Regular Boundary Research Articles

The Relative Index Theorem for General First-Order Elliptic Operators

The relative index theorem is proved for general first-order elliptic operators that are complete and coercive at infinity over measured manifolds. This extends the original result by Gromov–Lawson for generalised Dirac operators as well as the result of Bär–Ballmann for Dirac-type operators. The theorem is seen through the point of view of boundary value problems, using the graphical decomposition of elliptically regular boundary conditions for general first-order elliptic operators due to Bär–Bandara. Splitting, decomposition and the Phi-relative index theorem are proved on route to the relative index theorem.

Interior and boundary regularity criteria for the 6D steady Navier-Stokes equations

It is shown in this paper that suitable weak solutions to the 6D steady incompressible Navier-Stokes are Hölder continuous at 0 provided that ∫B1|u(x)|3dx+∫B1|f(x)|qdx or ∫B1|∇u(x)|2dx+∫B1|∇u(x)|2dx(∫B1|u(x)|dx)2+∫B1|f(x)|qdx with q>3 is sufficiently small, which implies that the 2D Hausdorff measure of the set of singular points is zero. For the boundary case, we also obtain that 0 is regular provided that ∫B1+|u(x)|3dx+∫B1+|f(x)|3dx or ∫B1+|∇u(x)|2dx+∫B1+|f(x)|3dx is sufficiently small. These results improve previous regularity theorems by Dong-Strain ([8], Indiana Univ. Math. J., 2012), Dong-Gu ([7], J. Funct. Anal., 2014), and Liu-Wang ([27], J. Differential Equations, 2018), where either the smallness of the pressure or the smallness of the scaling invariant quantities on all balls is necessary.

Correct and Stable Algorithm for Numerical Solving Nonlocal Heat Conduction Problems with Not Strongly Regular Boundary Conditions

For a nonlocal initial-boundary value problem for a one-dimensional heat equation with not strongly regular boundary conditions of general type, an approximate difference scheme with weights is constructed. A correct and stable algorithm for the numerical solving of the difference problem is proposed. It is proven that the difference scheme with weights is stable and its solution converges to the exact solution of the differential problem in the grid L2h-norm. Stability conditions are established. An estimate of the numerical solution with respect to the initial data and the right-hand side of the difference problem is given.

Approximation of Green functions and domains with uniformly rectifiable boundaries of all dimensions

The present paper concerns divergence form elliptic and degenerate elliptic operators in a domain Ω⊂Rn, and establishes the equivalence between the uniform rectifiability of the boundary E=∂Ω and weak Carleson condition on the good approximation of the Green function G by affine, or distance, functions. There are two main original contexts for the results, elliptic operators in a non-tangential access domain with an n−1 dimensional boundary and degenerate elliptic operators adapted to a domain with an Ahlfors regular boundary of larger co-dimension. In both cases necessary and sufficient conditions are given, in the form of Carleson packing conditions on the collection of balls centered on E where G is not well approximated.(1)This is the first time the underlying property of the control of the Green function by affine functions, or by the distance to the boundary, in the sense of the Carleson prevalent sets, appears in the literature; some results established here are new even in the half space;(2)the results are optimal, providing a full characterization of uniform rectifiability under the (standard) mild topological assumptions;(3)to the best of the authors' knowledge, even in traditional domains with (n−1)-dimensional boundaries, this is the first free boundary result applying to all elliptic operators, without any restriction on the coefficients (the direct one assumes the standard, and necessary, Carleson measure condition);(4)this is the first free boundary result in higher co-dimensional setting and as such, the first PDE characterization of uniform rectifiability for a set of dimension d, d<n−1, in Rn. The paper offers a general way to deal with related issues considerably beyond the scope of the aforementioned theorem, including the question of approximability of the gradient of the Green function, and the comparison of the Green function to a certain version of the distance to the original set rather than distance to the hyperplanes.

Finite element model for aeroelastic instability analysis of rectangular flat plates in supersonic airflow under various regular and irregular boundary conditions

Finite element model for aeroelastic instability analysis of rectangular flat plates in supersonic airflow under various regular and irregular boundary conditions

Half-Space Solutions with 7/2 Frequency in the Thin Obstacle Problem

For the thin obstacle problem in 3d, we show that half-space solutions form an isolated family in the space of $7/2$-homogeneous solutions. For a general solution with one blow-up profile in this family, we establish the rate of convergence to this profile. As a consequence, we obtain regularity of the free boundary near such contact points.

The classical obstacle problem with Hölder continuous coefficients

Weiss’ and Monneau’s type quasi-monotonicity formulas are established for quadratic energies having matrix of coefficients which are Dini, double-Dini continuous, respectively. Free boundary regularity for the corresponding classical obstacle problems under Hölder continuity assumptions is then deduced.

