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1.
In this paper we continue with our work in Lederman and Wolanski (Ann Math Pura Appl 187(2):197–220, 2008) where we developed
a local monotonicity formula for solutions to an inhomogeneous singular perturbation problem of interest in combustion theory.
There we proved local monotonicity formulae for solutions ue{{u^\varepsilon}} to the singular perturbation problem and for u=limue{u=\lim{u^\varepsilon}} , assuming that both ue{{u^\varepsilon}} and u were defined in an arbitrary domain D{\mathcal{D}} in
\mathbbRN+1{\mathbb{R}^{N+1}} . In the present work we obtain global monotonicity formulae for limit functions u that are globally defined, while ue{{u^\varepsilon}} are not. We derive such global formulae from a local one that we prove here. In particular, we obtain a global monotonicity
formula for blow up limits u
0 of limit functions u that are not globally defined. As a consequence of this formula, we characterize blow up limits u
0 in terms of the value of a density at the blow up point. We also present applications of the results in this paper to the
study of the regularity of ∂{u > 0} (the flame front in combustion models). The fact that our results hold for the inhomogeneous singular perturbation problem
allows a very wide applicability, for instance to problems with nonlocal diffusion and/or transport. 相似文献
2.
A. L. Karakhanyan C. E. Kenig H. Shahgholian 《Calculus of Variations and Partial Differential Equations》2007,28(1):15-31
We show that the free boundary ∂{u > 0}, arising from the minimizer(s) u, of the functional
approaches the (smooth) fixed boundary ∂Ω tangentially, at points where the Dirichlet data vanishes along with its gradient.
相似文献
3.
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
4.
Jorge García-Melián José C. Sabina De Lis Julio D. Rossi 《NoDEA : Nonlinear Differential Equations and Applications》2007,14(5-6):499-525
We deal with positive solutions of Δu = a(x)u
p
in a bounded smooth domain subject to the boundary condition ∂u/∂v = λu, λ a parameter, p > 1. We prove that this problem has a unique positive solution if and only if 0 < λ < σ1 where, roughly speaking, σ1 is finite if and only if |∂Ω ∩ {a = 0}| > 0 and coincides with the first eigenvalue of an associated eigenvalue problem. Moreover, we find the limit profile
of the solution as λ → σ1.
Supported by DGES and FEDER under grant BFM2001-3894 (J. García-Melián and J. Sabina) and ANPCyT PICT No. 03-05009 (J. D.
Rossi). J.D. Rossi is a member of CONICET. 相似文献
5.
In this paper we consider the heat flow of harmonic maps between two compact Riemannian Manifolds M and N (without boundary)
with a free boundary condition. That is, the following initial boundary value problem ∂1,u −Δu = Γ(u)(∇u, ∇u) [tT Tu
uN, on M × [0, ∞), u(t, x) ∈ Σ, for x ∈ ∂M, t > 0, ∂u/t6n(t, x) ⊥u Tu(t,x) Σ, for x ∈ ∂M, t > 0, u(o,x) = uo(x), on M, where Σ is a smooth submanifold without boundary in N and n is a unit normal vector field of M along ∂M.
Due to the higher nonlinearity of the boundary condition, the estimate near the boundary poses considerable difficulties,
even for the case N = ℝn, in which the nonlinear equation reduces to ∂tu-Δu = 0.
We proved the local existence and the uniqueness of the regular solution by a localized reflection method and the Leray-Schauder
fixed point theorem. We then established the energy monotonicity formula and small energy regularity theorem for the regular
solutions. These facts are used in this paper to construct various examples to show that the regular solutions may develop
singularities in a finite time. A general blow-up theorem is also proven. Moreover, various a priori estimates are discussed
to obtain a lower bound of the blow-up time. We also proved a global existence theorem of regular solutions under some geometrical
conditions on N and Σ which are weaker than KN <-0 and Σ is totally geodesic in N. 相似文献
6.
Piotr Kot 《Czechoslovak Mathematical Journal》2009,59(2):371-379
We solve the following Dirichlet problem on the bounded balanced domain with some additional properties: For p > 0 and a positive lower semi-continuous function u on ∂Ω with u(z) = u(λ z) for |λ| = 1, z ∈ ∂Ω we construct a holomorphic function f ∈ (Ω) such that for z ∈ ∂Ω, where = {λ ∈ ℂ: |λ| < 1}.
相似文献
7.
