Liouville Type Theorems for Elliptic Operators in Divergence Form

Vertical square functions and other operators associated with an elliptic operator

Let A be a strongly elliptic operator of order 2m in divergence form with Holder continuous coefficients of exponent $${\sigma \in (0,1)}$$ defined in a uniformly C 1+σ domain Ω of $${\mathbb{R}^n}$$ . Regarding A as an operator from the Holder space of order m + σ associated with the Dirichlet data to the Holder space of order −m + σ, we show that the inverse (A − λ)−1 exists for λ in a suitable angular region of the complex plane and estimate its operator norms. As an application, we give a regularity theorem for elliptic equations.

First, we will apply the method to study W 1,p regularity for a nonlinear elliptic operator in divergence form. We would like to point out that in the particular case of a linear elliptic equation, this method gives an alternative proof to the classical one, which uses the general theory of singular integrals. The utility of the method that we describe below is that hypotheses (A) and (B) are obtained directly by studying the deviation of the “coefficients” of A from the “coefficients” of A0, and this is usually not a difficult task. A more interesting application of this kind of approximation method is to elliptic homogenization problems for which we obtain results that give W 1,p estimates with a weak hypothesis of regularity in the coefficients. For instance, we are able to study the following cases:

Representation formulae of solutions of second order elliptic inequalities

We consider elliptic operators in divergence form with variable coefficients defined through an accretive sesquilinear form on the whole space. The coefficients of leading order are supposed to be lipschitzian. We show how wavelets bases allow us to explicitly compute the inverse of such operators. The first main ingredient is a detailed study, of independant interest, of the convergence of the usual Galerkin approximations. The second main ingredient is the notion of paraproduct, suitably adapted to our context.

Let ( X , d , μ ) be a metric measure space with a local regular Dirichlet form. We give necessary and sufficient conditions for a parabolic Harnack inequality with global space-time scaling exponent β ≥ 2 to hold. We show that this parabolic Harnack inequality is stable under rough isometries. As a consequence, once such a Harnack inequality is established on a metric measure space, then it holds for any uniformly elliptic operator in divergence form on a manifold naturally defined from the graph approximation of the space.

We study a shape optimization problem for the first eigenvalue of an elliptic operator in divergence form, with nonconstant coefficients, over a fixed domain $\Omega$. Dirichlet conditions are imposed along $\partial \Omega$ and, additionally, along a set $\Sigma$ of prescribed length (one-dimensional Hausdorff measure). We look for the best shape and position for the supplementary Dirichlet region $\Sigma$ in order to maximize the first eigenvalue. The limit distribution of the optimal sets, as their prescribed length tends to infinity, is characterized via $\Gamma$-convergence of suitable functionals defined over varifolds; the use of varifolds, as opposed to probability measures, allows one to keep track of the local orientation of the optimal sets (which comply with the anisotropy of the problem), and not just of their limit distribution.

In this note, we obtain sharp bounds for the Green’s function of the linearized Monge–Ampère operators associated to convex functions with either Hessian determinant bounded away from zero and infinity or Monge–Ampère measure satisfying a doubling condition. Our result is an affine invariant version of the classical result of Littman–Stampacchia–Weinberger for uniformly elliptic operators in divergence form. We also obtain the L p integrability for the gradient of the Green’s function in two dimensions. As an application, we obtain a removable singularity result for the linearized Monge–Ampère equation.

Given a complex, elliptic coefficient function we investigate for which values of p the corresponding second-order divergence form operator, complemented with Dirichlet, Neumann or mixed boundary conditions, generates a strongly continuous semigroup on \(\mathrm {L}^p(\Omega )\). Additional properties like analyticity of the semigroup, \(\mathrm {H}^\infty \)-calculus and maximal regularity are also discussed. Finally, we prove a perturbation result for real coefficients that gives the whole range of p’s for small imaginary parts of the coefficients. Our results are based on the recent notion of p-ellipticity, reverse Hölder inequalities and Gaussian estimates for the real coefficients.

Let Ω a regular open set ofRn, μ a measure with compact support and L a second order elliptic operator in divergence form. If L is coercive we prove a theorem of existence and uniqueness for the solution of Lu=μ, u∈H01+H01,p′(Ω)where p′ is the conjugate of p∈[n, ∞].

Let L be a uniformly elliptic linear second order differential operator in divergence form with bounded measurable real coefficients in a bounded domain G ⊂ ℝn (n ⩾ 2). We define classes of continuous functions in G that contain generalized solutions of the equation Lƒ = 0 and have the property that the compact sets removable for such solutions in these classes can be completely described in terms of Hausdorff measures.

On the atomic decomposition for Hardy spaces on Lipschitz domains of [formula omitted

We propose and analyze several multilevel algorithms for the fast simulation of possibly nonstationary Gaussian random fields (GRFs) indexed, for example, by the closure of a bounded domain [Formula: see text] or, more generally, by a compact metric space [Formula: see text] such as a compact [Formula: see text]-manifold [Formula: see text]. A colored GRF [Formula: see text], admissible for our algorithms, solves the stochastic fractional-order equation [Formula: see text] for some [Formula: see text], where [Formula: see text] is a linear, local, second-order elliptic self-adjoint differential operator in divergence form and [Formula: see text] is white noise on [Formula: see text]. We thus consider GRFs on [Formula: see text] with covariance operators of the form [Formula: see text]. The proposed algorithms numerically approximate samples of [Formula: see text] on nested sequences [Formula: see text] of regular, simplicial partitions [Formula: see text] of [Formula: see text] and [Formula: see text], respectively. Work and memory to compute one approximate realization of the GRF [Formula: see text] on the triangulation [Formula: see text] of [Formula: see text] with consistency [Formula: see text], for some consistency order [Formula: see text], scale essentially linearly in [Formula: see text], independent of the possibly low regularity of the GRF. The algorithms are based on a sinc quadrature for an integral representation of (the application of) the negative fractional-order elliptic “coloring” operator [Formula: see text] to white noise [Formula: see text]. For the proposed numerical approximation, we prove bounds of the computational cost and the consistency error in various norms.

Estimates of the gaps between consecutive eigenvalues for a class of elliptic differential operators in divergence form on Riemannian manifolds

We consider semilinear elliptic second-order partial differential inequalities of the form (*) in the whole space , where , q>0 and L is a linear elliptic second-order partial differential operator in divergence form. We assume that the coefficients of the operator L are measurable and locally bounded such that the quadratic form associated with the operator L is symmetric and non-negative definite. We obtain a Liouville comparison principle in terms of capacities associated with the operator L for solutions of (*) which are measurable and belong locally in to a Sobolev-type function space also associated with the operator L.