Order Elliptic Operators Research Articles

Positive solutions of Schrödinger equations and fine regularity of boundary points

Given a Lipschitz domain $\Omega $ in ${\mathbb R} ^N $ and a nonnegative potential $V$ in $\Omega $ such that $V(x)\, d(x,\partial \Omega)^2$ is bounded in $\Omega $ we study the fine regularity of boundary points with respect to the Schr\"odinger operator $L_V:= \Delta -V$ in $\Omega $. Using potential theoretic methods, several conditions equivalent to the fine regularity of $z \in \partial \Omega $ are established. The main result is a simple (explicit if $\Omega $ is smooth) necessary and sufficient condition involving the size of $V$ for $z$ to be finely regular. An essential intermediate result consists in a majorization of $\int_A | {\frac {u} {d(.,\partial \Omega)}} | ^2\, dx$ for $u$ positive harmonic in $\Omega $ and $A \subset \Omega $. Conditions for almost everywhere regularity in a subset $A $ of $ \partial \Omega $ are also given as well as an extension of the main results to a notion of fine ${\mathcal L}_1 | {\mathcal L}_0$-regularity, if ${\mathcal L}_j={\mathcal L}-V_j$, $V_0,\, V_1$ being two potentials, with $V_0 \leq V_1$ and ${\mathcal L}$ a second order elliptic operator.

Kernel and eigenfunction estimates for some second order elliptic operators

For a potential V such that V ( x ) ⩾ | x | α with α > 2 we prove that the heat kernel k t ( x , y ) associated to the uniformly elliptic operator A = − ∑ j , k = 1 n ∂ k ( a j k ∂ j ) + V satisfies the estimate k t ( x , y ) ⩽ C e − μ 0 t e c t − b ( e − 2 θ α + 2 | x | 1 + α 2 | x | α 4 + n − 1 2 ) ( e − 2 θ α + 2 | y | 1 + α 2 | y | α 4 + n − 1 2 ) for large x , y ∈ R n and all t > 0 . Here 0 < θ ⩽ 1 is an appropriate constant, b > α + 2 α − 2 and μ 0 is the first eigenvalue of A. We also obtain an estimate for large | x | of the eigenfunctions of A.

Sharp spectral stability estimates via the Lebesgue measure of domains for higher order elliptic operators

We prove sharp stability estimates for the variation of the eigenvalues of non-negative self-adjoint elliptic operators of arbitrary even order upon variation of the open sets on which they are defined. These estimates are expressed in terms of the Lebesgue measure of the symmetric difference of the open sets. Both Dirichlet and Neumann boundary conditions are considered.

Quantitative uniqueness for second order elliptic operators with strongly singular coefficients

In this paper we study the local behavior of a solution to second order elliptic operators with sharp singular coefficients in lower order terms. One of the main results is the bound on the vanishing order of the solution, which is a quantitative estimate of the strong unique continuation property. Our proof relies on Carleman estimates with carefully chosen phases. A key strategy in the proof is to derive doubling inequalities via three-sphere inequalities. Our method can also be applied to certain elliptic systems with similar singular coefficients.

Hp‐FEM for second moments of elliptic PDEs with stochastic data. I. Analytic regularity

