共查询到20条相似文献,搜索用时 15 毫秒
1.
We study compactness properties for solutions of a semilinear elliptic equation with critical nonlinearity. For high dimensions, we are able to show that any solutions sequence with uniformly bounded energy is uniformly bounded in the interior of the domain. In particular, singularly perturbed Neumann equations admit pointwise concentration phenomena only at the boundary. 相似文献
2.
Martin Schechter 《Nonlinear Analysis: Theory, Methods & Applications》2009,71(12):e234
We apply some of the methods of sandwich pairs to semilinear equations and systems. 相似文献
3.
Catherine Bandle 《Journal of Differential Equations》2011,251(8):2143-827
It is well known that the heat kernel in the hyperbolic space has a different behavior for large times than the one in the Euclidean space. The main purpose of this paper is to study its effect on the positive solutions of Cauchy problems with power nonlinearities. Existence and non-existence results for local solutions are derived. Emphasis is put on their long time behavior and on Fujita?s phenomenon. To have the same situation as for the Cauchy problem in RN, namely finite time blow up for all solutions if the exponent is smaller than a critical value and existence of global solutions only for powers above the critical exponent, we must introduce a weight depending exponentially on the time. In this respect the situation is similar to problems in bounded domains with Dirichlet boundary conditions. Important tools are estimates for the heat kernel in the hyperbolic space and comparison principles. 相似文献
4.
We consider both divergence and non-divergence parabolic equations on a half space in weighted Sobolev spaces. All the leading coefficients are assumed to be only measurable in the time and one spatial variable except one coefficient, which is assumed to be only measurable either in the time or the spatial variable. As functions of the other variables the coefficients have small bounded mean oscillation (BMO) semi-norms. The lower-order coefficients are allowed to blow up near the boundary with a certain optimal growth condition. As a corollary, we also obtain the corresponding results for elliptic equations. 相似文献
5.
D. Motreanu 《Set-Valued Analysis》1995,3(3):295-305
The paper proves the existence of critical points for a general locally Lipschitz functional usually arising in nonlinear elliptic problems. It extends and unifies various results in the critical point theory. The applications treat new situations involving discontinuous elliptic equations containing both sublinear and superlinear terms, integro-differential equations and nonlinear elliptic systems. 相似文献
6.
We develop the concept and the calculus of anti-self-dual (ASD) Lagrangians and their derived vector fields which seem inherent to many partial differential equations and evolutionary systems. They are natural extensions of gradients of convex functions – hence of self-adjoint positive operators – which usually drive dissipative systems, but also provide representations for the superposition of such gradients with skew-symmetric operators which normally generate unitary flows. They yield variational formulations and resolutions for large classes of non-potential boundary value problems and initial-value parabolic equations. Solutions are minima of newly devised energy functionals, however, and just like the self (and anti-self) dual equations of quantum field theory (e.g. Yang–Mills) the equations associated to such minima are not derived from the fact they are critical points of the functional I, but because they are also zeroes of suitably derived Lagrangians. The approach has many advantages: it solves variationally many equations and systems that cannot be obtained as Euler–Lagrange equations of action functionals, since they can involve non-self-adjoint or other non-potential operators; it also associates variational principles to variational inequalities, and to various dissipative initial-value first order parabolic problems. These equations can therefore be analyzed with the full range of methods – computational or not – that are available for variational settings. Most remarkable are the permanence properties that ASD Lagrangians possess making their calculus relatively manageable and their domain of applications quite broad. 相似文献
7.
We consider solutions of initial-boundary value problems for the heat equation on bounded domains in and their spatial critical points as in the previous paper [MS]. In Dirichlet, Neumann, and Robin homogeneous initial-boundary
value problems on bounded domains, it is proved that if the origin is a spatial critical point never moving for sufficiently
many compactly supported initial data being centrosymmetric with respect to the origin, then the domain must be centrosymmetric
with respect to the origin. Furthermore, we consider spatial zero points instead of spatial critical points, and prove some
similar symmetry theorems. Also, it is proved that these symmetry theorems hold for initial-boundary value problems for the
wave equation.
Received October 31, 1997; in final form February 3, 1998 相似文献
8.
