Second-order Elliptic Differential Operators Research Articles

An optimally convergent Fictitious Domain method for interface problems

We introduce a novel Fictitious Domain (FD) unfitted method for interface problems associated with a second-order elliptic linear differential operator, that achieves optimal convergence without the need for adaptive mesh refinements nor enrichments of the Finite Element spaces. The key aspect of the proposed method is that it extends the solution into the fictitious domain in a way that ensures high global regularity. Continuity of the solution across the interface is enforced through a boundary Lagrange multiplier. The subdomains coupling, however, is not achieved by means of the duality pairing with the Lagrange multiplier, but through an L2 product with the H1 Riesz representative of the latter, thus avoiding gradient jumps across the interface. Thanks to the enhanced regularity, the proposed method attains an increase, with respect to standard FD methods, of up to one order of convergence in energy norm. The Finite Element formulation of the method is presented, followed by its analysis. Numerical tests on a model problem demonstrate its effectiveness and its superior accuracy compared to standard unfitted methods.

Homogenization of elliptic and parabolic equations with periodic coefficients in a bounded domain under the Neumann condition

Let $\mathcal{O}\subset\mathbb{R}^d$ be a bounded domain of class $C^{1,1}$. In $L_2(\mathcal{O};\mathbb{C}^n)$, we consider a selfadjoint matrix elliptic second-order differential operator $B_{N,\varepsilon}$, $0<\varepsilon\leqslant1$, with the Neumann boundary condition. The principal part of this operator is given in a factorized form. The operator involves first-order and zero-order terms. The coefficients of $B_{N,\varepsilon}$ are periodic and depend on $\mathbf{x}/\varepsilon$. We study the generalized resolvent $(B_{N,\varepsilon}-\zeta Q_0( {\cdot} /\varepsilon))^{-1}$, where $Q_0$ is a periodic bounded and positive definite matrix-valued function, and $\zeta$ is a complex-valued parameter. We obtain approximations for the generalized resolvent in the operator norm on $L_2(\mathcal{O};\mathbb{C}^n)$ and in the norm of operators acting from $L_2(\mathcal{O};\mathbb{C}^n)$ to the Sobolev space $H^1(\mathcal{O};\mathbb{C}^n)$ with two-parametric error estimates (with respect to $\varepsilon$ and $\zeta$). The results are applied to study the behaviour of the solutions of the initial boundary value problem with the Neumann condition for the parabolic equation $Q_0(\mathbf{x}/\varepsilon) \partial_t \mathbf{u}_\varepsilon(\mathbf{x},t) = -(B_{N,\varepsilon} \mathbf{u}_\varepsilon)(\mathbf{x},t)$ in a cylinder $\mathcal{O} \times (0,T)$, where $0<T \le \infty$.

Inequalities for eigenvalues of operators in divergence form on Riemannian manifolds isometrically immersed in Euclidean space

In this paper, we compute universal inequalities of eigenvalues of a large class of second-order elliptic differential operators in divergence form, that includes, e.g., the Laplace-Beltrami and Cheng-Yau operators, on a bounded domain in a complete Riemannian manifolds isometrically immersed in Euclidean space. A key step in order to obtain the sequence of our estimates is to get the right Yang-type first inequality. We also prove some inequalities for manifolds supporting some special functions and tensors.

Equivalence of measures and asymptotically optimal linear prediction for Gaussian random fields with fractional-order covariance operators

We consider two Gaussian measures μ,μ˜ on a separable Hilbert space, with fractional-order covariance operators A−2β and A˜−2β˜, respectively, and derive necessary and sufficient conditions on A,A˜ and β,β˜>0 for I. equivalence of the measures μ and μ˜, and II. uniform asymptotic optimality of linear predictions for μ based on the misspecified measure μ˜. These results hold, e.g., for Gaussian processes on compact metric spaces. As an important special case, we consider the class of generalized Whittle–Matérn Gaussian random fields, where A and A˜ are elliptic second-order differential operators, formulated on a bounded Euclidean domain D⊂Rd and augmented with homogeneous Dirichlet boundary conditions. Our outcomes explain why the predictive performances of stationary and non-stationary models in spatial statistics often are comparable, and provide a crucial first step in deriving consistency results for parameter estimation of generalized Whittle–Matérn fields.

