Second-order Linear Elliptic Equations Research Articles

On the Boyarsky–Meyers Estimate for the Gradient of the Solution to the Dirichlet Problem for a Second-Order Linear Elliptic Equation with Drift: The Case of Critical Sobolev Exponent

On the Boyarsky–Meyers Estimate for the Gradient of the Solution to the Dirichlet Problem for a Second-Order Linear Elliptic Equation with Drift: The Case of Critical Sobolev Exponent

On the Boyarsky-Meyers estimate for the solution of the Dirichlet problem for a second-order linear elliptic equation with drift

We establish the increased integrability of the gradient of the solution to the Dirichlet problem for the Laplace operator with lower terms and prove the unique solvability of this problem.

On criticality theory for elliptic mixed boundary value problems in divergence form

The paper is devoted to the study of positive solutions of a second-order linear elliptic equation in divergence form in a domain [Formula: see text] that satisfy an oblique boundary condition on a portion of [Formula: see text]. First, we study weak solutions for the degenerate mixed boundary value problem [Formula: see text] where [Formula: see text] is a bounded Lipschitz domain, [Formula: see text] is a relatively open portion of [Formula: see text], and [Formula: see text] is an oblique (Robin) boundary operator defined on [Formula: see text] in a weak sense. In particular, we discuss the unique solvability of the above problem, the existence of a principal eigenvalue, and the existence of a minimal positive Green function. Then we establish a criticality theory for positive weak solutions of the operator [Formula: see text] in a general domain [Formula: see text] with no boundary condition on [Formula: see text] and no growth condition at infinity. The paper extends results obtained by Pinchover and Saadon for classical solutions of such a problem, where stronger regularity assumptions on the coefficients of [Formula: see text], and the boundary [Formula: see text] are assumed.

Inverse, ill-posed problems, essentially different solutions and explicit formulas for solutions

A request for an inverse problem, as well as for an incorrect problem produces tens of millions of answers in the Internet. In the past few decades, hundreds of international conferences on these topics have been held annually. Problems of this kind are quite involved, and their numerical analysis requires the development of special methods and numerical algorithms. Explicit formulas provide the main tool for testing these methods and numerical algorithms. The Cauchy problem for an elliptic equation is a classical ill-posed problem, which serves as a model for many inverse and incorrect problems. In the present paper we give a numerically realizable explicit formula for solving the Cauchy problem in a two-dimensional domain for a general second-order linear elliptic equation with analytic coefficients and the Cauchy analytic data on the analytic boundary.

New efficient time-stepping schemes for the Navier–Stokes–Cahn–Hilliard equations

In this paper, we construct and analyze a class of efficient time-stepping schemes for the Navier–Stokes–Cahn–Hilliard equation system stemming from phase field modeling of two-phase incompressible flows. The proposed schemes are based on an auxiliary variable approach for the Cahn–Hilliard equation and a delicate treatment of the terms coupling the Navier–Stokes equation and the Cahn–Hilliard equation. The former has been found to be an efficient tool for numerical integration of the Cahn–Hilliard equation in recent years. A detailed comparison with existing schemes is given, and the advantage of the new schemes are emphasized. In the theoretical aspect we rigorously prove that the designed schemes are unconditionally stable in the sense that some kind of energy remains bounded during the time stepping. In the implementation, we show how the schemes can be reformulated into six linear second-order elliptic equations with constant coefficients. The efficiency of the proposed algorithm is verified through several numerical examples.

On Solving Second-Order Linear Elliptic Equations

A method is presented for solving interior boundary-value problems for second-order elliptic equations by transition to ray variables. The domain is divided into cells within which the coefficients and sources have the smoothness and continuity properties necessary for the existence of a regular classical solution in the cell. The finite discontinuities of the coefficients (if any) are located on the cell boundaries. The regular solution in the cell is sought in the form of a superposition of the contributions made by volume and boundary sources placed on the rays arriving at the given point from the cell boundaries. Next, a finite analytic scheme for the numerical solution of the boundary value problem in a domain with discontinuous coefficients and sources is constructed by matching the regular solutions emerging from cells at the cell boundaries. The scheme exhibits no hard dependence of the accuracy of approximation on the sizes and shape of the cells, which is inherent in finite-difference schemes.

