The system of equations for mixed BVP with one Dirichlet boundary condition and three Neumann boundary conditions

Let Ω be a compact complex n + 1-dimensional Hermitian manifold with smooth boundary M. In [2] we proved the following.THEOREM 1. Suppose satisfies condition Z(q) with 0 ≦ q ≦ n. Let □p,q denote the -Laplacian on (p, q) forms onwhich satisfy the -Neumann boundary conditions. Then as t → 0;,(0.1)(If q = n + 1, the -Neumann boundary condition is the Dirichlet boundary condition and the corresponding result is classical.)Theorem 1 is a version for the -Neumann problem of results initiated by Minakshisundaram and Pleijel [8] for the Laplacian on compact manifolds and extended by McKean and Singer [7] to the Laplacian with Dirichlet or Neumann boundary conditions and by Greiner [5] and Seeley [9] to elliptic boundary value problems on compact manifolds with boundary. McKean and Singer go on to show that the coefficients in the trace expansion are integrals of local geometric invariants.

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On the Number of Solutions to Semilinear Boundary Value Problems

Advanced Nonlinear Studies ◽

10.1515/ans-2004-0301 ◽

2004 ◽

Vol 4 (3) ◽

Cited By ~ 4

Author(s):

Markus Kunze ◽

Rafael Ortega

Keyword(s):

Boundary Conditions ◽

Boundary Value Problems ◽

Upper Bound ◽

Dirichlet Boundary ◽

Neumann Boundary ◽

Neumann Boundary Conditions ◽

Dirichlet Boundary Conditions ◽

Number Of Solutions ◽

Semilinear Elliptic ◽

Semilinear Elliptic Problems

AbstractWe consider semilinear elliptic problems of the form Δu + g(u) = f(x) with Neumann boundary conditions or Δu+λ1u+g(u) = f(x) with Dirichlet boundary conditions, and we derive conditions on g and f under which an upper bound on the number of solutions can be obtained.

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A note on boundary conditions in Euclidean gravity

Reviews in Mathematical Physics ◽

10.1142/s0129055x21400043 ◽

2021 ◽

pp. 2140004

Author(s):

Edward Witten

Keyword(s):

Boundary Condition ◽

General Relativity ◽

Perturbation Theory ◽

Quantum Gravity ◽

Boundary Conditions ◽

Dirichlet Boundary Condition ◽

Dirichlet Boundary ◽

Conformal Geometry ◽

Extrinsic Curvature ◽

Euclidean Signature

We review what is known about boundary conditions in General Relativity on a spacetime of Euclidean signature. The obvious Dirichlet boundary condition, in which one specifies the boundary geometry, is actually not elliptic and in general does not lead to a well-defined perturbation theory. It is better-behaved if the extrinsic curvature of the boundary is suitably constrained, for instance if it is positive- or negative-definite. A different boundary condition, in which one specifies the conformal geometry of the boundary and the trace of the extrinsic curvature, is elliptic and always leads formally to a satisfactory perturbation theory. These facts might have interesting implications for semiclassical approaches to quantum gravity. April, 2018

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REINFORCEMENT OF THE POISSON EQUATION BY A THIN LAYER

Mathematical Models and Methods in Applied Sciences ◽

10.1142/s0218202511005283 ◽

2011 ◽

Vol 21 (05) ◽

pp. 1153-1192 ◽

Cited By ~ 5

Author(s):

JINGYU LI ◽

KAIJUN ZHANG

Keyword(s):

Boundary Condition ◽

Shear Modulus ◽

Poisson Equation ◽

Dirichlet Boundary Condition ◽

Dirichlet Boundary ◽

Neumann Boundary ◽

Robin Boundary Condition ◽

Oscillation Period ◽

The Poisson Equation ◽

Robin Boundary

We consider the problem of reinforcing an elastic medium by a strong, rough, thin external layer. This model is governed by the Poisson equation with homogeneous Dirichlet boundary condition. We characterize the asymptotic behavior of the solution as the shear modulus of the layer goes to infinity. We find that there are four types of behaviors: the limiting solution satisfies Poisson equation with Dirichlet boundary condition, Robin boundary condition or Neumann boundary condition, or the limiting solution does not exist. The specific type depends on the integral of the load on the medium, the curvature of the interface and the scaling relations among the shear modulus, the thickness and the oscillation period of the layer.

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Abelian mirror symmetry of $$ \mathcal{N} $$ = (2, 2) boundary conditions

Journal of High Energy Physics ◽

10.1007/jhep03(2021)163 ◽

2021 ◽

Vol 2021 (3) ◽

Author(s):

Tadashi Okazaki

Keyword(s):

Boundary Condition ◽

Boundary Conditions ◽

Mirror Symmetry ◽

Dirichlet Boundary ◽

Triangular Matrix ◽

Neumann Boundary ◽

Dirichlet Boundary Conditions ◽

Verma Modules ◽

Abelian Gauge ◽

Mirror Theory

Abstract We evaluate half-indices of $$ \mathcal{N} $$ N = (2, 2) half-BPS boundary conditions in 3d $$ \mathcal{N} $$ N = 4 supersymmetric Abelian gauge theories. We confirm that the Neumann boundary condition is dual to the generic Dirichlet boundary condition for its mirror theory as the half-indices perfectly match with each other. We find that a naive mirror symmetry between the exceptional Dirichlet boundary conditions defining the Verma modules of the quantum Coulomb and Higgs branch algebras does not always hold. The triangular matrix obtained from the elliptic stable envelope describes the precise mirror transformation of a collection of half-indices for the exceptional Dirichlet boundary conditions.