Operator-theoretic approach to the homogenization of Schrödinger-type equations with periodic coefficients

In $L_2(\mathbb{R}^d;\mathbb{C}^n)$, we consider a selfadjoint strongly elliptic second-order differential operator ${\mathcal A}_\varepsilon$. It is assumed that the coefficients of ${\mathcal A}_\varepsilon$ are periodic and depend on ${\mathbf x}/\varepsilon$, where $\varepsilon>0$. We study the behaviour of the operator exponential $e^{-i{\mathcal A}_\varepsilon\tau}$ for small $\varepsilon$ and $\tau \in \mathbb{R}$. The results are applied to the homogenization of solutions of the Cauchy problem for the Schrödinger-type equation $i\partial_\tau{\mathbf u}_\varepsilon({\mathbf x},\tau)=({\mathcal A}_\varepsilon{\mathbf u}_\varepsilon)({\mathbf x},\tau)$ with initial data from a special class. For fixed $\tau$, as $\varepsilon \to 0$, the solution converges in $L_2(\mathbb{R}^d;\mathbb{C}^n)$ to the solution of the homogenized problem; the error is of the order $O(\varepsilon)$. For fixed $\tau$ we obtain an approximation of the solution ${\mathbf u}_\varepsilon( \cdot ,\tau)$ in the $L_2(\mathbb{R}^d;\mathbb{C}^n)$-norm with error $O(\varepsilon^2)$, and also an approximation of the solution in the $H^1(\mathbb{R}^d;\mathbb{C}^n)$-norm with error $O(\varepsilon)$. In these approximations correctors are taken into account. The dependence of errors on the parameter $\tau$ is traced. Bibliography: 113 items.

Resolvent Approximations in L2-Norm for Elliptic Operators Acting in a Perforated Space

We study homogenization of a second-order elliptic differential operator Aε = −div a(x/ε) ×∇ acting in an ε-periodically perforated space, where ε is a small parameter. Coefficients of the operator Aε are measurable ε-periodic functions. The simplest case where coefficients of the operator are constant is also interesting for us. We find an approximation for the resolvent (Aε + 1)−1 with remainder term of order ε2 as ε → 0 in operator L2-norm on the perforated space. This approximation turns to be the sum of the resolvent (A0 + 1)−1 of the homogenized operator A0 = −div a0∇, a0 > 0 being a constant matrix, and some correcting operator εCε. The proof of this result is given by the modified method of the first approximation with the usage of the Steklov smoothing operator.

A Liouville comparison principle for semilinear elliptic second-order inequalities

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.

Heat Kernels Estimates for Hermitian Line Bundles on Manifolds of Bounded Geometry

We consider a family of semiclassically scaled second-order elliptic differential operators on high tensor powers of a Hermitian line bundle (possibly, twisted by an auxiliary Hermitian vector bundle of arbitrary rank) on a Riemannian manifold of bounded geometry. We establish an off-diagonal Gaussian upper bound for the associated heat kernel. The proof is based on some tools from the theory of operator semigroups in a Hilbert space, results on Sobolev spaces adapted to the current setting, and weighted estimates with appropriate exponential weights.

Anderson acceleration based on the [formula omitted] Sobolev norm for contractive and noncontractive fixed-point operators

Anderson acceleration (AA) is a technique for accelerating the convergence of fixed-point iterations. In this paper, we apply AA to a sequence of functions and modify the norm in its internal optimization problem to the H−s norm, for some positive integer s, to bias it towards low-frequency spectral content in the residual. We analyze the convergence of AA by quantifying its improvement over Picard iteration. We find that AA based on the H−2 norm is well-suited to solve fixed-point operators derived from second-order elliptic differential operators, including the Helmholtz equation.

Non-well-ordered lower and upper solutions for semilinear systems of PDEs

We prove existence results for systems of boundary value problems involving elliptic second-order differential operators. The assumptions involve lower and upper solutions, which may be either well-ordered, or not at all. The results are stated in an abstract framework, and can be translated also for systems of parabolic type.

Jordan chains of elliptic partial differential operators and Dirichlet-to-Neumann maps

Let \Omega \subset \mathbb{R}^d be a bounded open set with Lipschitz boundary \Gamma . It will be shown that the Jordan chains of m-sectorial second-order elliptic partial differential operators with measurable coefficients and (local or non-local) Robin boundary conditions in L_2(\Omega) can be characterized with the help of Jordan chains of the Dirichlet-to-Neumann map and the boundary operator from H^{1/2}(\Gamma) into H^{-1/2}(\Gamma) . This result extends the Birman–Schwinger principle in the framework of elliptic operators for the characterization of eigenvalues, eigenfunctions and geometric eigenspaces to the complete set of all generalized eigenfunctions and algebraic eigenspaces.

