共查询到20条相似文献,搜索用时 31 毫秒
1.
Guozhen Lu 《Advances in Mathematics》2008,217(4):1838-1868
We present the theory of the viscosity solutions of the inhomogeneous infinity Laplace equation in domains in Rn. We show existence and uniqueness of a viscosity solution of the Dirichlet problem under the intrinsic condition f does not change its sign. We also discover a characteristic property, which we call the comparison with standard functions property, of the viscosity sub- and super-solutions of the equation with constant right-hand side. Applying these results and properties, we prove the stability of the inhomogeneous infinity Laplace equation with nonvanishing right-hand side, which states the uniform convergence of the viscosity solutions of the perturbed equations to that of the original inhomogeneous equation when both the right-hand side and boundary data are perturbed. In the end, we prove the stability of the well-known homogeneous infinity Laplace equation , which states the viscosity solutions of the perturbed equations converge uniformly to the unique viscosity solution of the homogeneous equation when its right-hand side and boundary data are perturbed simultaneously. 相似文献
2.
Using the mappings which involve first‐order derivatives, the Burgers equation with linear damping and variable viscosity is linearized to several parabolic equations including the heat equation, by applying a method which is a combination of Lie’s classical method and Kawamota’s method. The independent variables of the linearized equations are not t, x but z(x, t), τ(t) , where z is the similarity variable. The linearization is possible only when the viscosity Δ(t) depends on the damping parameter α and decays exponentially for large t . And the linearization makes it possible to pose initial and/or boundary value problems for the Burgers equation with linear damping and exponentially decaying viscosity. Bäcklund transformations for the nonplanar Burgers equation with algebraically decaying viscosity are also reported. 相似文献
3.
4.
We show that viscosity solutions to the normalized p(x)-Laplace equation coincide with distributional weak solutions to the strong p(x)-Laplace equation when p is Lipschitz and \(\inf p>1\). This yields \(\smash {C^{1,\alpha }}\) regularity for the viscosity solutions of the normalized p(x)-Laplace equation. As an additional application, we prove a Radó-type removability theorem. 相似文献
5.
N. I. Insarova 《Mechanics of Composite Materials》1967,3(5):612-615
On the basis of Geisekus's rheological equation of state the viscosity of a non-Newtonian fluid is investigated in relation to the particular mode of deformation (pure shear, axisymmetric deformation, simple shear), The intrinsic viscosity is calculated for pure shear, the following model being used: rigid ellipsoids of revolution uniformly distributed in an incompressible viscous Newtonian fluid. The dependence of the intrinsic viscosity on the parameter =(2/3)(q/D) (q is strain rate, D is the rotational diffusion coefficient) is obtained in specific form for various ratios of the ellipsoid semiaxes.Mekhanika Polimerov, Vol. 3, No. 5, pp. 927–932, 1967 相似文献
6.
Yousong Luo 《Journal of Global Optimization》2008,40(1-3):155-160
We prove uniqueness of the viscosity solutions of the Dirichlet problem of the spectral equation where is the vector whose components are eigenvalues of a matrix associated with the unknown function u. 相似文献
7.
8.
Giulia Cavagnari Antonio Marigonda Khai T. Nguyen Fabio S. Priuli 《Set-Valued and Variational Analysis》2018,26(3):663-691
In this paper we formulate a time-optimal control problem in the space of probability measures. The main motivation is to face situations in finite-dimensional control systems evolving deterministically where the initial position of the controlled particle is not exactly known, but can be expressed by a probability measure on \(\mathbb {R}^{d}\). We propose for this problem a generalized version of some concepts from classical control theory in finite dimensional systems (namely, target set, dynamic, minimum time function...) and formulate an Hamilton-Jacobi-Bellman equation in the space of probability measures solved by the generalized minimum time function, by extending a concept of approximate viscosity sub/superdifferential in the space of probability measures, originally introduced by Cardaliaguet-Quincampoix in Cardaliaguet and Quincampoix (Int. Game Theor. Rev. 10, 1–16, 2008). We prove also some representation results linking the classical concept to the corresponding generalized ones. The main tool used is a superposition principle, proved by Ambrosio, Gigli and Savaré in Ambrosio et al. [3], which provides a probabilistic representation of the solution of the continuity equation as a weighted superposition of absolutely continuous solutions of the characteristic system. 相似文献
9.
Halina Frankowska 《Applied Mathematics and Optimization》1989,19(1):291-311
In this paper we study the existence of optimal trajectories associated with a generalized solution to the Hamilton-Jacobi-Bellman equation arising in optimal control. In general, we cannot expect such solutions to be differentiable. But, in a way analogous to the use of distributions in PDE, we replace the usual derivatives with contingent epiderivatives and the Hamilton-Jacobi equation by two contingent Hamilton-Jacobi inequalities. We show that the value function of an optimal control problem verifies these contingent inequalities.Our approach allows the following three results: (a) The upper semicontinuous solutions to contingent inequalities are monotone along the trajectories of the dynamical system. (b) With every continuous solutionV of the contingent inequalities, we can associate an optimal trajectory along whichV is constant. (c) For such solutions, we can construct optimal trajectories through the corresponding optimal feedback.They are also viscosity solutions of a Hamilton-Jacobi equation. Finally, we prove a relationship between superdifferentials of solutions introduced by Crandallet al. [10] and the Pontryagin principle and discuss the link of viscosity solutions with Clarke's approach to the Hamilton-Jacobi equation. 相似文献
10.
