Order Elliptic Equations Research Articles

A global random walk on grid algorithm for second order elliptic equations

Abstract In this paper we develop stochastic simulation methods for solving large systems of linear equations, and focus on two issues: (1) construction of global random walk algorithms (GRW), in particular, for solving systems of elliptic equations on a grid, and (2) development of local stochastic algorithms based on transforms to balanced transition matrix. The GRW method calculates the solution in any desired family of prescribed points of the gird in contrast to the classical stochastic differential equation based Feynman–Kac formula. The use in local random walk methods of balanced transition matrices considerably decreases the variance of the random estimators and hence decreases the computational cost in comparison with the conventional random walk on grids algorithms.

The Boundary Behavior of a Solution to the Dirichlet Problem for a Linear Degenerate Second Order Elliptic Equation

For an elliptic equation with a partially Muckenhoupt weight we find a sufficient condition for the regularity of a boundary point for the solution to the Dirichlet problem with a continuous boundary function and obtain an estimate for the modulus of continuity for the solution at this point.

Scale invariant regularity estimates for second order elliptic equations with lower order coefficients in optimal spaces

We show local and global scale invariant regularity estimates for subsolutions and supersolutions to the equation −div(A∇u+bu)+c∇u+du=−divf+g, assuming that A is elliptic and bounded. In the setting of Lorentz spaces, under the assumptions b,f∈Ln,1, d,g∈Ln2,1 and c∈Ln,q for q≤∞, we show that, with the surprising exception of the reverse Moser estimate, scale invariant estimates with “good” constants (that is, depending only on the norms of the coefficients) do not hold in general. On the other hand, assuming a necessary smallness condition on b,d or c,d, we show a maximum principle and Moser's estimate for subsolutions with “good” constants. We also show the reverse Moser estimate for nonnegative supersolutions with “good” constants, under no smallness assumptions when q<∞, leading to the Harnack inequality for nonnegative solutions and local continuity of solutions. Finally, we show that, in the setting of Lorentz spaces, our assumptions are the sharp ones to guarantee these estimates.

Second order elliptic equations and Hodge-Dirac operators

In this paper we show how a second order scalar uniformly elliptic equation in divergence form with measurable coefficients and Dirichlet boundary conditions can be transformed into a first order elliptic system with half-Dirichlet boundary condition. This first order system involves Hodge-Dirac operators and can be seen as a natural generalization of the Beltrami equation in the plane and we develop a theory for this equation, extending results from the plane to higher dimension. The reduction to a first order system applies both to linear as well as quasilinear second order equations and we believe this to be of independent interest. Using the first order system, we give a new representation formula of the solution of the Dirichlet problem both on simply and finitely connected domains. This representation formula involves only singular integral operators of convolution type and Neumann series there of, for which classical Calderón-Zygmund theory is applicable. Moreover, no use is made of any fundamental solution or Green's function beside fundamental solutions of constant coefficient operators. Remarkably, this representation formula applies also for solutions of the fully non-linear first order system. We hope that the representation formula could be used for numerically solving the equations. Using these tools we give a new short proof of Meyers' higher integrability theorem. Furthermore, we show that the solutions of the first order system are Hölder continuous with the same Hölder coefficient as the solutions of the second order equations. Finally, factorization identities and representation formulas for the higher dimensional Beurling-Ahlfors operator are proven using Clifford algebras, and certain integral estimates for the Cauchy transform is extended to higher dimensions.

Discrete maximum principle for the weak Galerkin method on triangular and rectangular meshes

In this paper, the discrete maximum principle (DMP) of the weak Galerkin (WG) method for the second order elliptic equations is considered. The WG schemes are proposed on triangular and rectangular meshes, respectively. For both schemes, the discrete maximum principle for non-negative source and general source are proved. Numerical experiments are also presented to verify the convergence and the DMP-preserving property.

Constrained semilinear elliptic systems on $\mathbb R^N$

The existence of solutions $u$ in $H^1(\mathbb R^N,\mathbb R^M)\cap H^2_{loc}(\mathbb R^N,\mathbb R^M)$ of a coupled semilinear system of second order elliptic partial differential equations on $\mathbb R^N$ of the form \[ \mathcal{P}[u] = f(x,u,\partial u), \quad x\in \mathbb R^N, \] under pointwise constraints is considered. The problem is studied via the constructed suitable topological invariant, the so-called constrained topological degree, which allows to get the existence of solutions of abstract problems considered as $L^2$-realizations of the approximating sequence of systems obtained by the truncation of the initial system to bounded subdomains. The key step of the proof consists in showing the relative $H^1$-compactness of the sequence of solutions to the truncated systems by the use of the so-called tail estimates. The constructions rely on the semigroup approach combined with topological methods, as well as invariance/viability techniques.

