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Here, we study a level-set forced mean curvature flow with the homogeneous Neumann boundary condition. We first show that the solution is Lipschitz in time and locally Lipschitz in space. Then, under an additional condition on the forcing term, we prove that the solution is globally Lipschitz. We obtain the large time behavior of the solution in this setting and study the large time profile in some specific situations. Finally, we give two examples demonstrating that the additional condition on the forcing term is sharp, and without it, the solution might not be globally Lipschitz. [ABSTRACT FROM AUTHOR]

In this paper, we give a probabilistic interpretation for solutions to the Neumann boundary problems for a class of semi-linear parabolic partial differential equations (PDEs for short) with singular non-linear divergence terms. This probabilistic approach leads to the study on a new class of backward stochastic differential equations (BSDEs for short). A connection between this class of BSDEs and semi-linear PDEs is established. [ABSTRACT FROM AUTHOR]

This paper investigates the simultaneous recovery for two time-dependent coefficients for heat equation under Neumann boundary condition. This problem is considered under extra conditions of nonlocal type. The main issue with this problem is the solution unstable to small contamination of noise in the input data. The Crank-Nicolson finite difference method is utilized to solve the direct problem whilst the inverse problem is viewed as nonlinear optimization problem. The later problem is solved numerically using optimization toolbox from MATLAB. We found that the numerical results are accurate and stable. [ABSTRACT FROM AUTHOR]

Let n ≥ 2 and Ω be a bounded Lipschitz domain in R n. In this article, the authors investigate global (weighted) norm estimates for the gradient of solutions to Neumann boundary value problems of second order elliptic equations of divergence form with real-valued, bounded, measurable coefficients in Ω. More precisely, for any given p ∈ (2 , ∞) , two necessary and sufficient conditions for W 1 , p estimates of solutions to Neumann boundary value problems, respectively, in terms of a weak reverse Hölder inequality with exponent p or weighted W 1 , q estimates of solutions with q ∈ [ 2 , p ] and some Muckenhoupt weights, are obtained. As applications, for any given p ∈ (1 , ∞) and ω ∈ A p (R n) (the class of Muckenhoupt weights), the authors establish weighted W ω 1 , p estimates for solutions to Neumann boundary value problems of second order elliptic equations of divergence form with small BMO coefficients on bounded (semi-)convex domains. As further applications, the global gradient estimates are obtained, respectively, in (weighted) Lorentz spaces, (Lorentz–)Morrey spaces, (weighted) Orlicz spaces, and variable Lebesgue spaces. [ABSTRACT FROM AUTHOR]

Based on the theory of the Neumann boundary value problem of geodesy on the geocentric reference ellipsoid as the boundary surface. In the Helmert space, derive the ellipsoidal series expansion of harmonic functions outside the referential ellipsoidal, the relationship of second kinds of associa-ted Legendre functions and its derivatives recurrence formula, Helmert disturbing potential and its integral kernel function with ellipsoidal harmonic formula,in order to research the Neumann boundary value problem of geodesy in the ellipsoidal coordinates. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence of boundary bubbling solutions for a supercritical Neumann problem with mixed nonlinearities under a suitable assumption on the mean curvature of the boundary ∂Ω, the behavior of the solutions is as a tower of k bubbles when the exponent goes to Sobolev critical exponent, and the blow-up point is a critical point of the mean curvature. [ABSTRACT FROM AUTHOR]

In this paper, we mainly employed the idea of the previous paper [34] to study the sharp uniform W 1 , p estimates with 1 < p ≤ ∞ for more general elliptic systems with the Neumann boundary condition on a bounded C 1 , η domain, arising in homogenization theory. Based on the skills developed by Z. Shen in [27] and by T. Suslina in [31,32] , we also established the L 2 convergence rates on a bounded C 1 , 1 domain and a Lipschitz domain, respectively. Here we found a “rough” version of the first order correctors (see (1.12) ), which can unify the proof in [27] and [32] . It allows us to skip the corresponding convergence results on R d that are the preconditions in [31,32] . Our results can be regarded as an extension of [23] developed by C. Kenig, F. Lin, Z. Shen, as well as of [32] investigated by T. Suslina. [ABSTRACT FROM AUTHOR]

The performance of a well installed under-specific situation can accurately be reviewed from pumping-test data with the use of analytical/semi-analytical solution derived for the said state. For verifying the performance of the derived solution, it is best to compare its results with the field observations. Nevertheless, in comparison to constant pumping-rate well (Neumann boundary problem) test results, availability of documented data from the field studies of constant draw-down/flowing wells (Dirichlet boundary problem) is very rare. Hence, a novel method is proposed by which the performance of an analytical/semi-analytical solution derived for constant draw-down/flowing well can be verified using constant pumpingrate well test data before its application to the actual field problems The proposed method is demonstrated by comparing and verifying the results of a standard semi-analytical solution for a constant-drawdown well (i.e. flowing well) with the constant pumping-rate well-test observations. [ABSTRACT FROM AUTHOR]

In the paper, properties of orbit functions are reviewed and further developed. Orbit functions on the Euclidean space E_n are symmetrized exponential functions. The symmetrization is fulfilled by a Weyl group corresponding to a Coxeter-Dynkin diagram. Properties of such functions will be described. An orbit function is the contribution to an irreducible character of a compact semisimple Lie group G of rank n from one of its Weyl group orbits. It is shown that values of orbit functions are repeated on copies of the fundamental domain F of the affine Weyl group (determined by the initial Weyl group) in the entire Euclidean space E_n. Orbit functions are solutions of the corresponding Laplace equation in E_n, satisfying the Neumann condition on the boundary of F. Orbit functions determine a symmetrized Fourier transform and a transform on a finite set of points.