L p -theory for second-order elliptic operators with unbounded coefficients in an endpoint class

The m-accretivity and m-sectoriality of the minimal and maximal realizations of second-order elliptic operators of the form ${Au=-{\rm div}(a \nabla u)+F\cdot \nabla u +Vu}$ in ${L^p(\mathbb{R}^N)}$ are shown, where the coefficients a, F and V are unbounded. The result may be regarded as an endpoint assertion of the previous result in Sobajima (J Evol Equ 12:957–971, 2012) and an improvement of that in Metafune et al. (Forum Math 22:583–601, 2010). Moreover, an L ^p -generalization of Kato’s self-adjoint problem in Kato (1981, Appendix 2) is discussed. The proof is based on Sobajima (J Evol Equ 12:957–971, 2012). As examples, the operators ${-\Delta \pm |x|^{\beta-1}x \cdot \nabla +c|x|^{\gamma}}$ are also dealt with, which are mentioned in Metafune et al. (Forum Math 22:583–601, 2010).     

L p -regularity for parabolic operators with unbounded time-dependent coefficients

We establish the maximal regularity for nonautonomous Ornstein–Uhlenbeck operators in L ^p -spaces with respect to a family of invariant measures, where ${p \in (1, +\infty)}We establish the maximal regularity for nonautonomous Ornstein–Uhlenbeck operators in L ^p -spaces with respect to a family of invariant measures, where p ? (1, +￥){p \in (1, +\infty)} . This result follows from the maximal L ^p -regularity for a class of elliptic operators with unbounded, time-dependent drift coefficients and potentials acting on L^p(\mathbbR^N ){L^{p}(\mathbb{R}^{N} )} with Lebesgue measure.     

$$L^{p}$$-theory of elliptic differential operators with unbounded coefficients

In this paper, we consider the generation of strongly continuous analytic semigroups on \(L^p((0,\omega ),\mu _{p}\, dx)\) and \(L^p((0,\omega ), dx), 1<p<\infty \), by a family of second order elliptic operators of the form

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As in [24], we shall prove the generation results on \(L^2\)-spaces using the sesquilinear forms. More general results are obtained by using interpolation procedure and Neuberger’s theorem.

     

Quasi-L p -contractive analytic semigroups generated by elliptic operators with complex unbounded coefficients on arbitrary domains

Let ?? be a domain in ? ^N and consider a second order linear partial differential operator A in divergence form on ?? which is not required to be uniformly elliptic and whose coefficients are allowed to be complex, unbounded and measurable. Under rather general conditions on the growth of the coefficients we construct a quasi-contractive analytic semigroup $(e^{-t A_{V}})_{t\geqslant0}$ on L ²(??,dx), whose generator A _V gives an operator realization of A under general boundary conditions. Under suitable additional conditions on the imaginary parts of the diffusion coefficients, we prove that for a wide class of boundary conditions, the semigroup $(e^{-t A_{V}})_{t\geqslant0}$ is quasi-L ^p -contractive for 1<p<??. Similar results hold for second order nondivergence form operators whose coefficients satisfy conditions similar to those on the coefficients of the operator A, except for some further requirements on the diffusion coefficients. Some examples where our results can be applied are provided.     

Cores for parabolic operators with unbounded coefficients

Let be a family of elliptic differential operators with unbounded coefficients defined in RN+1. In [M. Kunze, L. Lorenzi, A. Lunardi, Nonautonomous Kolmogorov parabolic equations with unbounded coefficients, Trans. Amer. Math. Soc., in press], under suitable assumptions, it has been proved that the operator G:=A−Ds generates a semigroup of positive contractions (Tp(t)) in Lp(RN+1,ν) for every 1?p<+∞, where ν is an infinitesimally invariant measure of (Tp(t)). Here, under some additional conditions on the growth of the coefficients of A, which cover also some growths with an exponential rate at ∞, we provide two different cores for the infinitesimal generator Gp of (Tp(t)) in Lp(RN+1,ν) for p∈[1,+∞), and we also give a partial characterization of D(Gp). Finally, we extend the results so far obtained to the case when the coefficients of the operator A are T-periodic with respect to the variable s for some T>0.     

