Non-smooth Domains Research Articles

Global gradient estimates for shear thinning-type Stokes system on the non-smooth domains

In this paper, we present the global $L^q$ estimates for the weak solution of the steady p-Stokes equations describing the motion of the shear-thinning flow under the nonslip boundary condition. Our interest is a non-smooth domain whose boundary can go beyond the Lipschitz category, and the coefficients belong to BMO(Bounded Mean Oscillation) space with sufficiently small BMO semi-norm.

Uniqueness, regularity, continuity, and stability of solutions of integral equations in irregular domains

This study explores the uniqueness, regularity, continuity, and stability of solutions to integral equations related to elliptical boundary value problems in irregular domains. Traditional methods usually consider soft boundaries, but this work extends these results to include domains with rough boundaries. By employing Sobolev spaces, especially fractional Sobolev spaces, we develop a comprehensive theoretical framework. Under appropriate conditions, the studied integral equation has a unique solution in fractional Sobolev spaces, maintaining the regularity properties of the input function. Two new theorems are introduced to strengthen the findings. The first theorem establishes that solutions are continuous with respect to perturbations in the input data. This means small changes in the input do not cause significant deviations in the solution, ensuring robustness in practical applications. The second theorem demonstrates the stability of the solutions when the domain geometry undergoes small changes. This stability is crucial for real-world scenarios where the exact shapes of domains may not be perfectly known or may vary slightly. These results significantly improve the mathematical understanding of boundary value problems in non-smooth domains. The proposed extended framework allows for more generalized applications, addressing challenges in various fields of physics and engineering. The ability to handle irregular domains with rough boundaries opens up new possibilities for modeling and solving complex problems where traditional soft boundary assumptions are not valid. This research provides a robust basis for future studies and applications, highlighting the importance of considering fractional Sobolev spaces in the analysis of integral equations.

An essay on the foundations of variational methods: Exploring Sobolev Spaces for boundary integral equations

This work addresses the uniqueness and regularity of solutions to integral equations associated with elliptic boundary value problems in irregular domains. Traditional results often assume smooth (Lipschitz) boundaries, but this study extends these results to more general domains with irregular boundaries. By leveraging Sobolev spaces, particularly fractional Sobolev spaces , and the properties of the Slobodetskii norm, we develop a robust theoretical framework. Our main theorem demonstrates that, under suitable conditions, has a unique solution in , and this solution inherits the regularity properties from the function . The results provide significant advancements in the mathematical understanding of boundary value problems in non-smooth domains, with potential applications in various fields of physics and engineering.

Nonlocal energy functionals and determinantal point processes on non-smooth domains

Nonlocal energy functionals and determinantal point processes on non-smooth domains

Elliptic equations with matrix weights and measurable nonlinearities on nonsmooth domains

We study general elliptic equations with singular/degenerate matrix weights and measurable nonlinearities on nonsmooth bounded domains to obtain a global Calderón-Zygmund type estimate under possibly minimal assumptions that the logarithm of the matrix weight has a small bounded mean oscillation (BMO) norm, the nonlinearity is allowed to be merely measurable in one variable but has a small BMO norm in the other variables and that the boundary of the domain is sufficiently flat in Reifenberg sense.

Fractional Besov spaces and Hardy inequalities on bounded non-smooth domains

Fractional Besov spaces and Hardy inequalities on bounded non-smooth domains

THE REGULARITY OF SOLUTIONS PROBLEMS TO NONLINEAR ELLIPTIC EQUATIONS

Abstract. We obtained estimates in generalized Morrey spaces are used to study global regularity of the solution of the boundary problems of nonlinear elliptic equations in divergence form over a bounded non-smooth domain. For these is we used of Calderon-Zygmund theory. The main goal of the article is to obtain Hölder estimates for solutions to the boundary problems for nonlinear elliptic equations.

Bounds on the Optimal Radius When Covering a Set with Minimum Radius Identical Disks

The problem of covering a two-dimensional bounded set with a fixed number of minimum-radius identical disks is studied in the present work. Bounds on the optimal radius are obtained for a certain class of nonsmooth domains, and an asymptotic expansion of the bounds as the number of disks goes to infinity is provided. The proof is based on the approximation of the set to be covered by hexagonal honeycombs and on the thinnest covering property of the regular hexagonal lattice arrangement in the whole plane. The dependence of the optimal radius on the number of disks is also investigated numerically using a shape-optimization approach, and theoretical and numerical convergence rates are compared. An initial point construction strategy is introduced, which, in the context of a multistart method, finds good-quality solutions to the problem under consideration. Extensive numerical experiments with a variety of polygonal regions and regular polygons illustrate the introduced approach. Funding: This work was supported by Fundação de Amparo à Pesquisa do Estado de São Paulo [Grants 2013/07375-0, 2016/01860-1, 2018/24293-0, and 2019/25258-7] and Conselho Nacional de Desenvolvimento Científico e Tecnológico [Grants 302682/2019-8, 303243/2021-0, 304258/2018-0, and 408175/2018-4].

Calderón-Zygmund estimates for nonlinear parabolic equations with matrix weights on nonsmooth domains

A wider class of nonlinear parabolic equations with degenerate/singular matrix weights is studied for Calderón-Zygmund type estimates of weak solutions in the setting of weighted Sobolev spaces. Minimal regularity assumptions on the associated nonlinearities as well as weights are investigated for such classical estimates.

