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1.
Alessandra Pagano 《Annali dell'Universita di Ferrara》1993,39(1):1-17
We consider a (possibly) vector-valued function u: Ω→R
N, Ω⊂R
n, minimizing the integral
, whereD
iu=∂u/∂x
i, or some more general functional retaining the same behaviour; we prove higher integrability forDu:D
1u,…,Dn−1u∈Lq, under suitable assumptions ona
i(x).
Sunto Consideriamo una funzione u: Ω→R N, Ω⊂R n che minimizzi l'integrale , doveD iu=∂u/∂xi, o un funzionale con un comportamento simile; sotto opportune ipotesi sua i(x), dimostriamo la seguente maggiore integrabilità perDu:D 1u,…,Dn−1uεLq.相似文献
2.
We address the question of attainability of the best constant in the following Hardy–Sobolev inequality on a smooth domain
Ω of
:
when
and when 0 is on the boundary ∂Ω. This question is closely related to the geometry of ∂Ω, as we extend here the main result
obtained in [GhK] by proving that at least in dimension n ≥ 4, the negativity of the mean curvature of ∂Ω at 0 is sufficient to ensure the attainability of μs(Ω). Key ingredients in our proof are the identification of symmetries enjoyed by the extremal functions corresponding to
the best constant in half-space, as well as a fine analysis of the asymptotic behaviour of appropriate minimizing sequences.
The result holds true also in dimension 3 but the more involved proof will be dealt with in a forthcoming paper [GhR2].
N.G.’s research partially supported by the Natural Sciences and Engineering Research Council of Canada. The first named author
gratefully acknowledges the hospitality and support of the Université de Nice where this work was initiated. F.R. gratefully
acknowledges the hospitality and support of the University of British Columbia where this work was completed.
Received: February 2005; Accepted: May 2005 相似文献
3.
Xu Chaojiang 《数学学报(英文版)》1992,8(4):362-374
We prove in this paper theC
∞ regularity for a “very strict” local minimum of classC
loc
ρ
, ρ>3, of functionals with genuine degenerate quasiconvex integrand
depending on a vector-valued function u. Such a minimum satisfies the condition: for all x∈Ω, there exists a neighbourhoodK(x) ofx in Ω andC
1
(x)>0,C
2
(x)>0,1≥ε(x)>0, such that
for all real ϕ∈c
0
∞
(K).
This work is supported by the National Natural Science Foundation of China and the Fok Ying Tung Education Foundation. 相似文献
4.
We extend the results for 2-D Boussinesq equations from ℝ2 to a bounded domain Ω. First, as for the existence of weak solutions, we transform Boussinesq equations to a nonlinear evolution
equation U
t
+ A(t, U) = 0. In stead of using the methods of fundamental solutions in the case of entire ℝ2, we study the qualities of F(u, υ) = (u · ▽)υ to get some useful estimates for A(t, U), which helps us to conclude the local-in-time existence and uniqueness of solutions. Second, as for blow-up criterions,
we use energy methods, Sobolev inequalities and Gronwall inequality to control
and
by
and
. Furthermore,
can control
by using vorticity transportation equations. At last,
can control
. Thus, we can find a blow-up criterion in the form of
.
相似文献
5.
Juha Lehrbäck 《manuscripta mathematica》2008,127(2):249-273
We establish necessary and sufficient conditions for a domain to admit the (p, β)-Hardy inequality , where d(x) = dist(x, ∂Ω) and . Our necessary conditions show that a certain dichotomy holds, even locally, for the dimension of the complement Ω
c
when Ω admits a Hardy inequality, whereas our sufficient conditions can be applied in numerous situations where at least
a part of the boundary ∂Ω is “thin”, contrary to previously known conditions where ∂Ω or Ω
c
was always assumed to be “thick” in a uniform way. There is also a nice interplay between these different conditions that
we try to point out by giving various examples.
The author was supported in part by the Academy of Finland. 相似文献
6.
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
7.
Menita Carozza Chiara Leone Antonia Passarelli di Napoli Anna Verde 《Calculus of Variations and Partial Differential Equations》2009,35(2):215-238
We prove a C
2,α
partial regularity result for local minimizers of polyconvex variational integrals of the type , where Ω is a bounded open subset of , and is a convex function, with subquadratic growth. 相似文献
8.
In this paper we consider the Lane–Emden problem adapted for the p-Laplacian
where Ω is a bounded domain in , n ≥ 2, λ > 0 and p < q < p* (with if p < n, and p* = ∞ otherwise). After some recalls about the existence of ground state and least energy nodal solutions, we prove that,
when q → p, accumulation points of ground state solutions or of least energy nodal solutions are, up to a “good” scaling, respectively
first or second eigenfunctions of −Δ
p
.
Received: 29 April 2008 相似文献
9.
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
10.
Mabel Cuesta Humberto Ramos Quoirin 《NoDEA : Nonlinear Differential Equations and Applications》2009,16(4):469-491
Let Δ
p
denote the p-Laplacian operator and Ω be a bounded domain in . We consider the eigenvalue problem
for a potential V and a weight function m that may change sign and be unbounded. Therefore the functional to be minimized is indefinite and may be unbounded from below.
The main feature here is the introduction of a value α(V, m) that guarantees the boundedness of the energy over the weighted sphere . We show that the above equation has a principal eigenvalue if and only if either m ≥ 0 and α(V, m) > 0 or m changes sign and α(V, m) ≥ 0. The existence of further eigenvalues is also treated here, mainly a second eigenvalue (to the right) and their dependence
with respect to V and m.
相似文献
11.