Lucio Damascelli Berardino Sciunzi 《Calculus of Variations and Partial Differential Equations》2006,25(2):139-159
We consider the Dirichlet problem for positive solutions of the equation −Δm (u) = f(u) in a bounded smooth domain Ω, with f positive and locally Lipschitz continuous. We prove a Harnack type inequality for the solutions of the linearized operator,
a Harnack type comparison inequality for the solutions, and exploit them to prove a Strong Comparison Principle for solutions
of the equation, as well as a Strong Maximum Principle for the solutions of the linearized operator. We then apply these results,
together with monotonicity results recently obtained by the authors, to get regularity results for the solutions. In particular
we prove that in convex and symmetric domains, the only point where the gradient of a solution u vanishes is the center of symmetry (i.e. Z≡{x∈ Ω ∨ D(u)(x) = 0 = {0} assuming that 0 is the center of symmetry). This is crucial in the study of m-Laplace equations, since Z is exactly the set of points where the m-Laplace operator is degenerate elliptic. As a corollary u ∈ C2(Ω∖{0}).
Supported by MURST, Project “Metodi Variazionali ed Equazioni Differenziali Non Lineari.”
Mathematics Subject Classification (1991) 35B05, 35B65, 35J70 相似文献
8.
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
9.
Reinhard Farwig 《Mathematische Zeitschrift》1992,210(1):449-464
Consider the Dirichlet problem −vΔu+k∂
1
u = f withv, k>0 in ℝ3 or in an exterior domain of ℝ3 where the skew-symmetric differential operator −1=∂/∂x1 is a singular perturbation of the Laplacian. Because of the inhomogeneity of the fundamental solution we study existence,
uniqueness and regularity in Sobolev spaces with anisotropic weights. In these spaces the operator ∂1 yields an additional positive definite term giving better results than in Sobolev spaces with radial weights. The elliptic
equation −vΔu +k∂1
u=f can be taken as a model problem for the Oseen equations, a linearized form of the Navier-Stokes equations.
Supported by the Sonderforschungsbereich 256 of the Deutsche Forschungsgemeinschaft at the University of Bonn 相似文献
10.
L. Sobrero 《Annali di Matematica Pura ed Applicata》1935,14(1):139-148
Sunto Due funzionif(x, y) e ϕ(x, y), biarmoniche (e cioè soddisfacenti all'equazione ΔΔ=0), rispettivamente definite nei semipianix>0 edx<0, le cui derivate seconde si annullano all'infinito, e tali che nei punti dell'assey risultif=ϕ e∂f/∂x=∂ϕ/∂x, si dicono l'una ? riflessa ? dell'altra attorno all'assey. Da ognuna delle due funzionif e ϕ l'altra si ottiene con sole operazioni di sostituzione e derivazione (indicando, precisamente, con{f}, {∂f/∂x} e{Δf} le funzioni che si ottengono daf, ∂f/∂x eΔf ponendo, in queste, in luogo dix il suo contrario −x, si ha ϕ={f}+2x{∂f/∂x}+x
2
{Δf} e, reciprocamente,f={ϕ}+2x{∂ϕ/∂x}+x
2{Δϕ}). In modo analogo si definisce una operazione di riflessione analitica attorno a un cerchio. La retta potendosi riguardare
come cerchio degenere (di raggio infinito) l'operazione di rifiessione analitica attorno alla retta viene ottenuta, nel testo,
come caso limite di quella di riflessione attorno al cerchio. L'operazione di riflessione analitica trova applicazione in
alcuni problemi di elasticità piana (perturbazione prodotta da un foro circolare nella sollecitazione di un sistema piano;
determinazione degli sforzi in un semipiano elastico sollecitato da una forza applicata in un punto interno). 相似文献
11.
Tiziana Giorgi Robert Smits 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2008,124(1):600-618
We consider the principal eigenvalue λ
1Ω(α) corresponding to Δu = λ (α) u in
W, \frac?u?v = au \Omega, \frac{\partial u}{\partial v} = \alpha u on ∂Ω, with α a fixed real, and W ì Rn\Omega \subset {\mathcal{R}}^n a C
0,1 bounded domain. If α > 0 and small, we derive bounds for λ
1Ω(α) in terms of a Stekloff-type eigenvalue; while for α > 0 large we study the behavior of its growth in terms of maximum curvature.
We analyze how domain monotonicity of the principal eigenvalue depends on the geometry of the domain, and prove that domains
which exhibit domain monotonicity for every α are calibrable. We conjecture that a domain has the domain monotonicity property for some α if and only if it is calibrable. 相似文献
12.
Marcelo Montenegro Olivâine S. de Queiroz Eduardo V. Teixeira 《Mathematische Annalen》2011,351(1):215-250
We establish existence and sharp regularity results for solutions to singular elliptic equations of the order u
−β
, 0 < β < 1, with gradient dependence and involving a forcing term λ f(x, u). Our approach is based on a singularly perturbed technique. We show that if the forcing parameter λ > 0 is large enough,
our solution is positive. For λ small solutions vanish on a nontrivial set and therefore they exhibit free boundaries. We
also establish regularity results for the free boundary and study the asymptotic behavior of the problem as
b\searrow 0{\beta\searrow 0} and
b\nearrow 1{\beta\nearrow 1}. In the former, we show that our solutions u
β
converge to a C
1,1 function which is a solution to an obstacle type problem. When
b\nearrow 1{\beta\nearrow 1} we recover the Alt-Caffarelli theory. 相似文献
13.
