Non-smooth Domains Research Articles

Second-Order Regularity Estimates for Singular Schrödinger Equations on Convex Domains

LetΩ⊂ℝnbe a nonsmooth convex domain and letfbe a distribution in the atomic Hardy spaceHatp(Ω); we study the Schrödinger equations-div⁡(A∇u)+Vu=finΩwith the singular potentialVand the nonsmooth coefficient matrixA. We will show the existence of the Green function and establish theLpintegrability of the second-order derivative of the solution to the Schrödinger equation onΩwith the Dirichlet boundary condition forn/(n+1)<p≤2. Some fundamental pointwise estimates for the Green function are also given.

The necessary and sufficient conditions for the solutions of elliptic problems

This is the necessary and sucient conditions to the regularity of solution of elliptic problems on nonsmooth domains in R3. I study a boundary value problem for elliptic partial dierential equation.I study the regularity of solution to the problem in non smooth domain. I obtain the necessary andsucient conditions of the problem to belong to Cm+2+:

On Neumann and oblique derivatives boundary conditions for nonlocal elliptic equations

Inspired by the penalization of the domain approach of Lions and Sznitman, we give a sense to Neumann and oblique derivatives boundary value problems for nonlocal, possibly degenerate elliptic equations. Two different cases are considered: (i) homogeneous Neumann boundary conditions in convex, possibly non-smooth and unbounded domains, and (ii) general oblique derivatives boundary conditions in smooth, bounded, and possibly non-convex domains. In each case we give appropriate definitions of viscosity solutions and prove uniqueness of solutions of the corresponding boundary value problems. We prove that these boundary value problems arise in the penalization of the domain limit from whole space problems and obtain as a corollary the existence of solutions of these problems.

General boundary value problems for pseudo-differential equations and related difference equations

The author develops the theory of pseudo-differential equations and boundary value problems in non-smooth domains. A model pseudo-differential equation in a canonical flat domain is reduced to a system of linear difference equations.

Asymptotic Behavior of Positive Harmonic Functions in Certain Unbounded Domains

We derive asymptotic estimates at infinity for positive harmonic functions in a large class of non-smooth unbounded domains. These include domains whose sections, after rescaling, resemble a Lipschitz cylinder or a Lipschitz cone, e.g., various paraboloids and horns.

Operator Calculus of Differential Chains and Differential Forms

We describe a subspace of de Rham currents called “differential chains,” given as an inductive limit of Banach spaces. We make use of the viewpoint introduced by Mackey and developed by Whitney in which currents are treated as chains instead of cochains. This approach gives our space and its topology an intrinsic geometric definition. Moreover, our sequence of norms gives fine control over convergence and compactness, providing viable alternatives to the weak and flat topologies on currents. Boundary is a continuous operator, as are operators that dualize to Hodge star, Lie derivative, pullback and interior product. Partitions of unity exist in this setting, and we are also able to define a Cartesian wedge product. We conclude with an application of this framework, generalizing the Reynolds’ Transport Theorem to nonsmooth domains.

Boundary value problems and integral operators for the bi-Laplacian in non-smooth domains

In this paper we explore the effectiveness of the classical method of layer potentials in the treatment of boundary value problems for the bi-Laplacian formulated in arbitrary Lipschitz domains, Lipschitz domains whose outward unit normal has small mean-oscillations, and domains of class \mathcal C^1 .

Weighted L p -estimates for Elliptic Equations with Measurable Coefficients in Nonsmooth Domains

We obtain a global weighted $L^p$ estimate for the gradient of the weak solutions to divergence form elliptic equations with measurable coefficients in a nonsmooth bounded domain. The coefficients are assumed to be merely measurable in one variable and to have small BMO semi-norms in the remaining variables, while the boundary of the domain is supposed to be Reifenberg flat, which goes beyond the category of domains with Lipschitz continuous boundaries. As consequence of the main result, we derive global gradient estimate for the weak solution in the framework of the Morrey spaces which implies global Hoelder continuity of the solution.

Asymptotic of solutions for second IBVP for hyperbolic systems in non-smooth domains

The purpose of this paper is to study the asymptotic behaviour of solutions of the second initial boundary value problem for hyperbolic systems in an infinite cylinder with the base containing conical points. Some new results on the asymptotics of solutions of elliptic problem depending on a parameter are also given.

On Serrin's symmetry result in nonsmooth domains and its applications

The paper investigates overdetermined fully nonlinear problems on nonsmooth domains. Under natural regularity assumptions on solutions it is shown that overdetermined problems on reflectionally symmetric, bounded domains can have positive solutions only if the domain is a ball. These results are extensions of results of Serrin, who proved this statement for smooth solutions on smooth domains. The results for overdetermined problems are applied to a study of Dirichlet problems, specifically to the question when a nonnegative solution is positive or zero everywhere. As a consequence, an extension of symmetry results of Gidas--Ni--Nirenberg to nonnegative solutions is obtained.

Wolff–Denjoy theorems in nonsmooth convex domains

We give a short proof of Wolff–Denjoy theorem for (not necessarily smooth) strictly convex domains. With similar techniques we are also able to prove a Wolff–Denjoy theorem for weakly convex domains, again without any smoothness assumption on the boundary.