Li-Ming Yeh 《Journal of Differential Equations》2011,250(4):1828-1849
A priori estimate for non-uniform elliptic equations with periodic boundary conditions is concerned. The domain considered consists of two sub-regions, a connected high permeability region and a disconnected matrix block region with low permeability. Let ? denote the size ratio of one matrix block to the whole domain. It is shown that in the connected high permeability sub-region, the Hölder and the Lipschitz estimates of the non-uniform elliptic solutions are bounded uniformly in ?. But Hölder gradient estimate and Lp estimate of the second order derivatives of the solutions in general are not bounded uniformly in ?. 相似文献
9.
The Ginzburg-Landau-Allen-Cahn equation is a variational model for phase coexistence and for other physical problems. It contains a term given by a kinetic part of elliptic type plus a double-well potential. We assume that the functional depends on the space variables in a periodic way.We show that given a plane with rational normal, there are minimal solutions, satisfying the following properties. These solutions are asymptotic to the pure phases and are separated by an interface. The convergence to the pure phases is exponentially fast. The interface lies at a finite distance M from the chosen plane, where M is a universal constant. Furthermore, these solutions satisfy some monotonicity properties with respect to integer translations (namely, integer translations are always comparable to the function).We then show that all the interfaces of the global periodic minimizers satisfy similar monotonicity and plane-like properties.We also consider the case of possibly irrationally oriented planes. We show that either there is a one parameter family of minimizers whose graphs provide a field of extremals or there are at least two solutions, one which is a minimizer and another one which is not. These solutions also have interfaces bounded by a universal constant, they enjoy monotonicity properties with respect to integer translations and the nonminimal solutions are trapped inside a gap of the lamination induced by the minimizers. 相似文献
10.
We study the existence of multiple solutions for a quasilinear elliptic system of gradient type with the possibility of coupling on the critical and subcritical terms which are not necessarily homogeneous. The solutions are obtained from a version of the Symmetric Mountain Pass Theorem. A version of the Concentration-Compactness Principle for this class of systems allows us to verify that the Palais–Smale condition is satisfied below a certain level. 相似文献
11.
Dmitry Golovaty Leonid Berlyand 《Calculus of Variations and Partial Differential Equations》2002,14(2):213-232
Consider a class of Sobolev functions satisfying a prescribed degree condition on the boundary of a planar annular domain.
It is shown that, within this class, the Ginzburg-Landau functional possesses the unique, radially symmetric minimizer, provided
that the annulus is sufficiently narrow. This result is known to be false for wide annuli where vortices are energetically
feasible. The estimate for the critical radius below which the uniqueness of the minimizer is guaranteed is obtained as well.
Received: 19 July 2000 / Accepted: 23 February 2001/ Published online: 23 July 2001 相似文献
12.
Chad A.S. Mullikin 《Topology and its Applications》2007,154(14):2697-2708
The distortion of a curve is the supremum, taken over distinct pairs of points of the curve, of the ratio of arclength to spatial distance between the points. Gromov asked in 1981 whether a curve in every knot type can be constructed with distortion less than a universal constant C. Answering Gromov's question seems to require the construction of lower bounds on the distortion of knots in terms of some topological invariant. We attempt to make such bounds easier to construct by showing that pairs of points with high distortion are very common on curves of minimum length in the set of curves in a given knot type with distortion bounded above and distortion thickness bounded below. 相似文献
13.
In this paper we are concerned with multi-lump bound states of the nonlinear Schr?dinger equation
for sufficiently small , where for and for . V is bounded on . For any finite collection of nondegenerate critical points of V, we show the uniqueness of solutions of the form for , where u is positive on and is a small perturbation of a sum of one-lump solutions concentrated near , respectively for sufficiently small .
Received: 30 October 2001; in final form: 10 June 2002 /Published online: 2 December 2002
RID="*"
ID="*" Research supported by Alexander von Humboldt Foundation in Germany and NSFC in China 相似文献
14.