Rosa M. Migo-Roig 《manuscripta mathematica》1993,80(1):89-94
We show the following theorem of compensated compactness type: Ifu
n
⇁u weakly in the spaceH
1,p
(Ω, ℝ
k
) and if also
in the sense of distributions then ∂α(∣∇u∣
p-2∂α
u)=0. This result has applications in the partial regularity theory ofp-stationary mappings Ω→S
k
−1. 相似文献
14.
Marina GHISI Massimo GOBBINO 《数学学报(英文版)》2006,22(4):1161-1170
We consider the Cauchy problem εu^″ε + δu′ε + Auε = 0, uε(0) = uo, u′ε(0) = ul, where ε 〉 0, δ 〉 0, H is a Hilbert space, and A is a self-adjoint linear non-negative operator on H with dense domain D(A). We study the convergence of (uε) to the solution of the limit problem ,δu' + Au = 0, u(0) = u0.
For initial data (u0, u1) ∈ D(A1/2)× H, we prove global-in-time convergence with respect to strong topologies.
Moreover, we estimate the convergence rate in the case where (u0, u1)∈ D(A3/2) ∈ D(A1/2), and we show that this regularity requirement is sharp for our estimates. We give also an upper bound for |u′ε(t)| which does not depend on ε. 相似文献
15.
Tetsutaro Shibata 《Annali di Matematica Pura ed Applicata》2003,182(2):211-229
We consider the two-parameter nonlinear eigenvalue problem?−Δu = μu − λ(u + u
p
+ f(u)), u > 0 in Ω, u = 0 on ∂Ω,?where p>1 is a constant and μ,λ>0 are parameters. We establish the asymptotic formulas for the variational eigencurves λ=λ(μ,α) as
μ→∞, where α>0 is a normalizing parameter. We emphasize that the critical case from a viewpoint of the two-term asymptotics
of the eigencurve is p=3. Moreover, it is shown that p=5/3 is also a critical exponent from a view point of the three-term asymptotics when Ω is a ball or an annulus. This sort
of criticality for two-parameter problems seems to be new.
Received: February 9, 2002; in final form: April 3, 2002?Published online: April 14, 2003 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.
We consider the parabolic Anderson problem ∂
t
u = κΔu + ξ(x)u on ℝ+×ℝ
d
with initial condition u(0,x) = 1. Here κ > 0 is a diffusion constant and ξ is a random homogeneous potential. We concentrate on the two important cases
of a Gaussian potential and a shot noise Poisson potential. Under some mild regularity assumptions, we derive the second-order
term of the almost sure asymptotics of u(t, 0) as t→∞.
Received: 26 July 1999 / Revised version: 6 April 2000 / Published online: 22 November 2000 相似文献
18.
A. Malchiodi 《Geometric And Functional Analysis》2005,15(6):1162-1222
We prove new concentration phenomena for the equation −ɛ2 Δu + u = up in a smooth bounded domain
and with Neumann boundary conditions. Here p > 1 and ɛ > 0 is small. We show that concentration of solutions occurs at some geodesics of ∂Ω when ɛ → 0.
Received: September 2004 Accepted: March 2005 相似文献
19.
Tiziana Giorgi Robert Smits 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2008,59(4):600-618
We consider the principal eigenvalue λ
1Ω(α) corresponding to Δu = λ (α) u in on ∂Ω, with α a fixed real, and a C
0,1 bounded domain. If α > 0 and small, we derive bounds for λ
1Ω(α) in terms of a Stekloff-type eigenvalue; while for α > 0 large we study the behavior of its growth in terms of maximum curvature.
We analyze how domain monotonicity of the principal eigenvalue depends on the geometry of the domain, and prove that domains
which exhibit domain monotonicity for every α are calibrable. We conjecture that a domain has the domain monotonicity property for some α if and only if it is calibrable.
Robert Smits: This author was partially supported by a grant of the National Security Agency, grant #H98230-05-1-0060. 相似文献
20.
In this paper, we study the ill-posdness of the Cauchy problem for semilinear wave equation with very low regularity, where
the nonlinear term depends on u and ∂
t
u. We prove a ill-posedness result for the “defocusing” case, and give an alternative proof for the supercritical “focusing”
case, which improves the result in Fang and Wang (Chin. Ann. Math. Ser. B 26(3), 361–378, 2005).
Supported by NSF of China 10571158. 相似文献