Albert Fathi Antonio Siconolfi 《Calculus of Variations and Partial Differential Equations》2005,22(2):185-228
We propose a PDE approach to the Aubry-Mather theory using viscosity solutions. This allows to treat Hamiltonians (on the flat torus
) just coercive, continuous and quasiconvex, for which a Hamiltonian flow cannot necessarily be defined. The analysis is focused on the family of Hamilton-Jacobi equations
with a real parameter, and in particular on the unique equation of the family, corresponding to the so-called critical value a = c, for which there is a viscosity solution on
. We define generalized projected Aubry and Mather sets and recover several properties of these sets holding for regular Hamiltonians.Received: 23 November 2003, Accepted: 3 March 2004, Published online: 12 May 2004 相似文献
11.
Summary. We consider the finite-difference space semi-discretization of a locally damped wave equation, the damping being supported in a suitable subset of the domain under consideration, so that the energy of solutions of the damped wave equation decays exponentially to zero as time goes to infinity. The decay rate of the semi-discrete systems turns out to depend on the mesh size h of the discretization and tends to zero as h goes to zero. We prove that adding a suitable vanishing numerical viscosity term leads to a uniform (with respect to the mesh size) exponential decay of the energy of solutions. This numerical viscosity term damps out the high frequency numerical spurious oscillations while the convergence of the scheme towards the original damped wave equation is kept. We discuss this problem in 1D and 2D in the interval and the square respectively. Our method of proof relies on discrete multiplier techniques.
Mathematics Subject Classification (1991):65M06 相似文献
12.
We consider a generalization of the Stokes resolvent equation, where the constant viscosity is replaced by a general given
positive function. Such a system arises in many situations as linearized system, when the viscosity of an incompressible,
viscous fluid depends on some other quantities. We prove that an associated Stokes-like operator generates an analytic semi-group
and admits a bounded H
∞
-calculus, which implies the maximal L
q
-regularity of the corresponding parabolic evolution equation. The analysis is done for a large class of unbounded domains
with -boundary for some r > d with r ≥ q, q′. In particular, the existence of an L
q
-Helmholtz projection is assumed. 相似文献
13.
The initial-boundary-value problem for the Kadomtsev-Petviashvili equation in infinite space is considered. When formulated as an evolution equation, found that a symmetric integral is the appropriate choice in the nonlocal term; namely, . If one simply chooses , then an infinite number of constraints on the initial data in physical space are required, the first being . The conserved quantities are calculated, and it is shown that they must be suitably regularized from those that have been used when the constraints are imposed. 相似文献
14.
In this paper, we study a stochastic recursive optimal control problem in which the objective functional is described by the solution of a backward stochastic differential equation driven by \(G\) -Brownian motion. Under standard assumptions, we establish the dynamic programming principle and the related Hamilton–Jacobi–Bellman (HJB) equation in the framework of \(G\) -expectation. Finally, we show that the value function is the viscosity solution of the obtained HJB equation. 相似文献
15.
Goro Akagi Kazumasa Suzuki 《Calculus of Variations and Partial Differential Equations》2008,31(4):457-471
The existence, uniqueness and regularity of viscosity solutions to the Cauchy–Dirichlet problem are proved for a degenerate
nonlinear parabolic equation of the form , where denotes the so-called infinity-Laplacian given by . To do so, a coercive regularization of the equation is introduced and barrier function arguments are also employed to verify
the equi-continuity of approximate solutions. Furthermore, the Cauchy problem is also studied by using the preceding results
on the Cauchy–Dirichlet problem.
Dedicated to the memory of our friend Kyoji Takaichi.
The research of the first author was partially supported by Waseda University Grant for Special Research Projects, #2004A-366. 相似文献
16.
Summary The problem of the construction of an equilibrium surface taking the surface tension into account leads to Laplace-Young equation which is a nonlinear elliptic free-boundary problem. In contrast to Orr et al. where an iterative technique is used for direct solution of the equation for problems with simple geometry, we propose here an alternative approach based on shape optimization techniques. The shape of the domain of the liquid is varied to attain the optimality condition. Using optimal control theory to derive expressions for the gradient, a numerical scheme is proposed and simple model problems are solved to validate the scheme. 相似文献
17.
Bui An Ton 《Mathematical Methods in the Applied Sciences》1988,10(4):351-366
Let g is a positive increasing function with 1?g(0). The existence of a unique solution of the Navier-Stokes flow associated with Kε,γ and the convergence of the solution to that of the Euler equations as the viscosity goes to zero are established. 相似文献
18.
In this article, we are interested in uniqueness results for viscosity solutions of a general class of quasilinear, possibly degenerate, parabolic equations set in
. Using classical viscosity solutions' methods, we obtain a general comparison result for solutions with polynomial growths but with a restriction on the growth of the initial data. The main application is the uniqueness of solutions for the mean curvature equation for graphs which was only known in the class of uniformly continuous functions. An application to the mean curvature flow is given.Received: 1 December 2001, Accepted: 30 September 2002, Published online: 17 December 2002Mathematics Subject Classification:
35A05, 35B05, 35D05, 35K15, 35K55, 53C44This work was partially supported by the TMR program "Viscosity solutions and their applications." 相似文献
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20.
In this paper we present a linear representation of the
Landau-Lifshitz-Gilbert equation for describing the magnetization
of ferromagnetic materials. According to Lies theory, we prove
that this equation admits a superposition principle and its
formula is derived. The underlying vector space of the
Landau-Lifshitz-Gilbert equation is found to be a projective
Minkowski space denoted by
of which the
projective proper orthochronous Lorentz group
PSO
o(3,1)
left acts. By the Lie symmetry a group preserving scheme is developed,
which improves the computational accuracy and efficiency. 相似文献