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1.
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2.
Using an Orlicz–Sobolev Space setting, we consider an eigenvalue problem for a system of the form
We prove that the solution to a suitable minimizing problem, with a restriction, yields a solution to this problem for a certain λ. The differential operators involved lack homogeneity and in addition the Orlicz–Sobolev spaces needed may not be reflexive and the corresponding functional in the minimization problem is in general neither everywhere defined nor a fortiori C 1.  相似文献   

3.
For a bounded domain and , assume that is convex and coercive, and that has no interior points. Then we establish the uniqueness of viscosity solutions to the Dirichlet problem of Aronsson’s equation:
For H = H(p, x) depending on x, we illustrate the connection between the uniqueness and nonuniqueness of viscosity solutions to Aronsson’s equation and that of the Hamilton–Jacobi equation . Supported by NSF DMS 0601162. Supported by NSF DMS 0601403.  相似文献   

4.
Existence of a Solution “in the Large” for Ocean Dynamics Equations   总被引：1，自引：0，他引：1  
For the system of equations describing the large-scale ocean dynamics, an existence and uniqueness theorem is proved “in the large”. This system is obtained from the 3D Navier–Stokes equations by changing the equation for the vertical velocity component u 3 under the assumption of smallness of a domain in z-direction, and a nonlinear equation for the density function ρ is added. More precisely, it is proved that for an arbitrary time interval [0, T], any viscosity coefficients and any initial conditions
a weak solution exists and is unique and and the norms are continuous in t. The work was carried out under partial support of Russian Foundation for Basic Research (project 05-01-00864).  相似文献   

5.
We study the limit of the hyperbolic–parabolic approximation
The function is defined in such a way as to guarantee that the initial boundary value problem is well posed even if is not invertible. The data and are constant. When is invertible, the previous problem takes the simpler form
Again, the data and are constant. The conservative case is included in the previous formulations. Convergence of the , smallness of the total variation and other technical hypotheses are assumed, and a complete characterization of the limit is provided. The most interesting points are the following: First, the boundary characteristic case is considered, that is, one eigenvalue of can be 0. Second, as pointed out before, we take into account the possibility that is not invertible. To deal with this case, we take as hypotheses conditions that were introduced by Kawashima and Shizuta relying on physically meaningful examples. We also introduce a new condition of block linear degeneracy. We prove that, if this condition is not satisfied, then pathological behaviors may occur.  相似文献   

6.
We study the dynamics and regularity of level sets in solutions of the semilinear parabolic equation
where is a ring-shaped domain, a and μ are given positive constants, is the Heaviside maximal monotone graph: if s > 0, if s < 0. Such equations arise in climatology (the so-called Budyko energy balance model), as well as in other contexts such as combustion. We show that under certain conditions on the initial data the level sets are n-dimensional hypersurfaces in the (x, t)-space and show that the dynamics of Γ μ is governed by a differential equation which generalizes the classical Darcy law in filtration theory. This differential equation expresses the velocity of advancement of the level surface Γ μ through spatial derivatives of the solution u. Our approach is based on the introduction of a local set of Lagrangian coordinates: the equation is formally considered as the mass balance law in the motion of a fluid and the passage to Lagrangian coordinates allows us to watch the trajectory of each of the fluid particles.  相似文献   

7.
Let be the exterior of the closed unit ball. Consider the self-similar Euler system
Setting α = β = 1/2 gives the limiting case of Leray’s self-similar Navier–Stokes equations. Assuming smoothness and smallness of the boundary data on ∂Ω, we prove that this system has a unique solution , vanishing at infinity, precisely
The self-similarity transformation is v(x, t) = u(y)/(t* − t)α, y = x/(t* − t)β, where v(x, t) is a solution to the Euler equations. The existence of smooth function u(y) implies that the solution v(x, t) blows up at (x*, t*), x* = 0, t* < + ∞. This isolated singularity has bounded energy with unbounded L 2 − norm of curl v.  相似文献   