Luc Miller 《Bulletin des Sciences Mathématiques》2005,129(2):175-185
We make two remarks about the null-controllability of the heat equation with Dirichlet condition in unbounded domains. Firstly, we give a geometric necessary condition (for interior null-controllability in the Euclidean setting) which implies that one cannot go infinitely far away from the control region without tending to the boundary (if any), but also applies when the distance to the control region is bounded. The proof builds on heat kernel estimates. Secondly, we describe a class of null-controllable heat equations on unbounded product domains. Elementary examples include an infinite strip in the plane controlled from one boundary and an infinite rod controlled from an internal infinite rod. The proof combines earlier results on compact manifolds with a new lemma saying that the null-controllability of an abstract control system and its null-controllability cost are not changed by taking its tensor product with a system generated by a non-positive self-adjoint operator. 相似文献
15.
Juncheng Wei Matthias Winter 《Calculus of Variations and Partial Differential Equations》2000,10(3):249-289
We study the Cahn-Hilliard equation in a bounded smooth domain without any symmetry assumptions. We prove that for any fixed
positive integer K there exist interior K–spike solutions whose peaks have maximal possible distance from the boundary and from one another. This implies that for
any bounded and smooth domain there exist interior K–peak solutions. The central ingredient of our analysis is the novel derivation and exploitation of a reduction of the energy
to finite dimensions (Lemma 5.5) with variables which are closely related to the location of the peaks. We do not assume nondegeneracy
of the points of maximal distance to the boundary but can do with a global condition instead which in many cases is weaker.
Received March 5, 1999 / Accepted June 11, 1999 相似文献
16.
In this article, we prove that there are three positive solutions of a semilinear elliptic equation in a bounded symmetric domain t for large t>0 in which one is axially symmetric and the other two are nonaxially symmetric. Main tools are the Palais-Smale theory. 相似文献
17.
We study the Hamilton-Jacobi equation for undiscounted exit time
control problems with general nonnegative Lagrangians using the
dynamic programming approach. We prove theorems characterizing the
value function as the unique bounded-from-below viscosity solution
of the Hamilton-Jacobi equation that is null on the target. The
result applies to problems with the property that all trajectories
satisfying a certain integral condition must stay in a bounded
set. We allow problems for which the Lagrangian is not uniformly
bounded below by positive constants, in which the hypotheses of
the known uniqueness results for Hamilton-Jacobi equations are not
satisfied. We apply our theorems to eikonal equations from
geometric optics, shape-from-shading equations from image
processing, and variants of the Fuller Problem. 相似文献
18.
C.V.M Van der mee 《Applicable analysis》2013,92(1-4):89-110
For a general class of time dependent linear Boltzmann type equations with (i) an external, non divergence free force terma a ?u/?ξ (ii) a collision term which can be written as the difference of a gain term involving a general nonnegative "collision frequencyn h(x,ξ,t) and a loss term involving an arbitrary bounded linear operator J, and (iii) a general boundary operator K which is a (strict) contraction, the method of characteristics and perturbation techniques are used to obtain the well-posed- ness of the initial-boundary value problem, provided the divergence b of a is bounded above. The functional setting is Lp, 1
o-semigroup on Lp(Σdμ). The results are proven by generalizing a recently established theory of time dependent kinetic equations where the external force is divergence free with respect to velocity. Solutions on spaces of measures are discussed briefly. 相似文献
19.
Elves A. B. Silva Magda S. Xavier 《NoDEA : Nonlinear Differential Equations and Applications》2007,13(5-6):619-642
We study the existence of multiple solutions for a quasilinear elliptic system of gradient type with critical growth and the
possibility of coupling on the subcritical term. The solutions are obtained from a version of the Symmetric Mountain Pass
Theorem. The Concentration-Compactness Principle allows to verify that the Palais-Smale condition is satisfied below a certain
level.
The authors were partially supported by CNPq/Brazil 相似文献
20.
Joachim von Below Helmut Kaul 《Calculus of Variations and Partial Differential Equations》1998,7(1):41-51
We consider semilinear elliptic equations with a principal part degenerating on a boundary hyperplane. Weak existence, uniqueness
and regularity of solutions are established by variational methods and by reduction to uniformly elliptic equations. An important
application arises in the mathematical treatment of the rotating star problem in general relativity, where the axial symmetry
admits the reduction of one of the Einstein equations to a problem of the above form on a meridian half plane.
Received February 12, 1997 / Accepted May 15, 1997 相似文献