The Hardy–Littlewood lemma and the estimate of the [formula omitted]-Neumann problem in a general norm

We prove a generalized Hardy–Littlewood lemma on a non-smooth domain in the “f-norm” and give an application to a corresponding estimate for the ∂̄-Neumann problem by means of suitable weights.

Global gradient estimates for general nonlinear parabolic equations in nonsmooth domains

We establish the natural Calderón–Zygmund theory for a nonlinear parabolic equation of p-Laplacian type in divergence form,(0.1)ut−diva(Du,x,t)=div(|F|p−2F)in ΩT, by essentially proving that(0.2)|F|p∈Lq(ΩT)⇒|Du|p∈Lq(ΩT), for every q∈[1,∞). The equation under consideration is of general type and not necessarily of variation form, the involved nonlinearity a=a(ξ,x,t) is assumed to have a small BMO semi-norm with respect to (x,t)-variables and the lateral boundary ∂Ω of the domain is assumed to be δ-Reifenberg flat. As a consequence, we are able to not only relax the known regularity requirements on the nonlinearity for such a regularity theory, but also extend local results to a global one in a nonsmooth domain whose boundary has a fractal property. We also find an optimal regularity estimate in Orlicz–Sobolev spaces for such nonlinear parabolic problems.

Further analysis of multilevel Monte Carlo methods for elliptic PDEs with random coefficients

We consider the application of multilevel Monte Carlo methods to elliptic PDEs with random coefficients. We focus on models of the random coefficient that lack uniform ellipticity and boundedness with respect to the random parameter, and that only have limited spatial regularity. We extend the finite element error analysis for this type of equation, carried out in Charrier et al. (SIAM J Numer Anal, 2013), to more difficult problems, posed on non-smooth domains and with discontinuities in the coefficient. For this wider class of model problem, we prove convergence of the multilevel Monte Carlo algorithm for estimating any bounded, linear functional and any continuously Fréchet differentiable non-linear functional of the solution. We further improve the performance of the multilevel estimator by introducing level dependent truncations of the Karhunen–Loève expansion of the random coefficient. Numerical results complete the paper.

Weighted W 1,p estimates for solutions of non-linear parabolic equations over non-smooth domains

Weighted <i>W</i> ^1,<i>p</i> estimates for solutions of non-linear parabolic equations over non-smooth domains

A note on the regularity of thermally coupled viscous flows with critical growth in nonsmooth domains

We consider a coupled model for steady flows of viscous incompressible heat‐conducting fluids with dissipative and adiabatic heating and temperature‐dependent material coefficients in a plane bounded domain. The existence of a strong solution is proven by a fixed‐point technique based on Schauder theorem for sufficiently small external forces. Copyright © 2012 John Wiley &amp; Sons, Ltd.

An Accurate Boundary Value Problem Solver Applied to Scattering From Cylinders With Corners

In this paper we consider the classic problems of scattering of waves from perfectly conducting cylinders with piecewise smooth boundaries. The scattering problems are formulated as integral equations and solved using a Nystr\"om scheme where the corners of the cylinders are efficiently handled by a method referred to as Recursively Compressed Inverse Preconditioning (RCIP). This method has been very successful in treating static problems in non-smooth domains and the present paper shows that it works equally well for the Helmholtz equation. In the numerical examples we specialize to scattering of E- and H-waves from a cylinder with one corner. Even at a size kd=1000, where k is the wavenumber and d the diameter, the scheme produces at least 13 digits of accuracy in the electric and magnetic fields everywhere outside the cylinder.

Concentration of solutions for a singularly perturbed mixed problem in non-smooth domains

We consider a singularly perturbed problem with mixed Dirichlet and Neumann boundary conditions in a bounded domain Ω⊂Rn whose boundary has an (n−2)-dimensional singularity. Assuming 1<p<n+2n−2, we prove that, under suitable geometric conditions on the boundary of the domain, there exist solutions which approach the intersection of the Neumann and the Dirichlet parts as the singular perturbation parameter tends to zero.

Principal Eigenvalues for Generalised Indefinite Robin Problems

We consider the principal eigenvalue of generalised Robin boundary value problems on non-smooth domains, where the zero order coefficient of the boundary operator is negative or changes sign. We provide conditions so that the related eigenvalue problem has a principal eigenvalue. We work with the framework involving measure data on the boundary due to Arendt and Warma (Potential Anal 19:341–363, 2003). Examples of simple domains with cusps are used to illustrate all possible phenomena.

Parameter identification of Bouc-Wen model for MR fluid dampers using adaptive charged system search optimization

In this article, the charged system search (CSS) optimization method is improved to identify the parameters of a non-linear hysteretic Bouc-Wen differential model. The CSS is suitable for those optimization problems involving non-smooth or non-convex domains. Bouc-Wen is a well-established non-linear model which has been used to portray the hysteretic and high non-linear real behavior of numerous physical and mechanical systems. To improve the effectiveness and adaptability of the CSS algorithm, it is combined with sub-optimization mechanism. The obtained results show that the adaptive CSS embodies great robustness and accuracy to be successfully employed in such highly non-linear identification problems.