8.
In this paper, we consider the following PDE involving two Sobolev–Hardy critical exponents,
$ \label{0.1}\left\{\begin{aligned}& \Delta u + \lambda\frac{u^{2^*(s_1)-1}}{|x|^{s_1}} + \frac{u^{2^*(s_2)-1}}{|x|^{s_2}} =0 \quad \rm {in}\,\,\Omega,\quad\quad\quad(0.1)\\ & u=0 \quad {\rm on }\,\,\Omega, \end{aligned} \right.$ \label{0.1}\left\{\begin{aligned}& \Delta u + \lambda\frac{u^{2^*(s_1)-1}}{|x|^{s_1}} + \frac{u^{2^*(s_2)-1}}{|x|^{s_2}} =0 \quad \rm {in}\,\,\Omega,\quad\quad\quad(0.1)\\ & u=0 \quad {\rm on }\,\,\Omega, \end{aligned} \right.  相似文献   

9.
We are concerned with the existence of a weak solution to the degenerate quasi-linear Dirichlet boundary value problem
It is assumed that 1  <  p  <  ∞, p  ≠  2, Ω is a bounded domain in is a given function, and λ stands for the (real) spectral parameter near the first (smallest) eigenvalue λ1 of the positive p-Laplacian  − Δ p , where . Eigenvalue λ1 being simple, let φ1 denote the eigenfunction associated with it. We show the existence of a solution for problem (P) when f “nearly” satisfies the orthogonality condition ∫Ω f φ1  dx  =  0 and λ  ≤  λ1  +  δ (with δ >  0 small enough). Moreover, we obtain at least three distinct solutions if either p < 2 and λ1  −  δ ≤  λ  <  λ1, or else p > 2 and λ1  <  λ  ≤  λ1  +  δ. The proofs use a minimax principle for the corresponding energy functional performed in the orthogonal decomposition induced by the inner product in L 2(Ω). First, the global minimum is taken over , and then either a local minimum or a local maximum over lin {φ1}. If the latter is a local minimum, the local minimizer in thus obtained provides a solution to problem (P). On the other hand, if it is a local maximum, one gets only a pair of sub- and supersolutions to problem (P), which is then used to obtain a solution by a topological degree argument.  相似文献   

10.
For , we consider a family of damped wave equations , where − Λ denotes the Laplacian with zero Dirichlet boundary condition in L 2(Ω). For a dissipative nonlinearity f satisfying a suitable growth restrictions these equations define on the phase space semigroups which have global attractors A η, . We show that the family , behaves upper and lower semicontinuously as the parameter η tends to 0+.  相似文献   

11.
Let E be a Banach space. We prove the instability of the trivial solution of the differential equation
where f: [0, +∞) × E → ℝ is a continuous mapping for which
__________ Translated from Neliniini Kolyvannya, Vol. 8, No. 3, pp. 404–414, July–September, 2005.  相似文献   

12.
We establish a general weak* lower semicontinuity result in the space BD(Ω) of functions of bounded deformation for functionals of the form
$ {ll} \,\mathcal{F}(u) := &\int_\Omega f (x, \mathcal{E} u) \;{\rm d} x + \int_\Omega f^\infty \left( x, \frac{{\rm d} E^s u}{{\rm d} |{E^s u}|} \right) \;{\rm d} |{E^s u}| \\ &+ \int_{\partial \Omega} f^\infty \left( x, u|_{\partial \Omega} \odot n_\Omega \right) \;{\rm d} \mathcal{H}^{d-1}, \qquad u \in {\rm BD}(\Omega). $ \begin{array}{ll} \,\mathcal{F}(u) := &\int_\Omega f (x, \mathcal{E} u) \;{\rm d} x + \int_\Omega f^\infty \left( x, \frac{{\rm d} E^s u}{{\rm d} |{E^s u}|} \right) \;{\rm d} |{E^s u}| \\ &+ \int_{\partial \Omega} f^\infty \left( x, u|_{\partial \Omega} \odot n_\Omega \right) \;{\rm d} \mathcal{H}^{d-1}, \qquad u \in {\rm BD}(\Omega). \end{array}  相似文献   