Functional calculi of second-order elliptic partial differential operators with bounded measurable coefficients

Consider a second-order elliptic partial differential operatorL in divergence form with real, symmetric, bounded measurable coefficients, under Dirichlet or Neumann conditions on the boundary of a strongly Lipschitz domain Ω. Suppose that 1 <p < ∞ and μ > 0. ThenL has a bounded H_∞ functional calculus in L^p(Ω), in the sense that ¦¦f (L +cI)u¦¦_p ≤C sup_¦arλ¦<μ ^¦f¦ ¦‖u¦‖_p for some constantsc andC, and all bounded holomorphic functionsf on the sector ¦ argλ¦ < μ that contains the spectrum ofL +cI. We prove this by showing that the operatorsf(L + cI) are Calderón-Zygmund singular integral operators.     

Gaussian estimates for elliptic operators with unbounded drift

We consider a strictly elliptic operator

     

Fredholm differential operators with unbounded coefficients

We prove that a first-order linear differential operator G with unbounded operator coefficients is Fredholm on spaces of functions on with values in a reflexive Banach space if and only if the corresponding strongly continuous evolution family has exponential dichotomies on both and and a pair of the ranges of the dichotomy projections is Fredholm, and that the Fredholm index of G is equal to the Fredholm index of the pair. The operator G is the generator of the evolution semigroup associated with the evolution family. In the case when the evolution family is the propagator of a well-posed differential equation u′(t)=A(t)u(t) with, generally, unbounded operators , the operator G is a closure of the operator . Thus, this paper provides a complete infinite-dimensional generalization of well-known finite-dimensional results by Palmer, and by Ben-Artzi and Gohberg.     

Dirichlet boundary conditions for elliptic operators with unbounded drift

We study the realisation of the operator in with Dirichlet boundary condition, where is a possibly unbounded open set in , is a semi-convex function and the measure lets be formally self-adjoint. The main result is that at is a dissipative self-adjoint operator in .

     

Symmetry results for nonlinear elliptic operators with unbounded drift

We prove the one-dimensional symmetry of solutions to elliptic equations of the form ?div(e ^G(x) a(|?u|)?u) = f(u) e ^G(x), under suitable energy conditions. Our results holds without any restriction on the dimension of the ambient space.     

Efficient inversion of the Galerkin matrix of general second-order elliptic operators with nonsmooth coefficients

This article deals with the efficient (approximate) inversion of finite element stiffness matrices of general second-order elliptic operators with -coefficients. It will be shown that the inverse stiffness matrix can be approximated by hierarchical matrices ( -matrices). Furthermore, numerical results will demonstrate that it is possible to compute an approximate inverse with almost linear complexity.

     

The ergodic problem for some subelliptic operators with unbounded coefficients

We study existence and uniqueness of the invariant measure for stochastic processes with degenerate diffusion, whose infinitesimal generators are linear subelliptic operators in the whole space \({{\mathbb{R}}^N}\) with possibly unbounded coefficients. Such a measure together with a Liouville-type theorem will play a crucial role in two applications: the ergodic problem studied through stationary problems with vanishing discount and the long time behavior of the solution to a parabolic Cauchy problem. In both cases, the constants will be characterized in terms of the invariant measure.     

Estimates of the derivatives for parabolic operators with unbounded coefficients

We consider a class of second-order uniformly elliptic operators with unbounded coefficients in . Using a Bernstein approach we provide several uniform estimates for the semigroup generated by the realization of the operator in the space of all bounded and continuous or Hölder continuous functions in . As a consequence, we obtain optimal Schauder estimates for the solution to both the elliptic equation (0$">) and the nonhomogeneous Dirichlet Cauchy problem . Then, we prove two different kinds of pointwise estimates of that can be used to prove a Liouville-type theorem. Finally, we provide sharp estimates of the semigroup in weighted -spaces related to the invariant measure associated with the semigroup.

     

L p -theory for the Stefan problem

Local solvability of the one-phase Stefan problem is established in anisotropic Sobolev spaces. There is no loss of regularity. Hanzawa transformation of the Stefan problem to a problem in a domain with a fixed boundary is modified. Bibliography: 27 titles. Dedicated to the memory of A. P. Oskolkov Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 243, 1997, pp. 299–323. Translated by E. V. Frolova.