AbstractFor a linear second order elliptic partial differential operator A : V → V′, we consider the boundary value problems Au = f with stationary Gaussian random data f over the dual V′ of the separable Hilbert space V in which the solution u is sought. The operator A is assumed to be deterministic and bijective. The unique solution u = A−1f is a Gaussian random field over V. It is characterized by its mean field Eu and its covariance Cu ∈ V ⊗ V. For a class of Gaussian data f with piecewise analytic covariance kernels Cf ∈ V′ ⊗ V′, we characterize analytic regularity of the covariance Cu of the Gaussian solution u in families of countably normed, weighted Sobolev spaces. To this end, we investigate shift theorems for the (nonhypoelliptic) deterministic tensor PDEs (A ⊗ A)Cu = Cf proposed by Schwab and Todor, (Numer Math 95 (2003), 707–734) for the computation of covariance Cu of the random solution u. The nonhypoelliptic nature of A ⊗ A implies that sing supp(Cu) is in general strictly larger than sing supp(Cf). For a model problem, analyticity and singular support of the solution is characterized completely. Based on our regularity results, we outline an hp‐finite element strategy (Pentenrieder, Ph.D thesis, ETHZurich, Dissertation No. 18729, 2009; Pentenrieder and Schwab, Research Report 2010‐08, Seminar for Applied Mathematics, ETH Zürich, Submitted) to approximate Cu stemming from covariances of stationary Gaussian data f. In the second part (Pentenrieder and Schwab, Research Report 2010–08, Seminar for Applied Mathematics, ETH Zürich, Submitted) of this work, we prove that this discretization gives exponential rates of convergence of the FE approximations, in terms of the number of degrees of freedom. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2012

Hp‐FEM for second moments of elliptic PDEs with stochastic data. II: Exponential convergence for stationary singular covariance functions

AbstractWe prove exponential rates of convergence of a class of hp Galerkin Finite Element approximations of solutions to a model tensor nonhypoelliptic equation in the unit square □ = (0, 1)2 which exhibit singularities on ∂□ and on the diagonal Δ = {(x, y) ∈ □ : x = y}, but are otherwise analytic in □. As we explained in the first part (Pentenrieder and Schwab, Research Report, Seminar for Applied Mathematics, 2010) of this work, such problems arise as deterministic second moment equations of linear, second order elliptic operator equations Au = f with Gaussian random field data f. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2012

On uniqueness problems related to elliptic equations for measures

We consider equations of the form L*μ = 0 for bounded measures on $$ {\mathbb{R}^{d}} $$ , where L is a second order elliptic operator, for example, Lu = Δu + (b,∇u), and the equation is understood as the identity $$ \int {Lud{\mu} = 0} $$ for all compactly supported smooth functions u. Stationary Kolmogorov equations for invariant measures of diffusion processes belong to this type. Solutions are considered in the class of probability measures and in the class of signed measures with integrable densities. We discuss the following problems: When is a probability solution to this equation unique? When does a given probability solution have the property that any integrable solution is a multiple of it? Which dimension can have the simplex of probability solutions? We present some recent positive results, give counterexamples and formulate open problems. Bibliography: 19 titles.

Sharp three sphere inequality for perturbations of a product of two second order elliptic operators and stability for the Cauchy problem for the anisotropic plate equation

We prove a sharp three sphere inequality for solutions to third order perturbations of a product of two second order elliptic operators with real coefficients. Then we derive various kinds of quantitative estimates of unique continuation for the anisotropic plate equation. Among these, we prove a stability estimate for the Cauchy problem for such an equation and we illustrate some applications to the size estimates of an unknown inclusion made of different material that might be present in the plate. The paper is self-contained and the Carleman estimate, from which the sharp three sphere inequality is derived, is proved in an elementary and direct way based on standard integration by parts.

A Borg–Levinson Theorem for Higher Order Elliptic Operators

We establish the Borg–Levinson theorem for elliptic operators of higher order with constant coefficients. The case of incomplete spectral data is also considered.

Numerical and analytical aspects of the pinning of martensitic phase boundaries

AbstractWe study the pinning and depinning behavior of interfaces immersed in a heterogeneous medium. For a continuum elasticicity model of the martensitic phase transformation, we numerically estimate the critical depinning stress of a phase boundary intersecting a non‐transforming inclusion in the material. In the limit of a nearly flat phase boundary, the elastic energy of the phase boundary can be approximated by an elliptic operator of order 1. For such an approximation we study the depinning transition near the critical point. Finally, we prove existence of a pinned solution for a parabolic model for the evolution of phase boundaries in a random environment (© 2011 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Second order elliptic operators with complex bounded measurable coefficients in $L^p$, Sobolev and Hardy spaces