Homogenization of the parabolic equation with periodic coefficients at the edge of a spectral gap

In , consider a second-order elliptic differential operator , , of the form with periodic coefficients. For small ε, we study the behavior of the semigroup , t>0, cut by the spectral projection of the operator for the interval . Here is the right edge of a spectral gap for the operator . We obtain approximation for the ‘cut semigroup’ in the operator norm in with error , and also a more accurate approximation with error (after singling out the factor ). The results are applied to homogenization of the Cauchy problem , , with the initial data from a special class.

Operator error estimates for homogenization of the nonstationary Schrödinger-type equations: sharpness of the results

In , we consider a self-adjoint matrix strongly elliptic second-order differential operator with periodic coefficients depending on . We find approximations of the exponential , , for small ε in the ()-operator norm with suitable s. The sharpness of the error estimates with respect to τ is discussed. The results are applied to study the behavior of the solution of the Cauchy problem for the Schrödinger-type equation .

On resolvent approximations of elliptic differential operators with periodic coefficients

We consider resolvents of elliptic second-order differential operators in with ε-periodic measurable matrix and study the asymptotic behaviour of , as the period ε goes to zero. We provide a construction for the leading terms of the ‘operator asymptotics’ of in the sense of -operator-norm convergence and prove order remainder estimates. We apply the modified method of the first approximation with the usage of Steklov's smoothing. The class of operators covered by our analysis includes uniformly elliptic families with bounded coefficients and also with unbounded coefficients from the John–Nirenberg space BMO (bounded mean oscillation).

Resolvent Approximations in L2-Norm for Elliptic Operators Acting in a Perforated Space

We study homogenization of a second-order elliptic differential operator Aε = - div a(x/ε)∇ acting in an ε-periodically perforated space, where ε is a small parameter. Coefficients of the operator Aε are measurable ε-periodic functions. The simplest case where coefficients of the operator are constant is also interesting for us. We find an approximation for the resolvent (Aε + 1)-1 with remainder term of order ε2 as ε → 0 in operator L2-norm on the perforated space. This approximation turns to be the sum of the resolvent (A0 + 1)-1 of the homogenized operator A0 = - div a0 ∇, a0 > 0 being a constant matrix, and some correcting operator εCε. The proof of this result is given by the modified method of the first approximation with the usage of the Steklov smoothing operator.

Regularity and convergence analysis in Sobolev and H\xf6lder spaces for generalized Whittle\u2013Mat\xe9rn fields

We analyze several types of Galerkin approximations of a Gaussian random field mathscr {Z}:mathscr {D}times varOmega rightarrow mathbb {R} indexed by a Euclidean domain mathscr {D}subset mathbb {R}^d whose covariance structure is determined by a negative fractional power L^{-2beta } of a second-order elliptic differential operator L:= -nabla cdot (Anabla ) + kappa ^2. Under minimal assumptions on the domain mathscr {D}, the coefficients A:mathscr {D}rightarrow mathbb {R}^{dtimes d}, kappa :mathscr {D}rightarrow mathbb {R}, and the fractional exponent beta >0, we prove convergence in L_q(varOmega ; H^sigma (mathscr {D})) and in L_q(varOmega ; C^delta (overline{mathscr {D}})) at (essentially) optimal rates for (1) spectral Galerkin methods and (2) finite element approximations. Specifically, our analysis is solely based on H^{1+alpha }(mathscr {D})-regularity of the differential operator L, where 0<alpha le 1. For this setting, we furthermore provide rigorous estimates for the error in the covariance function of these approximations in L_{infty }(mathscr {D}times mathscr {D}) and in the mixed Sobolev space H^{sigma ,sigma }(mathscr {D}times mathscr {D}), showing convergence which is more than twice as fast compared to the corresponding L_q(varOmega ; H^sigma (mathscr {D}))-rate. We perform several numerical experiments which validate our theoretical results for (a) the original Whittle–Matérn class, where Aequiv mathrm {Id}_{mathbb {R}^d} and kappa equiv {text {const.}}, and (b) an example of anisotropic, non-stationary Gaussian random fields in d=2 dimensions, where A:mathscr {D}rightarrow mathbb {R}^{2times 2} and kappa :mathscr {D}rightarrow mathbb {R} are spatially varying.

