic functions in the unit disk Δ and define the class
For a functionf εA the Alexander transformF
0 is given by
Our main object is to establish a sharp relation betweenβ andγ such thatf εP
β implies thatF
0 is starlike of orderγ, 0 ≤γ ≤ 1/2. A corresponding result for the Libera transformF
1(z) = 2∫
0
1
f(tz)dt is also given. 相似文献
10.
D. F. Miller 《Journal of Optimization Theory and Applications》2007,134(3):413-432
This paper develops boundary integral representation formulas for the second variations of cost functionals for elliptic domain
optimization problems. From the collection of all Lipschitz domains Ω which satisfy a constraint ∫
Ω
g(x) dx=1, a domain is sought which maximizes either
, fixed x
0∈Ω, or ℱ(Ω)=∫
Ω
F(x,u(x)) dx, where u solves the Dirichlet problem Δu(x)=−f(x), x∈Ω, u(x)=0, x∈∂Ω. Necessary and sufficient conditions for local optimality are presented in terms of the first and second variations of the
cost functionals
and ℱ. The second variations are computed with respect to domain variations which preserve the constraint. After first summarizing
known facts about the first variations of u and the cost functionals, a series of formulas relating various second variations of these quantities are derived. Calculating
the second variations depends on finding first variations of solutions u when the data f are permitted to depend on the domain Ω. 相似文献
11.
A. V. Demyanov 《Journal of Mathematical Sciences》2006,136(2):3706-3717
The problem of establishing necessary and sufficient conditions for l.s.c. under PDE constraints is studied for a special
class of functionals:
with respect to the convergence un → u in measure, vn ⇀ v in Lp(Ω;ℝd)
in W−1,p(Ω), and χn ⇀ χ in Lp(Ω), where χn ∈ Z:= {χ ∈ L∞(Ω): 0 ≤ χ(x) ≤ 1 for a.e. x}. Here
is a constant-rank partial differential operator. The main result is that if the characteristic cone of
has the full dimension, then the l.s.c. is equivalent to the fact that the F± are both
-quasiconvex and
for a.e. x ∈ Ω and for all u ∈ ℝd. As a corollary, we obtain several results for the functional
with respect to the same convergence. We show that this functional is l.s.c. iff
Bibliography: 14 titles.
Published in Zapiski Nauchnykh Seminarov POMI, Vol. 318, 2004, pp. 100–119. 相似文献
12.
Luisa Fattorusso 《Czechoslovak Mathematical Journal》2008,58(1):113-129
Let Ω be a bounded open subset of ℝ
n
, n > 2. In Ω we deduce the global differentiability result
for the solutions u ∈ H
1 (Ω, ℝ
n
) of the Dirichlet problem
with controlled growth and nonlinearity q = 2.
The result was obtained by first extending the interior differentiability result near the boundary and then proving the global
differentiability result making use of a covering procedure. 相似文献
13.
Annalisa Malusa Luigi Orsina 《Calculus of Variations and Partial Differential Equations》2006,27(2):179-202
We study the limit as n goes to +∞ of the renormalized solutions u
n
to the nonlinear elliptic problems
where Ω is a bounded open set of ℝ
N
, N≥ 2, and μ is a Radon measure with bounded variation in Ω. Under the assumption of G-convergence of the operators , defined for , to the operator , we shall prove that the sequence (u
n
) admits a subsequence converging almost everywhere in Ω to a function u which is a renormalized solution to the problem
相似文献
14.
Let p
*
=n/(n−2) and n≥3. In this paper, we first classify all non-constant solutions of
We then establish a sup + inf and a Moser-Trudinger type inequalities for the equation −Δu=u
+
p*
. Our results illustrate that this equation is much closer to the Liouville problem −Δu=e
u
in dimension two than the usual critical exponent equation, namely is.
Received: 11 March 2002; in final form: 8 July 2002 /
Published online: 16 May 2003 相似文献
15.
The authors consider quasilinear parabolic systems
in two space dimensions. The function a has p-growth behaviour, 1< p < ∞, and the ellipticity “constant” behaves like (1+|∇u|)
p
− 2. The author prove full regularity of the weak solution on interior subdomains, but globally in time. The key idea in the
proof is a technique to obtain boundedness of the gradient based on logarithmic estimates.
Received: 8 July 1998 / Revised version: 19 February 1999 相似文献
16.
In this paper we study the Dirichlet problem in Q
T
= Ω × (0, T) for degenerate equations of porous medium-type with a lower order term:
The principal part of the operator degenerates in u = 0 according to a nonnegative increasing real function α(u), and the term grows quadratically with respect to the gradient. We prove an existence result for solutions to this problem in the framework
of the distributional solutions under the hypotheses that both f and the initial datum u
0 are bounded nonnegative functions. Moreover as further results we get an existence result for the model problem
in the case that the principal part of the operator is of fast-diffusion type, i.e. α(u) = u
m
, with −1 < m < 0.
相似文献
17.
Hui Yin Shuyue Chen Jing Jin 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2008,59(6):969-1001
This paper is concerned with the large time behavior of traveling wave solutions to the Cauchy problem of generalized Benjamin–Bona–Mahony–Burgers
equations
with prescribed initial data
Here v( > 0), β are constants, u
± are two given constants satisfying u
+ ≠ u
− and the nonlinear function f(u) ∈C
2(R) is assumed to be either convex or concave. An algebraic time decay rate to traveling waves of the solutions of the Cauchy
problem of generalized Benjamin-Bona-Mahony-Burgers equation is obtained by employing the weighted energy method developed
by Kawashima and Matsumura in [6] to discuss the asymptotic behavior of traveling wave solutions to the Burgers equation.
revised: May 23 and August 8, 2007 相似文献
18.
Let Ω be a bounded smooth domain inR
2. Letf:R→R be a smooth non-linearity behaving like exp{s
2} ass→∞. LetF denote the primitive off. Consider the functionalJ:H
0
1
(Ω)→R given by
It can be shown thatJ is the energy functional associated to the following nonlinear problem: −Δu=f(u) in Ω,u=0 on ρΩ. In this paper we consider the global compactness properties ofJ. We prove thatJ fails to satisfy the Palais-Smale condition at the energy levels {k/2},k any positive integer. More interestingly, we show thatJ fails to satisfy the Palais-Smale condition at these energy levels along two Palais-Smale sequences. These two sequences
exhibit different blow-up behaviours. This is in sharp contrast to the situation in higher dimensions where there is essentially
one Palais-Smale sequence for the corresponding energy functional. 相似文献
19.
Consider a bounded planar domain D, an instance h of the Gaussian free field on D, with Dirichlet energy (2π)−1∫
D
∇h(z)⋅∇h(z)dz, and a constant 0≤γ<2. The Liouville quantum gravity measure on D is the weak limit as ε→0 of the measures
|
eg2/2 eghe(z)dz,\varepsilon^{\gamma^2/2} e^{\gamma h_\varepsilon(z)}dz, 相似文献
20.
This paper deals with the existence of positive solutions for the nonlinear system
|