Unconditionally energy stable large time stepping method for the L[formula omitted]-gradient flow based ternary phase-field model with precise nonlocal volume conservation

We consider numerical approximations for solving a new, Allen–Cahn type ternary phase-field model. We first formulate a phase-field model from an energetic variational formulation based on the L2-gradient flow and add three nonlocal Lagrange multipliers to the system to conserve the volume for each phase. Then, by combining the SAV approach with the stabilization technique, where a crucial linear stabilization term is added to enhance the stability and keep the required accuracy thus allowing large time steps, we arrive at a decoupled, linear, non-iterative, and volume-conserved scheme. This scheme is unconditionally energy stable and requires solving only four linear second-order elliptic equations with constant coefficients. We further prove energy stability and present numerous 2D and 3D numerical simulations to show accuracy and stability.

Convergence and quasi-optimality of an adaptive continuous interior multi-penalty finite element method

An adaptive continuous interior multi-penalty finite element method (ACMPFEM) for symmetric second-order linear elliptic equations is considered. The proposed ACMPFEM is defined using a bilinear form which not only penalize the jumps of the first order normal derivatives across the element edges but also the jumps of all normal derivatives up to order p. In addition, compared with the analyses for adaptive finite element method, extra works are done to overcome the difficulties caused by the additional penalty terms. To be precise, the first difficulty is that the variational formulation of the ACMPFEM is not consistent with the continuous problem for weak solution u just in and the second difficulty lies in the proof of the optimality of our ACMPFEM. Furthermore, to prove the convergence and quasi-optimality of the ACMPFEM, an approximate Galerkin orthogonality in which the penalty term is regarded as a perturbation, and estimating the perturbation very carefully is used and the definition of our approximation class has to be involved with the penalty terms. Numerical tests are provided to verify the theoretical results and show advantages of the ACMPFEM.

Goal-oriented a posteriori error estimation for conforming and nonconforming approximations with inexact solvers

We derive a unified framework for goal-oriented a posteriori estimation covering in particular higher-order conforming, nonconforming, and discontinuous Galerkin finite element methods, as well as the finite volume method. The considered problem is a model linear second-order elliptic equation with inhomogeneous Dirichlet and Neumann boundary conditions and the quantity of interest is given by an arbitrary functional composed of a volumetric weighted mean value (source) term and a surface weighted mean (Dirichlet boundary) flux term. We specifically do not request the primal and dual discrete problems to be resolved exactly, allowing for inexact solves. Our estimates are based on H(div)-conforming flux reconstructions and H1-conforming potential reconstructions and provide a guaranteed upper bound on the goal error. The overall estimator is split into components corresponding to the primal and dual discretization and algebraic errors, which are then used to prescribe efficient stopping criteria for the employed iterative algebraic solvers. Numerical experiments are performed for the finite volume method applied to the Darcy porous media flow problem in two and three space dimensions. They show excellent effectivity indices even in presence of primal and dual algebraic errors and enable to spare a large percentage of unnecessary algebraic iterations.

The Doubly Periodic “Jump” Problem for a Second-Order Linear Elliptic Equation With Constant Coefficients

For a homogeneous differential second-order equation of elliptic type with constant coefficients, we study the boundary-value problem of finding doubly periodic solutions for given jumps of the solutions themselves and their derivatives on a smooth as well as on a non-rectifiable closed contour. We get the necessary and sufficient conditions for the solvability of the problem and obtain explicit formulas for its solution.