Exact Green's formula for the fractional Laplacian and perturbations

Let Ω be an open, smooth, bounded subset of $ \mathbb{R}^n $. In connection with the fractional Laplacian $(-\Delta )^a$ ($a&gt;0$), and more generally for a $2a$-order classical pseudodifferential operator (ψdo) $P$ with even symbol, one can define the Dirichlet value $\gamma _0^{a-1}u$, resp. Neumann value $\gamma _1^{a-1}u$ of $u(x)$, as the trace, resp. normal derivative, of $u/d^{a-1}$ on $\partial \Omega $, where $d(x)$ is the distance from $x\in \Omega $ to $\partial \Omega $; they define well-posed boundary value problems for $P$.&#x0D; A Green's formula was shown in a preceding paper, containing a generally nonlocal term $(B\gamma _0^{a-1}u,\gamma _0^{a-1}v)_{\partial \Omega }$, where $B$ is a first-order ψdo on $\partial \Omega $. Presently, we determine $B$ from $L$ in the case $P=L^a$, where $L$ is a strongly elliptic second-order differential operator. A particular result is that $B=0$ when $L=-\Delta $, and that $B$ is multiplication by a function (is local) when $L$ equals $-\Delta $ plus a first-order term. In cases of more general $L$, $B$ can be nonlocal.

Note on quantitative homogenization results for\xa0parabolic systems in {mathbb {R}}^d

In L_2({mathbb {R}}^d;{mathbb {C}}^n), we consider a semigroup e^{-tA_varepsilon }, tgeqslant 0, generated by a matrix elliptic second-order differential operator A_varepsilon geqslant 0. Coefficients of A_varepsilon are periodic, depend on {mathbf {x}}/varepsilon , and oscillate rapidly as varepsilon rightarrow 0. Approximations for e^{-tA_varepsilon } were obtained by Suslina (Funktsional Analiz i ego Prilozhen 38(4):86–90, 2004) and Suslina (Math Model Nat Phenom 5(4):390–447, 2010) via the spectral method and by Zhikov and Pastukhova (Russ J Math Phys 13(2):224–237, 2006) via the shift method. In the present note, we give another short proof based on the contour integral representation for the semigroup and approximations for the resolvent with two-parametric error estimates obtained by Suslina (2015).

On Resolvent Approximations of Elliptic Differential Operators with Locally Periodic Coefficients

We study the asymptotic behavior of the resolvents $$(A_{\varepsilon}+1)^{-1}$$ of elliptic second-order differential operators $$A_{\varepsilon}=-{\textrm{div}}a^{\varepsilon}(x)\nabla$$ in $$\mathbb{R}^{d}$$ with rapidly oscillating coefficients, as the small parameter $$\varepsilon$$ tends to zero. The matrix $$a^{\varepsilon}(x)=a(x,x/\varepsilon)$$ has the two-scale structure: it depends on the fast variable $$x/\varepsilon$$ and on the slow variable $$x$$ , with periodicity only in the fast variable. We provide a construction for the leading terms in the ‘‘operator asymptotics’’ of $$(A_{\varepsilon}+1)^{-1}$$ in the sense of $$L^{2}$$ -operator-norm convergence with order $$\varepsilon^{2}$$ remainder estimates. We apply the modified method of the first approximation with the usage of the shift proposed by V.V. Zhikov in Dokl. Math., 72:1 (2005).

The algebra of Wick polynomials of a scalar field on a Riemannian manifold

On a connected, oriented, smooth Riemannian manifold without boundary we consider a real scalar field whose dynamics is ruled by [Formula: see text], a second-order elliptic partial differential operator of Laplace type. Using the functional formalism and working within the framework of algebraic quantum field theory and of the principle of general local covariance, first we construct the algebra of locally covariant observables in terms of equivariant sections of a bundle of smooth, regular polynomial functionals over the affine space of the parametrices associated to [Formula: see text]. Subsequently, adapting to the case in hand a strategy first introduced by Hollands and Wald in a Lorentzian setting, we prove the existence of Wick powers of the underlying field, extending the procedure to smooth, local and polynomial functionals and discussing in the process the regularization ambiguities of such procedure. Subsequently we endow the space of Wick powers with an algebra structure, dubbed E-product, which plays in a Riemannian setting the same role of the time-ordered product for field theories on globally hyperbolic spacetimes. In particular, we prove the existence of the E-product and we discuss both its properties and the renormalization ambiguities in the underlying procedure. As the last step, we extend the whole analysis to observables admitting derivatives of the field configurations and we discuss the quantum Møller operator which is used to investigate interacting models at a perturbative level.