Second-order Linear Elliptic Operators Research Articles

A spectral characterization and an approximation scheme for the Hessian eigenvalue

We revisit the $k$-Hessian eigenvalue problem on a smooth, bounded, $(k-1)$-convex domain in $\mathbb R^n$. First, we obtain a spectral characterization of the $k$-Hessian eigenvalue as the infimum of the first eigenvalues of linear second-order elliptic operators whose coefficients belong to the dual of the corresponding Garding cone. Second, we introduce a non-degenerate inverse iterative scheme to solve the eigenvalue problem for the $k$-Hessian operator. We show that the scheme converges, with a rate, to the $k$-Hessian eigenvalue for all $k$. When $2\leq k\leq n$, we also prove a local $L^1$ convergence of the Hessian of solutions of the scheme. Hyperbolic polynomials play an important role in our analysis.

The Principal Eigenvalue Problems for Perturbed Fractional Laplace Operators

We study a variety of basic properties of the principal eigenvalue of a perturbed fractional Laplace operator and weakly coupled cooperative systems involving fractional Laplace operators. Our work extends a number of well-known properties regarding the principal eigenvalues of linear second-order elliptic operators with Dirichlet boundary condition to perturbed fractional Laplace operators. The establish results are also utilized to investigate the spatio-temporary dynamics of population models.

Some improvements of invertibility verifications for second-order linear elliptic operators

This paper presents some computer-assisted procedures to prove the invertibility of a second-order linear elliptic operator and to compute a bound for the norm of its inverse. These approaches are based on constructive L2-norm estimates of the Laplacian and improve on previous procedures that use projection and a priori error estimations. Several examples which confirm the actual effectiveness of the procedures are reported.

Stability of Low-Rank Tensor Representations and Structured Multilevel Preconditioning for Elliptic PDEs

Folding grid value vectors of size 2^L into Lth-order tensors of mode size 2times cdots times 2, combined with low-rank representation in the tensor train format, has been shown to result in highly efficient approximations for various classes of functions. These include solutions of elliptic PDEs on nonsmooth domains or with oscillatory data. This tensor-structured approach is attractive because it leads to highly compressed, adaptive approximations based on simple discretizations. Standard choices of the underlying bases, such as piecewise multilinear finite elements on uniform tensor product grids, entail the well-known matrix ill-conditioning of discrete operators. We demonstrate that, for low-rank representations, the use of tensor structure itself additionally introduces representation ill-conditioning, a new effect specific to computations in tensor networks. We analyze the tensor structure of a BPX preconditioner for a second-order linear elliptic operator and construct an explicit tensor-structured representation of the preconditioner, with ranks independent of the number L of discretization levels. The straightforward application of the preconditioner yields discrete operators whose matrix conditioning is uniform with respect to the discretization parameter, but in decompositions that suffer from representation ill-conditioning. By additionally eliminating certain redundancies in the representations of the preconditioned discrete operators, we obtain reduced-rank decompositions that are free of both matrix and representation ill-conditioning. For an iterative solver based on soft thresholding of low-rank tensors, we obtain convergence and complexity estimates and demonstrate its reliability and efficiency for discretizations with up to 2^{50} nodes in each dimension.

On families of optimal Hardy-weights for linear second-order elliptic operators

We construct families of optimal Hardy-weights for a subcritical linear second-order elliptic operator using a one-dimensional reduction. More precisely, we first characterize all optimal Hardy-weights with respect to one-dimensional subcritical Sturm-Liouville operators on (a,b), ∞≤a<b≤∞, and then apply this result to obtain families of optimal Hardy inequalities for general linear second-order elliptic operators in higher dimensions. As an application, we prove a new Rellich inequality.

Extrapolation for the Lp Dirichlet Problem in Lipschitz Domains

Let L be a second-order linear elliptic operator with complex coefficients. It is shown that if the Lp Dirichlet problem for the elliptic system L(u) = 0 in a fixed Lipschitz domain Ω in ℝd is solvable for some $$1 < p = {p_0} < \frac{{2\left( {d - 1} \right)}}{{d - 2}},$$ then it is solvable for all p satisfying $${p_0} < p < \frac{{2\left( {d - 1} \right)}}{{d - 2}}+\varepsilon.$$ The proof is based on a real-variable argument. It only requires that local solutions of L(u) = 0 satisfy a boundary Cacciopoli inequality.