Boundary regularity for strongly degenerate operators of Grushin type

Boundary regularity for strongly degenerate operators of Grushin type

Boundary Regularity for Minimal Graphs of Higher Codimensions

Boundary Regularity for Minimal Graphs of Higher Codimensions

Properties of the free boundaries for the obstacle problem of the porous medium equations

Abstract In this paper, we study the existence and interior W 2 , p {W^{2,p}} -regularity of the solution, and the regularity of the free boundary ∂ ⁡ { u &gt; ϕ } {\partial\{u&gt;\phi\}} to the obstacle problem of the porous medium equation, u t = Δ ⁢ u m {u_{t}=\Delta u^{m}} ( m &gt; 1 {m&gt;1} ) with the obstacle function ϕ. The penalization method is applied to have the existence and interior regularity. To deal with the interaction between two free boundaries ∂ ⁡ { u &gt; ϕ } {\partial\{u&gt;\phi\}} and ∂ ⁡ { u &gt; 0 } {\partial\{u&gt;0\}} , we consider two cases on the initial data which make the free boundary ∂ ⁡ { u &gt; ϕ } {\partial\{u&gt;\phi\}} separate from the free boundary ∂ ⁡ { u &gt; 0 } {\partial\{u&gt;0\}} . Then the problem is converted into the obstacle problem for a fully nonlinear operator. Hence, the C 1 {C^{1}} -regularity of the free boundary ∂ ⁡ { u &gt; ϕ } {\partial\{u&gt;\phi\}} is obtained by the regularity theory of a class of obstacle problems for the general fully nonlinear operator.

Weighted Iterative CD-Spline for Mitigating Occlusion Effects on Building Boundary Regularization Using Airborne LiDAR Data.

Building occlusions usually decreases the accuracy of boundary regularization. Thus, it is essential that modeling methods address this problem, aiming to minimize its effects. In this context, we propose a weighted iterative changeable degree spline (WICDS) approach. The idea is to use a weight function for initial building boundary points, assigning a lower weight to the points in the occlusion region. As a contribution, the proposed method allows the minimization of errors caused by the occlusions, resulting in a more accurate contour modeling. The conducted experiments are performed using both simulated and real data. In general, the results indicate the potential of the WICDS approach to model a building boundary with occlusions, including curved boundary segments. In terms of Fscore and PoLiS, the proposed approach presents values around 99% and 0.19 m, respectively. Compared with the previous iterative changeable degree spline (ICDS), the WICDS resulted in an improvement of around 6.5% for completeness, 4% for Fscore, and 0.24 m for the PoLiS metric.

Optimal boundary regularity for viscosity solutions of fully nonlinear elliptic equations

We establish optimal boundary regularity results for viscosity solutions to second order fully nonlinear uniformly elliptic equations in the form F(D2u(x),Du(x),x)=f(x) in Ω. In particular, we obtain sharp estimates for borderline cases f∈Ln(Ω) and f∈BMO(Ω). For source functions in BMO(Ω), we obtain C1,Log−Lip interior regularity for flat solutions of non-convex elliptic equations. As a consequence, we obtain C1,Log−Lip estimates near the boundary; which again is an optimal estimate.

Abstract representation of the SMGTJ equation under rough boundary controls: Optimal interior regularity

We consider the linearized third order SMGTJ equation defined on a sufficiently smooth boundary domain in and subject to either Dirichlet or Neumann rough boundary control. Filling a void in the literature, we present a direct general system approach based on the vector state solution {position, velocity, acceleration}. It yields, in both cases, an explicit representation formula: input solution, based on the s.c. group generator of the boundary homogeneous problem and corresponding elliptic Dirichlet or Neumann map. It is close to, but also distinctly and critically different from, the abstract variation of parameter formula that arises in more traditional boundary control problems for PDEs L‐T.6. Through a duality argument based on this explicit formula, we provide a new proof of the optimal regularity theory: boundary control {position, velocity, acceleration} with low regularity boundary control, square integrable in time and space.

Boundary regularity of mixed local-nonlocal operators and its application

Let $$\Omega$$ be a bounded $$C^2$$ domain in $${\mathbb {R}^{n}}$$ and $$u\in C({\mathbb {R}^{n}})$$ solves $$\begin{aligned} \begin{aligned} \Delta u + a Iu + C_0|Du| \ge -K\quad \text {in}\; \Omega , \quad \Delta u + a Iu - C_0|Du|\le K \quad \text {in}\; \Omega , \quad u=0\quad \text {in}\; \Omega ^c, \end{aligned} \end{aligned}$$ in the viscosity sense, where $$0\le a\le A_0$$ , $$C_0, K\ge 0$$ , and I is a suitable nonlocal operator. We show that $$u/\delta$$ is in $$C^{\upkappa }(\bar{\Omega })$$ for some $$\upkappa \in (0,1)$$ , where $$\delta (x)=\mathop {\mathrm {dist}}\limits (x, \Omega ^c)$$ . Using this result, we also establish that $$u\in C^{1, \gamma }(\bar{\Omega })$$ . Finally, we apply these results to study an overdetermined problem for mixed local-nonlocal operators.