Let L be a second order divergence form elliptic operator with complex bounded measurable coefficients. The operators arising in connection with L, such as the heat semigroup and Riesz transform, are not, in general, of CalderonZygmund type and exhibit behavior different from their counterparts built upon the Laplacian. The current paper aims at a thorough description of the properties of such operators in Lp, Sobolev, and some new Hardy spaces naturally associated to L. First, we show that the known ranges of boundedness in Lp for the heat semigroup and Riesz transform of L, are sharp. In particular, the heat semigroup e−tL need not be bounded in Lp if p < [2n/(n+2), 2n/(n−2)]. Then we provide a complete description of all Sobolev spaces in which L admits a bounded functional calculus, in particular, where e−tL is bounded. Secondly, we develop a comprehensive theory of Hardy and Lipschitz spaces associated to L, that serves the range of p beyond [2n/(n + 2), 2n/(n − 2)]. It includes, in particular, characterizations by the sharp maximal function and the Riesz transform (for certain ranges of p), as well as the molecular decomposition and duality and interpolation theorems.

A comparison principle for a Sobolev gradient semi-flow

We consider gradient descent equations for energy functionals of the type $S(u) = \frac{1}{2} _{L^2} + \int_{\Omega} V(x,u) dx$, where $A$ is a uniformly elliptic operator of order 2, with smooth coefficients. The gradient descent equation for such a functional depends on the metric under consideration. We consider the steepest descent equation for $S$ where the gradient is an element of the Sobolev space $H^{\beta}$, $\beta \in (0,1)$, with a metric that depends on $A$ and a positive number $\gamma >$sup$|V_{2 2}|$. We prove a weak comparison principle for such a gradient flow. We extend our methods to the case where $A$ is a fractional power of an elliptic operator, and provide an application to the Aubry-Mather theory for partial differential equations and pseudo-differential equations by finding plane-like minimizers of the energy functional.

Preconditioning a class of fourth order problems by operator splitting

We develop preconditioners for systems arising from finite element discretizations of parabolic problems which are fourth order in space. We consider boundary conditions which yield a natural splitting of the discretized fourth order operator into two (discrete) linear second order elliptic operators, and exploit this property in designing the preconditioners. The underlying idea is that efficient methods and software to solve second order problems with optimal computational effort are widely available. We propose symmetric and non-symmetric preconditioners, along with theory and numerical experiments. They both document crucial properties of the preconditioners as well as their practical performance. It is important to note that we neither need H s -regularity, s > 1, of the continuous problem nor quasi-uniform grids.

Homogenization of elliptic boundary value problems in Lipschitz domains

In this paper we study the L p boundary value problems for $${\mathcal{L}(u)=0}$$ in $${\mathbb{R}^{d+1}_+}$$ , where $${\mathcal{L}=-{\rm div} (A\nabla )}$$ is a second order elliptic operator with real and symmetric coefficients. Assume that A is periodic in x d+1 and satisfies some minimal smoothness condition in the x d+1 variable, we show that the L p Neumann and regularity problems are uniquely solvable for 1 < p < 2 + δ. We also present a new proof of Dahlberg’s theorem on the L p Dirichlet problem for 2 − δ < p < ∞ (Dahlberg’s original unpublished proof is given in the Appendix). As the periodic and smoothness conditions are imposed only on the x d+1 variable, these results extend directly from $${\mathbb{R}^{d+1}_+}$$ to regions above Lipschitz graphs. Consequently, by localization techniques, we obtain uniform L p estimates for the Dirichlet, Neumann and regularity problems on bounded Lipschitz domains for a family of second order elliptic operators arising in the theory of homogenization. The results on the Neumann and regularity problems are new even for smooth domains.