An improved method for verifying the existence and bounds of the inverse of second-order linear elliptic operators mapping to dual space

This paper presents an improved method for determining the invertibility of second-order linear elliptic operators with a bound on the norm of their inverses by computers in a mathematically rigorous sense. This approach is an improvement on a previous method (Nakao et al. in Jpn J Ind Appl Math 32:19–32, 2015) which used a projection and constructive a priori error estimates. Several examples confirming the effectiveness of the proposed procedure are reported.

Virtual Element Method for general second-order elliptic problems on polygonal meshes

We consider the discretization of a boundary value problem for a general linear second-order elliptic operator with smooth coefficients using the Virtual Element approach. As in [A. H. Schatz, An observation concerning Ritz–Galerkin methods with indefinite bilinear forms, Math. Comput. 28 (1974) 959–962] the problem is supposed to have a unique solution, but the associated bilinear form is not supposed to be coercive. Contrary to what was previously done for Virtual Element Methods (as for instance in [L. Beirão da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L. D. Marini and A. Russo, Basic principles of virtual element methods, Math. Models Methods Appl. Sci. 23 (2013) 199–214]), we use here, in a systematic way, the [Formula: see text]-projection operators as designed in [B. Ahmad, A. Alsaedi, F. Brezzi, L. D. Marini and A. Russo, Equivalent projectors for virtual element methods, Comput. Math. Appl. 66 (2013) 376–391]. In particular, the present method does not reduce to the original Virtual Element Method of [L. Beirão da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L. D. Marini and A. Russo, Basic principles of virtual element methods, Math. Models Methods Appl. Sci. 23 (2013) 199–214] for simpler problems as the classical Laplace operator (apart from the lowest-order cases). Numerical experiments show the accuracy and the robustness of the method, and they show as well that a simple-minded extension of the method in [L. Beirão da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L. D. Marini and A. Russo, Basic principles of virtual element methods, Math. Models Methods Appl. Sci. 23 (2013) 199–214] to the case of variable coefficients produces, in general, sub-optimal results.

Some considerations of the invertibility verifications for linear elliptic operators

This paper presents three computer-assisted procedures for verifying the invertibility of second-order linear elliptic operators and for computing a bound on the norm of its inverse. One of these procedures is an improvement of a theorem by Nakao et al. (Computing 75:1–14, 2005) that uses projection and constructive a priori error estimates and was proposed by two of the authors of this paper. Results verifying these procedures are presented for several numerical examples.

Adaptive FEM with Optimal Convergence Rates for a Certain Class of Nonsymmetric and Possibly Nonlinear Problems

We analyze adaptive mesh-refining algorithms for conforming finite element discretizations of certain non-linear second-order partial differential equations. We allow continuous polynomials of arbitrary, but fixed polynomial order. The adaptivity is driven by the residual error estimator. We prove convergence even with optimal algebraic convergence rates. In particular, our analysis covers general linear second-order elliptic operators. Unlike prior works for linear non-symmetric operators, our analysis avoids the interior node property for the refinement, and the differential operator has to satisfy a G\r{a}rding inequality only. If the differential operator is uniformly elliptic, no additional assumption on the initial mesh is posed.

An identification problem with evolution on the boundary of hyperbolic type

We consider an equation of the type $A(u+k*u)=f$, where $A$ is a linear second-order elliptic operator, $k$ is a scalar function depending on time only and $k*u$ denotes the standard time convolution of functions defined in $(-\infty,T)$ with their supports in $[0,T]$. The previous equation is endowed with second-order dynamical boundary conditions. Assuming that the kernel $k$ is unknown and a supplementary condition is given, $k$ can be recovered and global existence, uniqueness and continuous dependence results can be shown.

A lower bound for principal eigenvalues in parabola-shaped regions

When a domain is inside a parabola-shaped region, we show that the principal eigenvalues for Dirichlet problems of a linear second-order elliptic operator in non-divergent form are bounded from below by a positive constant that is independent of the domain.

An identification problem with evolution on the boundary of parabolic type

We consider an equation of the type $A(u+k*u)=f$, where $A$ is a linear second-order elliptic operator, $k$ is a scalar function depending on time only and $k*u$ denotes the standard time convolution of functions defined on ${{\bf R}}$ with their supports in $[0,T]$. The previous equation is endowed with dynamical boundary conditions. Assuming that the kernel $k$ is unknown and information is given, under suitable additional conditions $k$ can be recovered and global existence, uniqueness and continuous dependence results can be shown.