13.
In this paper, first a class of fractional differential equations are obtained by using the fractional variational principles. We find a fractional Lagrangian L(x(t), where a c D t α x(t)) and 0<α<1, such that the following is the corresponding Euler–Lagrange
(1)
At last, exact solutions for some Euler–Lagrange equations are presented. In particular, we consider the following equations
(2)
(3)
where g(t) and f(t) are suitable functions. D. Baleanu is on leave of absence from Institute of Space Sciences, P.O. BOX MG-23, 76900 Magurele-Bucharest, Romania. e-mail: baleanu@venus.nipne.ro.  相似文献   

14.
Consider the problem where Ω is a bounded convex domain in , N > 2, with smooth boundary . We study the asymptotic behaviour of the least energy solutions of this system as . We show that the solution remain bounded for p large. In the limit, we find that the solution develops one or two peaks away from the boundary, and when a single peak occurs, we have a characterization of its location.This research was supported by FONDECYT 1061110 and 3040059.  相似文献   

15.
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16.
We investigate the dynamics of the semiflow φ induced on H01(Ω) by the Cauchy problem of the semilinear parabolic equation
on Ω. Here is a bounded smooth domain, and has subcritical growth in u and satisfies . In particular we are interested in the case when f is definite superlinear in u. The set
of attraction of 0 contains a decreasing family of invariant sets
distinguished by the rate of convergence . We prove that the Wk’s are global submanifolds of , and we find equilibria in the boundaries . We also obtain results on nodal and comparison properties of these equilibria. In addition the paper contains a detailed exposition of the semigroup approach for semilinear equations, improving earlier results on stable manifolds and asymptotic behavior near an equilibrium.Supported by DFG Grant BA 1009/15-1.  相似文献   

17.
Consider an incompressible, hyperelastic material occupying the unit ball B⊂ℝ n in its reference state. Suppose that the deformation u:B→ℝ n is specified on the boundary by
where λ>1 is a given constant. In this paper, isoperimetric arguments are used to prove that the radial deformation, producing a spherical cavity, is the energy minimiser in a general class of isochoric mappings that are discontinuous at the centre of the ball and produce a (possibly non-symmetric) cavity in the deformed body. This result has implications for the study of cavitation in certain polymers. This work was supported in part by the National Science Foundation under Grant No. DMS–0405646.  相似文献   

18.
We consider the nonlinear elliptic system
where and is the unit ball. We show that, for every and , the above problem admits a radially symmetric solution (u β , v β ) such that u β − v β changes sign precisely k times in the radial variable. Furthermore, as , after passing to a subsequence, u β → w + and v β → w − uniformly in , where w = w +− w − has precisely k nodal domains and is a radially symmetric solution of the scalar equation Δw − w + w 3 = 0 in , w = 0 on . Within a Hartree–Fock approximation, the result provides a theoretical indication of phase separation into many nodal domains for Bose–Einstein double condensates with strong repulsion.  相似文献   

19.
We consider the Cauchy problem for a strictly hyperbolic, N × N quasilinear system in one-space dimension
where , is a smooth matrix-valued map and the initial data is assumed to have small total variation. We present a front tracking algorithm that generates piecewise constant approximate solutions converging in to the vanishing viscosity solution of (1), which, by the results in [6], is the unique limit of solutions to the (artificial) viscous parabolic approximation
as . In the conservative case where A(u) is the Jacobian matrix of some flux function F(u) with values in , the limit of front tracking approximations provides a weak solution of the system of conservation laws u t + F(u) x = 0, satisfying the Liu admissibility conditions. These results are achieved under the only assumption of strict hyperbolicity of the matrices A(u), . In particular, our construction applies to general, strictly hyperbolic systems of conservation laws with characteristic fields that do not satisfy the standard conditions of genuine nonlinearity or of linear degeneracy in the sense of Lax[17], or in the generalized sense of Liu[23]. Dedicated to Prof. Tai Ping Liu on the occasion of his 60 th birthday  相似文献   

20.
This paper examines the asymptotic behavior of measure valued solutions to the initial value problem for the nonlinear heat conduction equation
  相似文献   

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