A global random walk on grid algorithm for second order elliptic equations

Abstract We suggest in this paper a global random walk on grid (GRWG) method for solving second order elliptic equations. The equation may have constant or variable coefficients. The GRWS method calculates the solution in any desired family of m prescribed points of the gird in contrast to the classical stochastic differential equation based Feynman–Kac formula, and the conventional random walk on spheres (RWS) algorithm as well. The method uses only N trajectories instead of mN trajectories in the RWS algorithm and the Feynman–Kac formula. The idea is based on the symmetry property of the Green function and a double randomization approach.

Growth theorems for metric spaces with applications to PDE

The paper is devoted to some extensions of the joint results by N. V. Krylov and the author on the Harnack inequalities for second order elliptic and parabolic equations in nondivergence form to metric spaces of homogeneous type. The main tools are special Landis-type growth theorems.

Adaptive direct discontinuous Galerkin method for elliptic equations

In this paper, adaptive direct discontinuous Galerkin method for second order elliptic equations is studied in two dimensional. A numerical flux with general weighted averages and proper weights for interface problems is introduced. In addition to the optimal a priori error estimator, we propose a residual-type a posteriori error estimator. The global upper bound and local lower bound for the error in DG norm are established. Several numerical examples are carried out to confirm the reliability and efficiency of the proposed error estimator and the corresponding adaptive direct discontinuous Galerkin method.

$(1+s)$-Order Convergence Analysis of Weak Galerkin Finite Element Methods for Second Order Elliptic Equations

$(1+s)$-Order Convergence Analysis of Weak Galerkin Finite Element Methods for Second Order Elliptic Equations

High-frequency conductivity at Larmor-frequency in human brain using moving local window multilayer perceptron neural network.

Magnetic resonance electrical properties tomography (MREPT) aims to visualize the internal high-frequency conductivity distribution at Larmor frequency using the B1 transceive phase data. From the magnetic field perturbation by the electrical field associated with the radiofrequency (RF) magnetic field, the high-frequency conductivity and permittivity distributions inside the human brain have been reconstructed based on the Maxwell’s equation. Starting from the Maxwell’s equation, the complex permittivity can be described as a second order elliptic partial differential equation. The established reconstruction algorithms have focused on simplifying and/or regularizing the elliptic partial differential equation to reduce the noise artifact. Using the nonlinear relationship between the Maxwell’s equation, measured magnetic field, and conductivity distribution, we design a deep learning model to visualize the high-frequency conductivity in the brain, directly derived from measured magnetic flux density. The designed moving local window multi-layer perceptron (MLW-MLP) neural network by sliding local window consisting of neighboring voxels around each voxel predicts the high-frequency conductivity distribution in each local window. The designed MLW-MLP uses a family of multiple groups, consisting of the gradients and Laplacian of measured B1 phase data, as the input layer in a local window. The output layer of MLW-MLP returns the conductivity values in each local window. By taking a non-local mean filtering approach in the local window, we reconstruct a noise suppressed conductivity image while maintaining spatial resolution. To verify the proposed method, we used B1 phase datasets acquired from eight human subjects (five subjects for training procedure and three subjects for predicting the conductivity in the brain).

Two-step MPS-MFS ghost point method for solving partial differential equations

The fictitious boundary surrounding the domain is required in the implementation of the method of fundamental solutions (MFS). A similar approach of taking a portion of the centres of radial basis functions (RBFs) outside the domain in the context of the method of particular solutions (MPS) is proposed to further improve the performance of a two-step MPS-MFS method. Since the shape parameter of RBFs is problem dependent, several known procedures on how to determine a good shape parameter are introduced for solving various types of problems. The proposed method is not only highly accurate but also fairly stable due to the use of the particular solutions and fundamental solutions. To demonstrate the effectiveness of the proposed method, five numerical examples in highly complicate and irregular domains, which include second and fourth order elliptic partial differential equations in 2D and 3D, are presented.