Regularity for a Class of Elliptic Operators with Dini Continuous Coefficients

We obtain regularity results for solutions to Pu=f when P is a kth order elliptic differential operator with the property that both P and Pt have coefficients that are continuous, with a modulus of continuity satisfying a Dini-type condition. Operators of this form arise in the study of regularity of functions that are known to be regular along the leaves of several foliations, such as arise in Anosov systems. The results here complement some previous results of the author and J. Rauch.

The connections between Dirichlet, regularity and Neumann problems for second order elliptic operators with complex bounded measurable coefficients

The present paper discusses relations between regularity, Dirichlet, and Neumann problems. We investigate the boundary problems for block operators and prove, in particular, that the solvability of the regularity problem does not imply the solvability of the dual Dirichlet problem for general elliptic operators with complex bounded measurable coefficients. This is strikingly different from the case of real operators, for which such an implication was established in 1993 by C. Kenig, J. Pipher [Invent. Math. 113 (3) (1993) 447–509] and since then has served as an integral part of many results.

Modeling and computation of two phase geometric biomembranes using surface finite elements

Biomembranes consisting of multiple lipids may involve phase separation phenomena leading to coexisting domains of different lipid compositions. The modeling of such biomembranes involves an elastic or bending energy together with a line energy associated with the phase interfaces. This leads to a free boundary problem for the phase interface on the unknown equilibrium surface which minimizes an energy functional subject to volume and area constraints. In this paper we propose a new computational tool for computing equilibria based on an L 2 relaxation flow for the total energy in which the line energy is approximated by a surface Ginzburg–Landau phase field functional. The relaxation dynamics couple a nonlinear fourth order geometric evolution equation of Willmore flow type for the membrane with a surface Allen–Cahn equation describing the lateral decomposition. A novel system is derived involving second order elliptic operators where the field variables are the positions of material points of the surface, the mean curvature vector and the surface phase field function. The resulting variational formulation uses H 1 spaces, and we employ triangulated surfaces and H 1 conforming quadratic surface finite elements for approximating solutions. Together with a semi-implicit time discretization of the evolution equations an iterative scheme is obtained essentially requiring linear solvers only. Numerical experiments are presented which exhibit convergence and the power of this new method for two component geometric biomembranes by computing equilibria such as dumbbells, discocytes and starfishes with lateral phase separation.

Yang-type inequalities for weighted eigenvalues of a second order uniformly elliptic operator with a nonnegative potential

In this paper, we investigate the Dirichlet weighted eigenvalue problem of a second order uniformly elliptic operator with a nonnegative potential on a bounded domain Ω C ℝ n . First, we prove a general inequality of eigenvalues for this problem. Then, by using this general inequality, we obtain Yang-type inequalities which give universal upper bounds for eigenvalues. An explicit estimate for the gaps of any two consecutive eigenvalues is also derived. Our results contain and extend the previous results for eigenvalues of the Laplacian, the Schrodinger operator and the second order elliptic operator on a bounded domain Ω C ℝ n .

A domain description and Green's function estimates up to the boundary for elliptic operator with singular potential

We consider the Friedrichs extension of the operator A = A 0 + q ( x ) , defined on a bounded domain Ω in R n , n ⩾ 1 . For n = 1 , we assume that Ω = ] a , b [ . Here A 0 = A 0 ( x , D ) is an elliptic operator of order 2 m with bounded smooth coefficients and q a function in L p ( Ω ) . Under some assumptions for q we obtain the uniform up to the boundary estimates for the Green's function of the Friedrichs extension of the operator A + λ I , for λ sufficiently large. Under some stronger assumptions for q we give a description for the domain of the Friedrichs extension of A.

Analytic semigroups generated in L 1(Ω) by second order elliptic operators via duality methods

Given an open domain (possibly unbounded) Ω⊂Rn, we prove that uniformly elliptic second order differential operators, under nontangential boundary conditions, generate analytic semigroups in L1(Ω). We use a duality method, and, further, give estimates of first order derivatives for the resolvent and the semigroup, through properties of the generator in Sobolev spaces of negative order.