Elliptic Equations In Nondivergence Form Research Articles

Continuous finite elements satisfying a strong discrete Miranda–Talenti identity

Abstract This article introduces continuous $H^{2}$-nonconforming finite elements in two and three space dimensions that satisfy a strong discrete Miranda–Talenti inequality in the sense that the global $L^{2}$ norm of the piecewise Hessian is bounded by the $L^{2}$ norm of the piecewise Laplacian. The construction is based on globally continuous finite element functions with $C^{1}$ continuity on the vertices (2D) or edges (3D). As an application, these finite elements are used to approximate uniformly elliptic equations in nondivergence form under the Cordes condition without additional stabilization terms. For the biharmonic equation in three dimensions, the proposed methods has less degrees of freedom than existing nonconforming schemes of the same order. Numerical results in two and three dimensions confirm the practical feasibility of the proposed schemes.

Uniform <inline-formula><tex-math id="M1">$ C^{1,1} $</tex-math></inline-formula> estimates of homogenization of elliptic equations in non-divergence form with Dini conditions

Uniform <inline-formula><tex-math id="M1">$ C^{1,1} $</tex-math></inline-formula> estimates of homogenization of elliptic equations in non-divergence form with Dini conditions

On the Periodic Homogenization of Elliptic Equations in Nondivergence Form with Large Drifts

On the Periodic Homogenization of Elliptic Equations in Nondivergence Form with Large Drifts

Correction to: Monotone Meshfree Methods for Linear Elliptic Equations in Non-divergence Form via Nonlocal Relaxation

Correction to: Monotone Meshfree Methods for Linear Elliptic Equations in Non-divergence Form via Nonlocal Relaxation

Monotone Meshfree Methods for Linear Elliptic Equations in Non-divergence Form via Nonlocal Relaxation

Monotone Meshfree Methods for Linear Elliptic Equations in Non-divergence Form via Nonlocal Relaxation

A [formula omitted] finite element approximation of planar oblique derivative problems in non-divergence form

This paper proposes a C0 (non-Lagrange) primal finite element approximation of the linear elliptic equations in non-divergence form with oblique boundary conditions in planar, curved domains. As an extension of [Calcolo, 58 (2022), No. 9], the Miranda–Talenti estimate for oblique boundary conditions at a discrete level is established by enhancing the regularity on the vertices. Consequently, the coercivity constant for the proposed scheme is exactly the same as that from PDE theory. The quasi-optimal order error estimates are established by carefully studying the approximation property of the finite element spaces. Numerical experiments are provided to verify the convergence theory and to demonstrate the accuracy and efficiency of the proposed methods.

The dimensional estimates of exponential growth solutions to uniformly elliptic equations of non-divergence form

<p style='text-indent:20px;'>In this paper, we demonstrate that, for general second order uniformly elliptic equations of non-divergence form in unbounded cylinders with zero boundary condition, the space of solutions with a given exponential growth is of finite dimension. In particular, we generalize a result of Hang and Lin (1999) [<xref ref-type="bibr" rid="b4">4</xref>] in the non-divergence's setting by using a different approach. This allows us to consider the general lower order terms. Furthermore, a sharp estimate of the dimension is established by using the mean value inequality.</p>

Fractional elliptic equations in nondivergence form: definition, applications and Harnack inequality

We define the fractional powers Ls=(−aij(x)∂ij)s, 0<s<1, of nondivergence form elliptic operators L=−aij(x)∂ij in bounded domains Ω⊂Rn, under minimal regularity assumptions on the coefficients aij(x) and on the boundary ∂Ω. We show that these fractional operators appear in several applications such as fractional Monge–Ampère equations, elasticity, and finance. The solution u to the nonlocal Poisson problem{(−aij(x)∂ij)su=finΩu=0on∂Ω is characterized by a local degenerate/singular extension problem. We develop the method of sliding paraboloids in the Monge–Ampère geometry and prove the interior Harnack inequality and Hölder estimates for solutions to the extension problem when the coefficients aij(x) are bounded, measurable functions. This in turn implies the interior Harnack inequality and Hölder estimates for solutions u to the fractional problem.