Interior Schauder-Type Estimates for Higher-Order Elliptic Operators in Grand-Sobolev Spaces

In this paper an elliptic operator of the $m$-th order $L$ with continuous coefficients in the $n$-dimensional domain $Omega subset R^{n} $ in the non-standard Grand-Sobolev space $W_{q)}^{m} left(Omega right), $ generated by the norm $left| , cdot , right| _{q)} $ of the Grand-Lebesgue space $L_{q)} left(Omega right), $ is considered. Interior Schauder-type estimates play a very important role in solving the Dirichlet problem for the equation $Lu=f$. The considered non-standard spaces are not separable, and therefore, to use classical methods for treating solvability problems in these spaces, one needs to modify these methods. To this aim, based on the shift operator, separable subspaces of these spaces are determined, in which finite infinitely differentiable functions are dense. Interior Schauder-type estimates are established with respect to these subspaces. It should be noted that Lebesgue spaces $L_{q} left(Gright), $ are strict parts of these subspaces. This work is a continuation of the authors of the work cite{28}, which established the solvability in the small of higher order elliptic equations in grand-Sobolev spaces.

Harnack's inequality for quasilinear elliptic equations with generalized Orlicz growth

We prove Harnack's inequality for bounded weak solutions to quasilinear second order elliptic equations with generalized Orlicz growth conditions. Our approach covers new cases of variable exponent and (p,q) growth conditions.&#x0D; For more information see https://ejde.math.txstate.edu/Volumes/2021/27/abstr.html

Maximization Problems for Second Order Elliptic Equations

We investigate the maxima in classes of rearrangements of some functionals associated with solutions to Dirichlet problems for second order elliptic equations.

Superconvergent Flux Recovery of the Rannacher–Turek Nonconforming Element

This work presents superconvergence estimates of the nonconforming Rannacher--Turek element for second order elliptic equations on any cubical meshes in $\mathbb{R}^{2}$ and $\mathbb{R}^{3}$. In particular, a corrected numerical flux is shown to be superclose to the Raviart--Thomas interpolant of the exact flux. We then design a superconvergent recovery operator based on local weighted averaging. Combining the supercloseness and the recovery operator, we prove that the recovered flux superconverges to the exact flux. As a by-product, we obtain a superconvergent recovery estimate of the Crouzeix--Raviart element method for general elliptic equations.

The Neumann problem for higher order elliptic equations

We solve the Neumann problem in the half space for higher order elliptic differential equations with variable self-adjoint t-independent coefficients, and with boundary data in the negative smoothness space where Our arguments are inspired by an argument of Shen and build on known well posedness results in the case p = 2. We use the same techniques to establish nontangential and square function estimates on layer potentials with inputs in Lp or for a similar range of p, based on known bounds for p near 2; in this case we may relax the requirement of self-adjointess.

New spinorial approach to mass inequalities for black holes in general relativity

A new spinorial strategy for the construction of geometric inequalities involving the Arnowitt-Deser-Misner mass of black hole systems in general relativity is presented. This approach is based on a second order elliptic equation (the approximate twistor equation) for a valence 1 Weyl spinor. This has the advantage over other spinorial approaches to the construction of geometric inequalities based on the Sen-Witten-Dirac equation that it allows us to specify boundary conditions for the two components of the spinor. This greater control on the boundary data has the potential of giving rise to new geometric inequalities involving the mass. In particular, it is shown that the mass is bounded from below by an integral functional over a marginally outer trapped surface (MOTS) which depends on a freely specifiable valence 1 spinor. From this main inequality, by choosing the free data in an appropriate way, one obtains a new nontrivial bounds of the mass in terms of the inner expansion of the MOTS. The analysis makes use of a new formalism for the $1+1+2$ decomposition of spinorial equations.

Note on a sign-dependent regularity for the polyharmonic Dirichlet problem

A priori estimates for semilinear higher order elliptic equations usually have to deal with the absence of a maximum principle. This note presents some regularity estimates for the polyharmonic Dirichlet problem that will make a distinction between the influence on the solution of the positive and the negative part of the right-hand side.

Second order elliptic partial differential equations driven by Lévy white noise

This paper deals with linear stochastic partial differential equations with variable coefficients driven by Lévy white noise. First, an existence theorem for integral transforms of Lévy white noise is derived and the existence of generalized and mild solutions of second order elliptic partial differential equations is proved. Further, the generalized electric Schrödinger operator for different potential functions V is discussed.