Ground state alternative for p-Laplacian with potential term

Let Ω be a domain in $$\mathbb{R}^d$$ , d ≥ 2, and 1 < p < ∞. Fix $$V \in L_{\mathrm{loc}}^\infty(\Omega)$$ . Consider the functional Q and its Gâteaux derivative Q′ given by $$ Q(u) := \mathop \int_\Omega (|\nabla u|^p+V|u|^p){\rm d}x,\,\, \frac{1}{p}Q^\prime (u) := -\nabla\cdot(|\nabla u|^{p-2}\nabla u)+V|u|^{p-2}u.$$ If Q ≥ 0 on $$C_0^{\infty}(\Omega)$$ , then either there is a positive continuous function W such that $$\int W|u|^p\,\mathrm{d}x\leq Q(u)$$ for all $$u\in C_0^{\infty}(\Omega)$$ , or there is a sequence $$u_k\in C_0^{\infty}(\Omega)$$ and a function v > 0 satisfying Q′ (v) = 0, such that Q(u k ) → 0, and $$u_k\to v$$ in $$L^p_\mathrm{loc}(\Omega)$$ . In the latter case, v is (up to a multiplicative constant) the unique positive supersolution of the equation Q′ (u) = 0 in Ω, and one has for Q an inequality of Poincaré type: there exists a positive continuous function W such that for every $$\psi\in C_0^\infty(\Omega)$$ satisfying $$\int \psi v\,{\rm d}x \neq 0$$ there exists a constant C > 0 such that $$C^{-1}\int W|u|^p\,\mathrm{d}x\le Q(u)+C\left|\int u \psi\,\mathrm{d}x\right|^p$$ . As a consequence, we prove positivity properties for the quasilinear operator Q′ that are known to hold for general subcritical resp. critical second-order linear elliptic operators.

A Numerical Method to Verify the Invertibility of Linear Elliptic Operators with Applications to Nonlinear Problems

In this paper, we propose a numerical method to verify the invertibility of second-order linear elliptic operators. By using the projection and the constructive a priori error estimates, the invertibility condition is formulated as a numerical inequality based upon the existing verification method originally developed by one of the authors. As a useful application of the result, we present a new verification method of solutions for nonlinear elliptic problems, which enables us to simplify the verification process. Several numerical examples that confirm the actual effectiveness of the method are presented.

Elliptic problems on ?N with unbounded coefficients in classical Sobolev spaces

We discuss the existence and uniqueness in H1(ℝN) and the H2(ℝN) regularity of the solutions of Au=f when f ∈ L2(ℝN) and A is a second-order linear elliptic operator whose first and zeroth order coefficients may be unbounded at infinity. We also investigate whether −A generates a C0 or analytic semigroup on L2. The approach in this nonweighted setting is based on a new and general method. The idea consists in embedding A into a suitable one-parameter family of operators (As)s∈ ℝ with A0=A. The properties of As when s≠0 make it possible to prove that the boundary integrals arising from simple integration by parts over balls with increasing radius tend to 0 at infinity. This provides the needed estimates for uniqueness and regularity.

Harnack inequality for nondivergent elliptic operators on Riemannian manifolds

We consider second-order linear elliptic operators of nondivergence type which are intrinsically defined on Riemannian manifolds. Cabre proved a global Krylov-Safonov Harnack inequality under the assumption that the sectional curvature is nonnegative. We improve Cabre's result and, as a consequence, we give another proof to the Harnack inequality of Yau for positive harmonic functions on Riemannian manifolds with nonnegative Ricci curvature using the nondivergence structure of the Laplace operator.

Nondivergent elliptic equations on manifolds with nonnegative curvature

We consider a class of second-order linear elliptic operators, intrinsically defined on Riemannian manifolds, that correspond to nondivergent operators in Euclidean space. Under the assumption that the sectional curvature is nonnegative, we prove a global Krylov-Safonov Harnack inequality and, as a consequence, a Liouville theorem for solutions of such equations. From the Harnack inequality, we obtain Alexandroff-Bakelman-Pucci estimates and maximum principles for subsolutions. © 1997 John Wiley & Sons, Inc.

Nondivergent elliptic equations on manifolds with nonnegative curvature

We consider a class of second-order linear elliptic operators, intrinsically defined on Riemannian manifolds, that correspond to nondivergent operators in Euclidean space. Under the assumption that the sectional curvature is nonnegative, we prove a global Krylov-Safonov Harnack inequality and, as a consequence, a Liouville theorem for solutions of such equations. From the Harnack inequality, we obtain Alexandroff-Bakelman-Pucci estimates and maximum principles for subsolutions. © 1997 John Wiley & Sons, Inc.

Tests for m-accretive closedness of a second-order linear elliptic operator

Tests for m-accretive closedness of a second-order linear elliptic operator