Affine Discontinuous Galerkin Method Approximation of Second-Order Linear Elliptic Equations in Divergence Form with Right-Hand Side in L1

We consider the standard affine discontinuous Galerkin method approximation of the second-order linear elliptic equation in divergence form with coefficients in L∞Ω and the right-hand side belongs to L1Ω; we extend the results where the case of linear finite elements approximation is considered. We prove that the unique solution of the discrete problem converges in W01,qΩ for every q with 1≤q<d/d-1 (d=2 or d=3) to the unique renormalized solution of the problem. Statements and proofs remain valid in our case, which permits obtaining a weaker result when the right-hand side is a bounded Radon measure and, when the coefficients are smooth, an error estimate in W01,qΩ when the right-hand side f belongs to LrΩ verifying Tkf∈H1Ω for every k>0, for some r>1.

On correctors for linear elliptic homogenization in the presence of local defects

We consider the corrector equation associated, in homogenization theory, to a linear second-order elliptic equation in divergence form , when the diffusion coefficient is a locally perturbed periodic coefficient. The question under study is the existence (and uniqueness) of the corrector, strictly sublinear at infinity, with gradient in Lr if the local perturbation is itself Lr, . This work follows up on previous works of ours, providing an alternative, more general and versatile approach, based on an a priori estimate, for this well-posedness result. Equations in non-divergence form such as are also considered, along with various extensions. The case of general advection–diffusion equations is postponed to a future work. An appendix contains a corrigendum to one of our earlier publication.

On correctors for linear elliptic homogenization in the presence of local defects: The case of advection–diffusion

We follow-up on our works devoted to homogenization theory for linear second-order elliptic equations with coefficients that are perturbations of periodic coefficients. We have first considered equations in divergence form in [6–8]. We have next shown, in our recent work [9], using a slightly different strategy of proof than in our earlier works, that we may also address the equation −aij∂iju=f. The present work is devoted to advection–diffusion equations: −aij∂iju+bj∂ju=f. We prove, under suitable assumptions on the coefficients aij, bj, 1≤i,j≤d (typically that they are the sum of a periodic function and some perturbation in Lp, for suitable p<+∞), that the equation admits a (unique) invariant measure and that this measure may be used to transform the problem into a problem in divergence form, amenable to the techniques we have previously developed for the latter case.

On Lp-Resolvent Estimates for Second-Order Elliptic Equations in Divergence Form

We consider the Dirichlet problems for second-order linear elliptic equations in divergence form. The leading coefficient A has small BMO semi-norm and first-order coefficient b belongs to Lr, where $n \leq r < \infty $ if n ≥ 3 and $2 < r < \infty $ if n = 2. We first establish Lp-resolvent estimates on bounded domains having small Lipschitz constant when $r/(r-1) < p < \infty $ . Under the additional assumption div A ∈ Lr, we also establish Lp-resolvent estimates on bounded domains with C1,1 boundary when 1 < p < r.

Stochastic Homogenization of Linear Elliptic Equations: Higher-Order Error Estimates in Weak Norms Via Second-Order Correctors

We are concerned with the homogenization of second-order linear elliptic equations with random coefficient fields. For symmetric coefficient fields with only short-range correlations, quantified through a logarithmic Sobolev inequality for the ensemble, we prove that when measured in weak spatial norms, the solution to the homogenized equation provides a higher-order approximation of the solution to the equation with oscillating coefficients. In the case of nonsymmetric coefficient fields, we provide a higher-order approximation (in weak spatial norms) of the solution to the equation with oscillating coefficients in terms of solutions to constant-coefficient equations. In both settings, we also provide optimal error estimates for the two-scale expansion truncated at second order. Our results rely on novel estimates on the second-order homogenization corrector, which we establish via sensitivity estimates for the second-order corrector and a large-scale $L^p$ theory for elliptic equations with random coefficients. Our results also cover the case of elliptic systems.

