Equations In Non-smooth Domains Research Articles

On approximations for reflected SDEs and SPDEs with Neumann boundary conditions

In this paper we study an approximation for reflected stochastic differential equations in non-smooth domains by replacing the driving Brownian motion with its smooth approximation and prove that the solutions of approximating equations converge in the mean square to the solution of the Stratonovich reflected stochastic differential equation. We also study the smooth approximation for Neumann-type semilinear stochastic partial differential equations, which is completed through establishing the convergence in $ L^2 $ of backward doubly stochastic differential equations coupled with reflected SDEs.

Mosco convergence for $H$ (curl) spaces, higher integrability for Maxwell's equations, and stability in direct and inverse EM scattering problems

This paper is concerned with the scattering problem for time-harmonic electromagnetic waves, due to the presence of scatterers and of inhomogeneities in the medium. We prove a sharp stability result for the solutions to the direct electromagnetic scattering problem, with respect to variations of the scatterer and of the inhomogeneity, under minimal regularity assumptions for both of them. The stability result leads to bounds on solutions to the scattering problems which are uniform for an extremely general class of admissible scatterers and inhomogeneities. These uniform bounds are a key step in tackling the challenging stability issue for the corresponding inverse electromagnetic scattering problem. In this paper we establish two optimal stability results of logarithmic type for the determination of polyhedral scatterers by a minimal number of electromagnetic scattering measurements. In order to prove the stability result for the direct electromagnetic scattering problem, we study two fundamental issues in the theory of Maxwell equations: Mosco convergence for $H$ (curl) spaces and higher integrability properties of solutions to Maxwell equations in nonsmooth domains.

On Integral Equation Methods for the First Dirichlet Problem of the Biharmonic and Modified Biharmonic Equations in NonSmooth Domains

Despite important applications in unsteady Stokes flow, a Fredholm second kind integral equation formulation modeling the first Dirichlet problem of the modified biharmonic equation in the plane ha...

Convergence of a decoupled mixed FEM for the dynamic Ginzburg–Landau equations in nonsmooth domains with incompatible initial data

In this paper, we propose a fully discrete mixed finite element method for solving the time-dependent Ginzburg–Landau equations, and prove the convergence of the finite element solutions in general curved polyhedra, possibly nonconvex and multi-connected, without assumptions on the regularity of the solution. Global existence and uniqueness of weak solutions for the PDE problem are also obtained in the meantime. A decoupled time-stepping scheme is introduced, which guarantees that the discrete solution has bounded discrete energy, and the finite element spaces are chosen to be compatible with the nonlinear structure of the equations. Based on the boundedness of the discrete energy, we prove the convergence of the finite element solutions by utilizing a uniform $$L^{3+\delta }$$ regularity of the discrete harmonic vector fields, establishing a discrete Sobolev embedding inequality for the Nédélec finite element space, and introducing a $$\ell ^2(W^{1,3+\delta })$$ estimate for fully discrete solutions of parabolic equations. The numerical example shows that the constructed mixed finite element solution converges to the true solution of the PDE problem in a nonsmooth and multi-connected domain, while the standard Galerkin finite element solution does not converge.

On the Coefficients of the Singularities of the Solution of Maxwell’s Equations near Polyhedral Edges

The solution fields of Maxwell’s equations are known to exhibit singularities near corners, crack tips, edges, and so forth of the physical domain. The structures of the singular fields are well known up to some undetermined coefficients. In two-dimensional domains with corners and cracks, the unknown coefficients are real constants. However, in three-dimensional domains the unknown coefficients are functions defined along the corresponding edges. This paper proposes explicit formulas for the computation of these coefficients in the case of two-dimensional domains with corners and three-dimensional domains with straight edges. The coefficients of the singular fields along straight edges of three-dimensional domains are represented in terms of Fourier series. The formulas presented are aimed at the numerical approximation of the coefficients of the singular fields. They can also be used for the construction of adaptiveH1-nodal finite-element procedures for the efficient numerical treatment of Maxwell’s equations in nonsmooth domains.

An efficient fully linearized semi-implicit Galerkin-mixed FEM for the dynamical Ginzburg–Landau equations of superconductivity

The paper focuses on numerical study of the time-dependent Ginzburg–Landau (TDGL) equations under the Lorentz gauge. The proposed method is based on a fully linearized backward Euler scheme in temporal direction, and a mixed finite element method (FEM) in spatial direction, where the magnetic field σ=curlA is introduced as a new variable. The linearized Galerkin-mixed FEM enjoys many advantages over existing methods. In particular, at each time step the scheme only requires the solution of two linear systems for ψ and (σ,A), respectively, with constant coefficient matrices. These two matrices can be pre-assembled at the initial time step and these two linear systems can be solved simultaneously. Moreover, the method provides the same order of optimal accuracy for the density function ψ, the magnetic potential A, the magnetic field σ=curlA, the electric potential divA and the current curlσ. Extensive numerical experiments in both two- and three-dimensional spaces, including complex geometries with defects, are presented to illustrate the accuracy and stability of the scheme. Our numerical results also show that the proposed method provides more realistic predictions for the vortex dynamics of the TDGL equations in nonsmooth domains, while the vortex motion influenced by a defect of the domain is of high interest in the study of superconductors.

On Uniqueness of Large Solutions of Nonlinear Parabolic Equations in Nonsmooth Domains

Abstract We study the existence and uniqueness of the positive solutions of the problem (P): Әtu - Δu + uq = 0 (q > 1) in Ω × (0, ∞), u = ∞ on ӘΩ × (0, ∞) and u(., 0) ∈ L1(Ω), when Ω is a bounded domain in ℝN. We construct a maximal solution, prove that this maximal solution is a large solution whenever q < N/(N - 2) and it is unique if ӘΩ = ӘΩ̅c. If ӘΩ has the local graph property, we prove that there exists at most one solution to problem (P).