Regularity estimates for fully nonlinear elliptic PDEs with general Hamiltonian terms and unbounded ingredients

We develop an optimal regularity theory for $$L^p$$ L p -viscosity solutions of fully nonlinear uniformly elliptic equations in nondivergence form whose gradient growth is described through a Hamiltonian function with measurable and possibly unbounded coefficients. Our approach treats both superlinear and sublinear gradient regimes in a unified way. We show $$C^{0,\alpha }$$ C 0 , α , $$C^{0,\text {Log-Lip}}$$ C 0 , Log-Lip , $$C^{1,\alpha }$$ C 1 , α , $$ C^{1,\text {Log-Lip}}$$ C 1 , Log-Lip and $$C^{2,\alpha }$$ C 2 , α regularity estimates, by displaying the growth allowed to the Hamiltonian in order to deal with an unbounded nonlinear gradient coefficient, whose integrability in turn gets worse as we approach the quadratic regime. These results may be seen as natural extensions of Teixeira (Arch Ration Mech Anal 211(3):911–927, 2014) and Nornberg (J Math Pures Appl 128(9):297–329, 2019). Moreover, we find proper compatibility conditions for which our regularity results depend intrinsically on the integrability of the underlying source term. As a byproduct of our methods, we prove a priori BMO estimates; sharp regularity to associated recession and flat profiles under relaxed convexity assumptions; improved regularity for a class of singular PDEs; and a Perron type result under unbounded ingredients.

A Sequential Least Squares Method for Elliptic Equations in Non-Divergence Form

We develop a new least squares method for solving the second-order elliptic equations in non-divergence form. Two least-squares-type functionals are proposed for solving the equations in two steps. We first obtain a numerical approximation to the gradient in a piecewisely irrotational polynomial space. Then together with the numerical gradient, we seek a numerical solution of the primitive variable in continuous finite element space. The variational setting naturally provides a posteriori error which could be used in an adaptive refinement algorithm. The error estimates in $L^2$ norm and energy norms for both two unknowns are derived. By a series of numerical experiments, we verify the convergence rates and show the efficiency of the adaptive algorithm.

Approximations to Problems of Optimal Control of Leading Coefficients of Elliptic Equations in Nondivergence Form with an Unbounded Nonlinearity in the Coefficients

Approximations to Problems of Optimal Control of Leading Coefficients of Elliptic Equations in Nondivergence Form with an Unbounded Nonlinearity in the Coefficients

Estimates for Green's Functions of Elliptic Equations in Non-Divergence Form with Continuous Coefficients

We present a new method for the existence and pointwise estimates of a Green's function of non-divergence form elliptic operator with Dini mean oscillation coefficients. We also present a sharp comparison with the corresponding Green's function for constant coefficients equations.

C0 finite element approximations of linear elliptic equations in non-divergence form and Hamilton–Jacobi–Bellman equations with Cordes coefficients

This paper is concerned with C0 (non-Lagrange) finite element approximations of the linear elliptic equations in non-divergence form and the Hamilton–Jacobi–Bellman (HJB) equations with Cordes coefficients. Motivated by the Miranda–Talenti estimate, a discrete analog is proved once the finite element space is C0 on the \((n-1)\)-dimensional subsimplex (face) and \(C^1\) on \((n-2)\)-dimensional subsimplex. The main novelty of the non-standard finite element methods is to introduce an interior stabilization term to argument the PDE-induced variational form of the linear elliptic equations in non-divergence form or the HJB equations. As a distinctive feature of the proposed methods, no stabilization parameter is involved in the variational forms. As a consequence, the coercivity constant (resp. monotonicity constant) for the linear elliptic equations in non-divergence form (resp. the HJB equations) at discrete level is exactly the same as that from PDE theory. The quasi-optimal order error estimates as well as the convergence of the semismooth Newton method are established. Numerical experiments are provided to validate the convergence theory and to illustrate the accuracy and computational efficiency of the proposed methods.