Liouville Principles and a Large-Scale Regularity Theory for Random Elliptic Operators on the Half-Space

We consider the large-scale regularity of solutions to second-order linear elliptic equations with random coefficient fields. In contrast to previous works on regularity theory for random elliptic operators, our interest is in the regularity at the boundary: We consider problems posed on the half-space with homogeneous Dirichlet boundary conditions and derive an associated $C^{1,\alpha}$-type large-scale regularity theory in the form of a corresponding decay estimate for the homogenization-adapted tilt-excess. This regularity theory entails an associated Liouville-type theorem. The results are based on the existence of homogenization correctors adapted to the half-space setting, which we construct---by an entirely deterministic argument---as a modification of the homogenization corrector on the whole space. This adaption procedure is carried out inductively on larger scales, crucially relying on the regularity theory already established on smaller scales.

Cell Averaging Two-Scale Convergence: Applications to Periodic Homogenization

The aim of the paper is to introduce an alternative notion of two-scale convergence which gives a more natural modeling approach to the homogenization of partial differential equations with periodically oscillating coefficients: while removing the bother of the admissibility of test functions, it nevertheless simplifies the proof of all the standard compactness results which made classical two-scale convergence very worthy of interest. Bounded sequences in $L^2_{\sharp} [ {Y}, L^2 (\Omega) ]$ and $L^2_{\sharp} [ {Y}, H^1 (\Omega) ]$ are proven to be relatively compact with respect to this new type of convergence. The strengths of the notion are highlighted on the classical homogenization problem of linear second-order elliptic equations for which first-order boundary corrector-type results are also established. Eventually, possible weaknesses of the method are pointed out on a nonlinear problem: the weak two-scale compactness result for ${\bf S}^2$-valued stationary harmonic maps.

Convergence Analysis of Discontinuous Finite Volume Methods for Elliptic Optimal Control Problems

In this paper, we discuss the convergence analysis of discontinuous finite volume methods applied to distribute the optimal control problems governed by a class of second-order linear elliptic equations. In order to approximate the control, two different methodologies are adopted: one is the method of variational discretization and second is piecewise constant discretization technique. For variational discretization method, optimal order of convergence in the [Formula: see text]-norm for state, adjoint state and control variables is derived. Moreover, optimal order of convergence in discrete [Formula: see text]-norm is also derived for state and adjoint state variables. Whereas, for piecewise constant approximation of control, first order convergence is derived for control, state and adjoint state variables in the [Formula: see text]-norm. In addition to that, optimal order of convergence in discrete [Formula: see text]-norm is derived for state and adjoint state variables. Also, some numerical experiments are conducted to support the derived theoretical convergence rate.

A Class of Second-Order Linear Elliptic Equations with Drift: Renormalized Solutions, Uniqueness and Homogenization

In this paper a class of N-dimensional second-order linear elliptic equations with a drift is studied. When the drift belongs to L2 the existence of a renormalized solution is proved. There is also uniqueness in the class of the renormalized solutions modulo \(L^{\infty }\), but the uniqueness is violated when the drift equation is regarded in the distributions sense. Then, considering a sequence of oscillating drifts which converges weakly in L2 to a limit drift in Lq, with q > N, the homogenization process makes appear an extra zero-order term involving a non-negative Radon measure which does not load the zero capacity sets. This extends the homogenization result obtained in [3] by relaxing the equi-integrability of the drifts in L2.

Study of a degenerate elliptic equation in an optimal consumption problem under partial information

We consider an optimal consumption/investment problem under partial information. By using a known reduction scheme, the problem can be transformed to a standard stochastic control problem. The related Hamilton–Jacobi–Bellman (HJB) equation is, after suitable transformation, a second-order linear degenerate elliptic equation. Although the problem is somewhat simple and classical, and the method of solution is well known, it is not obvious that the related HJB equation has a unique classical solution. Here, we prove the existence, uniqueness, and boundary regularity of the solution to the HJB equation. The main components of the proof are the construction of an appropriate weak solution and the verification of the Hormander condition. Using the classical solution, we also prove the verification theorem for the original optimal consumption/investment problem.