1,721,008 research outputs found
Rate of convergence for multiscale homogenization of Hamilton-Jacobi equations
No full textThis paper concerns the homogenization problem for fully nonlinear first-order equations of Hamilton-Jacobi type with a finite number of scales. Under some coercivity and periodicity assumptions we provide an estimate of the rate of convergence. Finally, some examples arising from optimal control and deterministic differential games are discussed
Well-posedness for parabolic equations of arbitrary order
No full textThis paper deals with the well-posedness of the Cauchy problem for higher order parabolic equations. Our aim is to show existence and uniqueness of the solution belonging to a suitable weighted Sobolev space, provided that the weight function satisfies some appropriate differential inequality (the "dual" one). Under some restrictions on the growth of the coefficients as \x\-->infinity (see conditions (A(1))-(A(4)) below), we obtain a simplified dual inequality; we deduce a well-posedness result which extends results known in literature. In Appendix, dropping any growth condition on the coefficients, we extend our result, but the dual inequality is complicated
The Cauchy problem for the Heat Equation with a Singular Potential
No full textThe aim of this paper is to investigate the well-posedness of
the Cauchy problem
∂_t u=∆u +V(x) u in R^N × (0, T ), N ≥ 3
u(x,0)=u_0(x) on R^N
where the potential V is defined by V=V(x) := λ/|x|^2 , 0≤λ 0, the solution shall present
a lack of regularity in the origin which is due only to the presence of the
singular potential
Continuous dependence estimates for the ergodic problem of Bellman equation with an application to the rate of convergence for the homogenization problem
No full textThis paper is devoted to establish continuous dependence estimates for the ergodic problem for Bellman operators (namely, estimates of (v1 −v2 ) where v1 and v2 solve two equations with different coefficients). We shall obtain an estimate of ||v1 − v2||_∞ with an explicit dependence on the L^∞ -distance between the coefficients and an explicit characterization of the constants and also, under some regularity conditions, an estimate of (v1 − v2) in C 2(Rn )-norm .
Afterwards, the former result will be crucial in the estimate of the rate of convergence for the homogenization of Bellman equations. In some regular cases, we shall obtain the same rate of convergence established in the monographs by Bensoussan et al. (Asymptotic analysis for periodic structures, North-Holland, Amsterdam, 1978) and by Jikov et al. (Homogenization of differential operators and integral functionals, Springer, Berlin, 1994) for regular linear problems
On the convergence of singular perturbations of Hamilton-Jacobi equations
No full textThis paper is devoted to singular perturbation problems for first order equations. Under some coercivity and periodicity assumptions, we establish the uniform convergence and we provide an estimate of the rate of convergence, which we consider the main result of the paper.
We shall also show that our results apply to the homogenization problem for coercive and periodic equations. Finally, some examples arising in optimal control and differential games theory will be discussed
Homogenization for fully nonlinear parabolic equations
No full textThis paper concerns the homogenization of fully nonlinear parabolic equations of the form
partial derivative(t)u(epsilon) + H (t, x, t/epsilon(2), x/epsilon, D(2)u(epsilon)) = 0 in (0, T)xR-n,
where the Hamiltonian H(t, x, tau, xi, X) is periodic both in tau and xi. Our aim is to establish sufficient conditions for the convergence (as epsilon goes to 0) of u(epsilon) to a solution a to the effective equation
partial derivative(t)u + H(t, x, D(2)u) = 0 in (0, T)xR-n,
where the effective Hamiltonian H is obtained by a parabolic equation called cell problem. We shall prove that H inherits several properties of H. We also consider the case that: u(epsilon)(0, x) = h (x, x/epsilon) on R-n; we point out a sufficient condition for having u(0, x) = h(x) on R-n,with an effective initial datum h given by the asymptotic behaviour of the solution to the recession problem (a parabolic Cauchy problem related to (1.1))
Continuous dependence estimates for the ergodic problem of Bellman-Isaacs operators via the parabolic Cauchy problem
Full text linkThis paper concerns continuous dependence estimates for Hamilton-Jacobi-Bellman-Isaacs operators. We establish such an estimate for the parabolic Cauchy problem in the whole space [0, +∞)×R^n and, under some periodicity and either ellipticity or controllability assumptions, we deduce a similar estimate for the ergodic constant associated to the operator. An interesting byproduct of the latter result will be the local uniform convergence for some classes of singular perturbation problems
Stationary Mean Field Games Systems Defined on Networks
Full text linkWe consider a stationary mean field games system constrained on a network. According to the optimal control interpretation of the problem, some transition conditions on the vertices are imposed. We prove separately the well-posedness for each of the two equations composing the system. Finally, we prove existence and uniqueness of the solution of the mean field games system
On the convergence rate in multiscale homogenization of fully nonlinear elliptic problems
No full textThis paper concerns periodic multiscale homogenization for fully nonlinear equations of the form u(epsilon) + H(epsilon) (x, x/epsilon, ..., x/epsilon(h), Du(epsilon), D(2)u(epsilon)) = 0. The operators H(epsilon) are a regular perturbations of some uniformly elliptic, convex operator H(epsilon). As epsilon goes to 0, the solutions u(epsilon) converge locally uniformly to the solution u of a suitably defined effective problem. The purpose of this paper is to obtain an estimate of the corresponding rate of convergence. Finally, some examples are discussed
Rates of convergence in periodic homogenization offully nonlinear uniformly elliptic PDEs
No full textWe consider periodic homogenization of the fully nonlinear uniformly elliptic equation u(epsilon) + H(x, x/epsilon, Du(epsilon), D(2)u(epsilon)) = 0. We give an estimate of the rate of convergence of u(epsilon) to the solution u of the homogenized problem u + H(x, Du, D(2)u) = 0. Moreover we describe a numerical scheme for the approximation of the effective nonlinearity H and we estimate the corresponding rate of convergence
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