Green's Function for Nondivergence Elliptic Operators in Two Dimensions

We construct the Green function for second-order elliptic equations in non-divergence form when the mean oscillations of the coefficients satisfy the Dini condition. We show that the Green's function is BMO in the domain and establish logarithmic pointwise bounds. We also obtain pointwise bounds for first and second derivatives of the Green's function.

Regularity of solutions to a class of variable–exponent fully nonlinear elliptic equations

In the present paper, we propose the investigation of variable-exponent, degenerate/singular elliptic equations in non-divergence form. This current endeavor parallels the by now well established theory of functionals satisfying nonstandard growth condition, which in particular encompasses problems ruled by the p(x)-laplacian operator. Under rather general conditions, we prove viscosity solutions to variable exponent fully nonlinear elliptic equations are locally of class C1,κ for a universal constant 0<κ<1. A key feature of our estimates is that they do not depend on the modulus of continuity of exponent coefficients, and thus may be employed to investigate a variety of problems whose ellipticity degenerates and/or blows-up in a discontinuous fashion.

A Least Square Based Weak Galerkin Finite Element Method for Second Order Elliptic Equations in Non-Divergence Form

This article is devoted to establishing a least square based weak Galerkin method for second order elliptic equations in non-divergence form using a discrete weak Hessian operator. Naturally, the resulting linear system is symmetric and positive definite, and thus the algorithm is easy to implement and analyze. Convergence analysis in the H2 equivalent norm is established on an arbitrary shape regular polygonal mesh. A superconvergence result is proved when the coefficient matrix is constant or piecewise constant. Numerical examples are performed which not only verify the theoretical results but also reveal some unexpected superconvergence phenomena.

Remarks on optimal rates of convergence in periodic homogenization of linear elliptic equations in non-divergence form

We study and characterize the optimal rates of convergence in periodic homogenization of linear elliptic equations in non-divergence form. We obtain that the optimal rate of convergence is either \(O(\varepsilon )\) or \(O(\varepsilon ^2)\) depending on the diffusion matrix A, source term f, and boundary data g. Moreover, we show that the set of diffusion matrices A that give optimal rate \(O(\varepsilon )\) is open and dense in the set of \(C^2\) periodic, symmetric, and positive definite matrices, which means that generically, the optimal rate is \(O(\varepsilon )\).

Classical solutions of oblique derivative problem in nonsmooth domains with mean Dini coefficients

We consider second-order elliptic equations in nondivergence form with oblique derivative boundary conditions. We show that any strong solutions to such problems are twice continuously differentiable up to the boundary when the boundary can be locally represented by a C 1 C^1 function whose first derivatives are Dini continuous and the mean oscillations of coefficients satisfy the Dini condition. This improves a recent result by Dong, Lee, and Kim. To the best of our knowledge, such a result is new even for the Poisson equation. An extension to concave fully nonlinear elliptic equations is also presented.

Hessian Estimates for Non-divergence form Elliptic Equations Arising from Composite Materials

In this paper, we prove that any $W^{2,1}$ strong solution to second-order non-divergence form elliptic equations is locally $W^{2,\infty}$ and piecewise $C^{2}$ when the leading coefficients and data are of piecewise Dini mean oscillation and the lower-order terms are bounded. Somewhat surprisingly here the interfacial boundaries are only required to be $C^{1,\text{Dini}}$. We also derive global weak-type $(1,1)$ estimates with respect to $A_{1}$ Muckenhoupt weights. The corresponding results for the adjoint operator are established. Our estimates are independent of the distance between these surfaces of discontinuity of the coefficients.

Stability of Patterns in the Abelian Sandpile

We show that the patterns in the Abelian sandpile are stable. The proof combines the structure theory for the patterns with the regularity machinery for non-divergence form elliptic equations. The stability results allow one to improve weak-$$*$$ convergence of the Abelian sandpile to pattern convergence for certain classes of solutions.