Futoshi Takahashi 《Archiv der Mathematik》2009,93(2):191-197
In this note, we consider the problem
on a smooth bounded domain Ω in for p > 1. Let u
p
be a positive solution of the above problem with Morse index less than or equal to . We prove that if u
p
further satisfies the assumption as p → ∞, then the number of maximum points of u
p
is less than or equal to m for p sufficiently large. If Ω is convex, we also show that a solution of Morse index one satisfying the above assumption has a
unique critical point and the level sets are star-shaped for p sufficiently large.
相似文献
12.
Riccardo De Arcangelis 《Annali dell'Universita di Ferrara》1989,35(1):135-145
Summary Letf: (x, z)∈R
n×Rn→f(x, z)∈[0, +∞] be measurable inx and convex inz.
It is proved, by an example, that even iff verifies a condition as|z|
p≤f(x, z)≤Λ(a(x)+|z|q) with 1<p<q,a∈L
loc
s
(R
n),s>1, the functional
that isL
1(Ω)-lower semicontinuous onW
1,1(Ω), does not agree onW
1,1(Ω) with its relaxed functional in the topologyL
1(Ω) given by inf
Riassunto Siaf: (x, z)∈R n×Rn→f(x, z)∈[0, +∞] misurabile inx e convessa inz. Si mostra con un esempio che anche sef verifica una condizione del tipo|z| p≤f(x, z)≤Λ(a(x)+|z|q) con 1<p<q,a∈L loc s (R n),s>1, il funzionale , che èL 1(Ω)-semicontinuo inferiormente suW 1,1(Ω), non coincide suW 1,1(Ω) con il suo funzionale rilassato nella topologiaL 1(Ω) definito da inf相似文献
13.
Yehuda Pinchover Kyril Tintarev 《Calculus of Variations and Partial Differential Equations》2007,28(2):179-201
Let Ω be a domain in , d ≥ 2, and 1 < p < ∞. Fix . Consider the functional Q and its Gateaux derivative Q′ given by If Q ≥ 0 on, then either there is a positive continuous function W such that for all, or there is a sequence and a function v > 0 satisfying Q′ (v) = 0, such that Q(u
k
) → 0, and in . In the latter case, v is (up to a multiplicative constant) the unique positive supersolution of the equation Q′ (u) = 0 in Ω, and one has for Q an inequality of Poincaré type: there exists a positive continuous function W such that for every satisfying there exists a constant C > 0 such that . As a consequence, we prove positivity properties for the quasilinear operator Q′ that are known to hold for general subcritical resp. critical second-order linear elliptic operators. 相似文献
14.
In this paper we study some optimization problems for nonlinear elastic membranes. More precisely, we consider the problem
of optimizing the cost functional
over some admissible class of loads f where u is the (unique) solution to the problem −Δ
p
u+|u|
p−2
u=0 in Ω with |∇
u|
p−2
u
ν
=f on ∂Ω.
Supported by Universidad de Buenos Aires under grant X078, by ANPCyT PICT No. 2006-290 and CONICET (Argentina) PIP 5478/1438.
J. Fernández Bonder is a member of CONICET. Leandro M. Del Pezzo is a fellow of CONICET. 相似文献
15.
Jens Habermann 《Mathematische Zeitschrift》2008,258(2):427-462
For weak solutions of higher order systems of the type , for all , with variable growth exponent p : Ω → (1,∞) we prove that if with , then . We should note that we prove this implication both in the non-degenerate (μ > 0) and in the degenerate case (μ = 0). 相似文献
16.
It is known [8] that, wheng ∈L
n
(Ω) (Ω open and bounded inR
n
, with ≪regular≫ boundary∂Ω), any minimizer (K, w) of the functional
among relatively closed subsetsC ofΩ and piecewise-constant functionsu onΩ/C, gives rise to a finite decomposition ofΩ/K. Here we exhibit a piecewise-constant functiong on the unit diskD ofR
2, with radial symmetry, such thatg ∈L
q
(D) for all 1 ⩽q < 2 and the unique minimizer of F has infinitely many components. We also fill a gap occurred in the proof of Proposition
5.2 of [8].
Sunto è noto [8] che quandog ∈L n (Ω (Ω aperto limitato diR n , con frontiera sufficientemente regolare) i minimi (K, w) del funzionale , doveC è relativamente chiuso in Ω eu è costante a tratti suΩ/C, danno luogo a decomposizioni finite diΩ/K. In questo lavoro mostriamo un controesempio relativo ad un datog ∈L q (D) per ogni 1 ⩽q < 2 (D è il disco unitario diR 2), a simmetria radiale e costante a tratti. Viene inoltre corretto un errore occorso nella dimostrazione della Prop. 5.2 di [8].相似文献
17.
We consider the following semilinear elliptic equation with singular nonlinearity:
where
and Ω is an open subset in
. Let u be a non-negative finite energy stationary solution and
be the rupture set of u. We show that the Hausdorff dimension of Σ is less than or equal to [(n−2) α+(n+2)]/(α +1). 相似文献
18.
Let u be a weak solution of (-△)mu = f with Dirichlet boundary conditions in a smooth bounded domain Ω Rn. Then, the main goal of this paper is to prove the following a priori estimate:‖u‖ Wω2 m,p(Ω) ≤ C ‖f‖ Lωp (Ω),where ω is a weight in the Muckenhoupt class Ap. 相似文献
19.
Martin Flucher 《Commentarii Mathematici Helvetici》1992,67(1):471-497
We prove that theTrudinger-Moser constant
相似文献
20.
Thomas Schmidt 《NoDEA : Nonlinear Differential Equations and Applications》2009,16(1):109-129
We consider multi-dimensional variational integrals
|