Existence and regularity of source-type self-similar solutions for stable thin-film equations

We investigate the existence and the boundary regularity of source-type self-similar solutions to the thin-film equation $h\_t=-(h^nh\_{zzz})z+(h^{n+3}){zz},$ $t>0,; z\in \mathbb{R};; h(0,z)= \omega \delta(z)$ where $n\in (3/2,3),; \omega > 0$ and $\delta$ is the Dirac mass at the origin. It is known that the leading order expansion near the edge of the support coincides with that of a traveling-wave solution for the standard thin-film equation: $h\_t=-(h^nh\_{zzz})\_z$. In this paper we sharpen this result, proving that the higher-order corrections are analytic with respect to three variables: the first one is just the {spatial} variable, whereas the second and the third (except for $n = 2$) are irrational powers of it. It is known that this third variable does not appear for the thin-film equation without gravity.

Prediction of elevation of substantial fluctuation in coal seam floor based on surface spline function and its derivative

With the increasing depth of coal mining, the geological stress and structure becomes more and more complex. The elevation of No.8 coal floor significantly undulates in the studied coal mine. Even though a lot of boreholes have been drilled, it remains difficult to predict the spatial distribution features and it becomes challenging to plan the tunnel for the working face. Consequently, there has been a great loss of production in the coal mine, therefore, it is of great significance to study the prediction method for coal floor elevation of the working face. The surface spline function can be regarded as an infinite flat plate deformation in pure bending. The fluctuation of the coal floor can be considered to be the result of multi-period tectonic stress applied on the coal seam, so, the prediction of elevation of the coal floor with surface spline method is feasible. An advantage of surface spline method is that any order differentiable smooth surface can be obtained without regular known lattice and boundary derivatives. In the paper, the complete expression of surface spline function is derived, which is used to predict the elevation of No.8 coal floor of the coal mine. The results show that the trend of elevation of the coal floor can be extrapolated by known data, and the maximum error is 20 m, the minimum error is 0 m, and the error of 80% of the data is less than 3 m. The general trend of the coal floor has been predicted well. However, local peaks and valleys could not be predicted correctly, therefore, the first and second order derivative are projected to predict the peaks and valleys in the head of the tunneling.

A Numerical Method for Singularly Perturbed Convection-diffusion Problems Posed on Smooth Domains

A finite difference method is constructed to solve singularly perturbed convection-diffusion problems posed on smooth domains. Constraints are imposed on the data so that only regular exponential boundary layers appear in the solution. A domain decomposition method is used, which uses a rectangular grid outside the boundary layer and a Shishkin mesh, aligned to the curvature of the outflow boundary, near the boundary layer. Numerical results are presented to demonstrate the effectiveness of the proposed numerical algorithm.

Interior and up to the boundary regularity for the fractional [formula omitted]-Laplacian: The convex case

We establish interior and up to the boundary Hölder regularity estimates for weak solutions of the Dirichlet problem for the fractional g−Laplacian with bounded right hand side and g convex. These are the first regularity results available in the literature for integro-differential equations in the context of fractional Orlicz–Sobolev spaces.

On the full regularity of the free boundary for minima of Alt–Caffarelli functionals in Orlicz spaces

In this paper, we discuss two issues about the full regularity of the free boundary for overdetermined Bernoulli-type problems in Orlicz spaces. First, we show that in dimension \(n = 2\) there are no singular points on the free boundary \(F(u) := \partial \{ u &gt; 0 \} \cap \Omega\) of minimizers of the Alt-Caffarelli functional&#x0D; \(J_G(u) := \int_{\Omega} \left( G(\vert \nabla u \vert) + \lambda \chi_{\{ u &gt; 0 \}} \right) dx\)&#x0D; for suitable N-functions \(G\). Next, we prove as a consequence of our main results that there exist a critical dimension \(5 \leq n_0 \leq 7\) and a universal constant \(\varepsilon_0 \in (0,1)\) such that if \(G(t)\) is "\(\varepsilon_0\)-close" of \(t^2\), then for \(2 \leq n &lt; n_0\), \(F(u)\) is a real analytic hypersurface.&#x0D;

Approximate solutions of the Fourth-Order Eigenvalue Problem

In this paper, the differential transformation (DTM) and the Adomian decomposition (ADM) methods are proposed for solving fourth order eigenvalue problem. This fourth order eigenvalue problem has nonstrongly regular boundary conditions. This the fourth order problem has been examined for p(t) = t, B = 0, a = 0,01 where p(t) ≠ 0 is a complex valued and a ≠ 0 The differential transformation and the Adomian decomposition methods are briefly described. An approximate solution is obtained by performing seven iterations with the Adomian decomposition method. The same number of iterations have been made in the differential transformation method. The approximation results obtained by both methods have been compared with each other. These data have been presented in table. The ADM and the DTM approximation solutions have been shown by plotting in Figure 1. Here, the approaches obtained by using the two methods are found to be in high agreement. Consequently, highly accurate approximate solutions of fourth order eigenvalue problem are obtained. Such good results also revealed that the Adomian decomposition and the differential transformation methods are fast, economical and motivating. The exact solution of the fourth order eigenvalue problem for nonstrongly regular can not be found in the literature. Therefore, this study will give an important idea to determine approximate solution behavior of this fourth order problem.