Lp boundedness of Forelli-Rudin type operators on the Hartogs triangle

The Hartogs triangle H={(z1,z2)∈C2:|z1|<|z2|<1}, compared with the unit ball, is a non-smooth pseudoconvex domain which reveals various remarkable phenomena in several complex variables. Motivated by the recent work of Kures and Zhu (2006) [14] over the unit ball setting, in this paper, we completely characterize Lp boundedness of Forelli-Rudin type integral operators on the Hartogs triangle for all 1≤p≤∞.

System of Nonlinear Second-Order Parabolic Partial Differential Equations with Interconnected Obstacles and Oblique Derivative Boundary Conditions on Non-Smooth Time-Dependent Domains

System of Nonlinear Second-Order Parabolic Partial Differential Equations with Interconnected Obstacles and Oblique Derivative Boundary Conditions on Non-Smooth Time-Dependent Domains

The coefficients in the asymptotic expansion of solutions of second‐order hyperbolic problems in polygonal domains

Solutions of boundary value problems for linear partial differential equations are known to exhibit singular behaviors near the boundary of nonsmooth domains. In this case, one is usually interested on the asymptotic behavior of the solutions near the geometric singularities. However, for both mathematical and engineering purposes, it is important to have extraction formulas for the coefficients in the asymptotic expansion. In this paper, we consider initial boundary value problems for the wave equation in two‐dimensional domains with corners and derive, by means of Fourier analysis, explicit computable expressions for the coefficients in the asymptotic expansion of the solution near the corners. The explicit structure of the singular solutions presented herein makes it fairly easy to compute numerically. Thus, the formulas can be used to construct postprocessing procedures for improving the accuracy of standard numerical techniques of approximating the solution and for improving the rates of convergence in the error estimates. The analysis presented herein would also apply to more general hyperbolic or parabolic equations with constant coefficients.

Geometric singularities and regularity of solution of the Stokes system in nonsmooth domains

This paper deals with the geometrical singularities of the weak solution of the mixed boundary value problem governed by the stationary Stokes system in two-dimensional nonsmooth domains with corner points and points at which the type of boundary conditions changes. The presence of these points on the boundary generally generates local singularities in the solution. We will see the impact of the geometrical singularities of the boundary or the mixed boundary conditions on the qualitative properties of the solution including its regularity. Moreover, the asymptotic singular representations for the solution which inherently depend on the zeros of certain transcendental functions are presented.

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.

On the relaxation of functionals with contact terms on non-smooth domains

We provide the integral representation formula for the relaxation in $BV(\Omega; \mathbb{R}^M)$ with respect to strong convergence in $L^1(\Omega; \mathbb{R}^M)$ of a functional with a boundary contact energy term. This characterization is valid for a large class of surface energy densities, and for domains satisfying mild regularity assumptions. Motivated by some classical examples where lower semicontinuity fails, we analyze the extent to which the geometry of the set enters the relaxation procedure.

Global gradient estimates in directional homogenization

&lt;abstract&gt;&lt;p&gt;In this research, we investigate a higher regularity result in periodic directional homogenization for divergence-form elliptic systems with discontinuous coefficients in a bounded nonsmooth domain. The coefficients are assumed to have small bounded mean oscillation (BMO) seminorms and the domain has the $ \delta $-Reifenberg property. Under these assumptions we derive global uniform Calderón-Zygmund estimates by proving that the gradient of the weak solution is as integrable as the given nonhomogeneous term.&lt;/p&gt;&lt;/abstract&gt;

Calderón–Zygmund theory for asymptotically regular nonlinear elliptic problems with double obstacles

ABSTRACT We study nonlinear elliptic double obstacles problems with an asymptotically regular nonlinearity in a non-smooth domain. Here, we obtain a global Calderón–Zygmund type estimate in the setting of Lorentz spaces by making use of the perturbation method. More precisely, we approximate the solutions of asymptotically regular double obstacles problems to the solutions of regular single obstacle problems when the gradients of solutions are close to infinity.

Stability Results for the Robin-Laplacian on Nonsmooth Domains

Stability Results for the Robin-Laplacian on Nonsmooth Domains

Passive earth pressures with sloping backfill based on a statically admissible stress field

Passive earth pressure is a fundamental problem in the field of soil-structure interaction. In this study, a fully plastic solution obeying the Mohr-Coulomb criterion was successfully developed to estimate passive earth pressure against rigid retaining walls with various sloping backfills, wall roughness, and wall rear face inclination. The numerical procedures using a shooting technique and the 5th order adaptive Runge-Kutta method were successfully developed for boundary-value and initial-value problems, respectively. The developed procedures successfully coped with the numerical integral in non-smooth domains then statically admissible stress fields in the sloping backfills are explored. The stress distributions together with the trajectories of principal stresses in various sloping backfills are presented to demonstrate the load transfer mechanisms of cohesionless soil in various geometries of the backfill. The effects of the wall roughness, the wall rear face inclination, and backfill inclination on the load transfer mechanism of the cohesionless soil behind the retaining wall are investigated. The obtained results of passive earth pressure coefficients (Kp) are validated by comparing with an exact solution and some well-known experimental works. Thus, highly accurate values of passive earth pressure coefficients on rigid retaining walls with various sloping backfills, wall roughness, and wall rear face inclinations are provided for engineering practice.

W 1,γ(·)-estimate to non-uniformly elliptic obstacle problems with borderline growth

ABSTRACT In this paper, we are devoted to a global -estimate for elliptic obstacle problem with double phase growth for the borderline setting in nonsmooth domain. More precisely, we consider the variational inequality whose characteristics of ellipticity and growth switch between a type of polynomial and its logarithm, then we prove a variable exponent Calderón-Zygmund estimate with an implication that provided that the nonlinearity satisfies a small BMO condition while the boundary of the underlying domain is flat in the sense of Reifenberg.