Hölder Regularity of Solutions to Second-Order Elliptic Equations in Nonsmooth Domains

We establish the global Holder estimates for solutions to second-order elliptic equations, which vanish on the boundary, while the right-hand side is allowed to be unbounded. For nondivergence elliptic equations in domains satisfying an cone condition, similar results were obtained by J. H. Michael, who in turn relied on the barrier techniques due to K. Miller. Our approach is based on special growth lemmas, and it works for both divergence and nondivergence, elliptic and parabolic equations, in domains satisfying a general exterior measure condition.

On the Dirichlet problem for reaction–diffusion equations in non-smooth domains

On the Dirichlet problem for reaction–diffusion equations in non-smooth domains

On the dimension of the attractor for the non-homogeneous Navier-Stokes equations in non-smooth domains

On the dimension of the attractor for the non-homogeneous Navier-Stokes equations in non-smooth domains

Griffith formulae for elasticity systems with unilateral conditions in domains with cracks

In this paper we consider the elasticity equations in nonsmooth domains in R n, n=2,3 . The domains have a crack whose length may change. At the crack faces, inequality type boundary conditions describing a mutual nonpenetration of the crack faces are prescribed. The derivative of the energy functional with respect to the crack length is obtained. The Griffith formulae are derived in 2D and 3D cases and the other properties of the solutions are established. In two-dimensional case the Rice–Cherepanov's integral over a closed curve is constructed. The path independence of the Rice–Cherepanov's integral is shown.

Hessian equations in non-smooth domains

Hessian equations in non-smooth domains

Spectral radius properties for layer potentials associated with the elastostatics and hydrostatics equations in nonsmooth domains

By producing a L2 convergent Neumann series, we prove the invertibility of the elastostatics and hydrostatics boundary layer potentials on arbitrary Lipschitz domains with small Lipschitz character and 3D polyhedra with large dihedral angles.

Maxwell's Equations and Dielectric Obstacles with Lipschitz Boundaries

The classical transmission problem for the scattering of time-harmonic electromagnetic waves by a dielectric object is solved using layer potential techniques. The surface of the object is only assumed to be a Lipschitz manifold. Unlike in the case of smoother surfaces, neither the calculus of pseudodifferential operators nor the compactness of certain boundary operators are available tools. Nevertheless, the particular representation of the solutions and the choice of the space of boundary data used in the paper make the problem tractable via harmonic analysis techniques for elliptical partial differential equations in nonsmooth domains.

Dini continuity of the first derivatives of generalized solutions to the Dirichlet problem for linear elliptic second order equations in nonsmooth domains

We consider generalized solutions to the Dirichlet problem for linear elliptic second order equations in a domain bounded by a Dini-Lyapunov surface and containing a conical point. For such solutions we derive Dini estimates for the first order generalize

Singular boundary value problems for nonlinear elliptic equations in nonsmooth domains

Let $\Omega$ be a piecewise-regular domain in $\Bbb R^N$ and O an irregular point on its boundary $\partial \Omega$. We study under what conditions on $q$ any solution $u$ of (E) $-\Delta u + g(x,u)=0$ where $g$ has a $q$-power-like growth at infinity ($q>1$) which coincides on $\partial \Omega \setminus {\{\text{O}}\}$ with a continuous function defined on whole $\partial \Omega$, can be extended as a continuous function in $\bar \Omega$.

Estimates for the Stokes operator in Lipschitz domains

We study the Stokes operator A in a three- dimensional Lipschitz domain Ω. Our main result asserts that the domain of A is contained in W 1,p 0 (Ω) ∩ W 3/2,2 (Ω) for some p> 3. Certain L ∞ -estimates are also established. Our results may be used to improve the regularity of strong solutions of Navier-Stokes equations in nonsmooth domains. In the ap- pendix we provide a simple proof of area integral estimates for solutions of Stokes equations.

A boundary integral method for parabolic equations in non-smooth domains

A boundary integral method for parabolic equations in non-smooth domains

A COUNTEREXAMPLE IN THE THEORY OF MIXED BOUNDARY VALUE PROBLEMS FOR ELLIPTIC EQUATIONS IN NON-SMOOTH DOMAINS

Article A COUNTEREXAMPLE IN THE THEORY OF MIXED BOUNDARY VALUE PROBLEMS FOR ELLIPTIC EQUATIONS IN NON-SMOOTH DOMAINS was published on April 1, 1993 in the journal Demonstratio Mathematica (volume 26, issue 2).

Boundary-value problems for partial differential equations in non-smooth domains

CONTENTS Introduction Chapter I. General elliptic boundary-value problems § 1. The solubility of general elliptic boundary-value problems in domains with conic points § 2. The asymptotic behaviour of the solutions of a general boundary-value problem in the neighbourhood of a conic boundary point § 3. General boundary-value problems in non-smooth domains Chapter II. Boundary-value problems for the equations of mathematical physics in non-smooth domains § 1. Boundary-value problems for the system of elasticity theory § 2. Problems of hydrodynamics in domains with a non-smooth boundary § 3. The biharmonic equation Chapter III. Second-order elliptic equations in domains with a non-smooth boundary § 1. Boundary-value problems for second-order elliptic equations in an arbitrary domain § 2. Boundary-value problems in domains with isolated non-regular points on the boundary § 3. Second-order elliptic equations in domains with edges § 4. Boundary-value problems in domains that are diffeomorphic to a polyhedron Chapter IV. Parabolic and hyperbolic equations and systems in non-smooth domains § 1. Parabolic equations and systems in non-smooth domains § 2. Hyperbolic equations and systems in domains with singular points on the boundary References