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On the abundance of supersymmetric strings in AdS 3 x S 3 x S 3 x S 1 describing BPS line operators

Journal of Physics A Mathematical and Theoretical ◽

10.1088/1751-8121/ac354d ◽

2021 ◽

Author(s):

Diego H Correa ◽

Victor I Giraldo-Rivera ◽

Martín Lagares

Keyword(s):

Fixed Point ◽

Boundary Conditions ◽

Dirichlet Boundary ◽

Neumann Boundary ◽

Open String ◽

Neumann Boundary Conditions ◽

Dirichlet Boundary Conditions ◽

Open Strings ◽

Straight Line ◽

Fermionic Fields

Abstract We study supersymmetric open strings in type IIB $AdS_3 \times S^3 \times S^3 \times S^1$ with mixed R-R and NS-NS fields. We focus on strings ending along a straight line at the boundary of $AdS_3$, which can be interpreted as line operators in a dual CFT$_2$. We study both classical configurations and quadratic fluctuations around them. We find that strings sitting at a fixed point in $S^3 \times S^3 \times S^1$, i.e. satisfying Dirichlet boundary conditions, are 1/2 BPS. We also show that strings sitting at different points of certain submanifolds of $S^3 \times S^3 \times S^1$ can still share some fraction of the supersymmetry. This allows to define supersymmetric smeared configurations by the superposition of them, which range from 1/2 BPS to 1/8 BPS. In addition to the smeared configurations, there are as well 1/4 BPS and 1/8 BPS strings satisfying Neumann boundary conditions. All these supersymmetric strings are shown to be connected by a network of interpolating BPS boundary conditions. Our study reveals the existence of a rich moduli of supersymmetric open string configurations, for which the appearance of massless fermionic fields in the spectrum of quadratic fluctuations is crucial.

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Why boundary conditions do not generally determine the universality class for boundary critical behavior

The European Physical Journal B ◽

10.1140/epjb/e2020-10422-9 ◽

2020 ◽

Vol 93 (10) ◽

Author(s):

Hans Werner Diehl

Keyword(s):

Boundary Condition ◽

Boundary Conditions ◽

Dirichlet Boundary ◽

Neumann Boundary ◽

Universality Class ◽

Field Theories ◽

Boundary Field ◽

Universality Classes ◽

The Continuum

Abstract Interacting field theories for systems with a free surface frequently exhibit distinct universality classes of boundary critical behaviors depending on gross surface properties. The boundary condition satisfied by the continuum field theory on some scale may or may not be decisive for the universality class that applies. In many recent papers on boundary field theories, it is taken for granted that Dirichlet or Neumann boundary conditions decide whether the ordinary or special boundary universality class is observed. While true in a certain sense for the Dirichlet boundary condition, this is not the case for the Neumann boundary condition. Building on results that have been worked out in the 1980s, but have not always been appropriately appreciated in the literature, the subtle role of boundary conditions and their scale dependence is elucidated and the question of whether or not they determine the observed boundary universality class is discussed. Graphical abstract

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Effects of Pressure Boundary on Dynamic Torque Behavior of Hydroviscous Drive

Journal of Tribology ◽

10.1115/1.4040376 ◽

2018 ◽

Vol 140 (6) ◽

Cited By ~ 1

Author(s):

Jianzhong Cui ◽

Jun Liu ◽

Fangwei Xie ◽

Cuntang Wang ◽

Pengliang Hou

Keyword(s):

Boundary Conditions ◽

Dirichlet Boundary ◽

Mixed Boundary ◽

Neumann Boundary ◽

Pressure Profile ◽

Neumann Boundary Conditions ◽

Torque Control ◽

Dirichlet Boundary Conditions ◽

Squeeze Film ◽

Soft Start

The object of this work is to investigate the effect of the change of film pressure resulting from axial squeeze-film motion between driving and driven disks on the performance of hydroviscous drive (HVD). A simplified mathematical model of the steady and laminar flow between parallel disks is established with consideration of three kinds of pressure boundary conditions. Some analytical solutions of film thickness, rotate speed of driven disk, viscous torque, and total torque are obtained. The numerical results show that the torque response depends on the relationship between the inlet pressure and the outlet pressure when considering the Dirichlet boundary conditions. The soft-start under Dirichlet boundary conditions and Mixed boundary conditions reflects the constant-torque startup and torque control startup, respectively. Compared with the two boundary conditions above, the soft-start under pressure profile boundary from Neumann boundary conditions has advantages for speed regulation. The effects of the ratio of inner and outer radius on the torque profiles and soft-start time are mainly related to Dirichlet boundary conditions and pressure profile boundary from Neumann boundary conditions.

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On the Solvability of Discrete Nonlinear Two-Point Boundary Value Problems

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/2012/927607 ◽

2012 ◽

Vol 2012 ◽

pp. 1-16 ◽

Cited By ~ 2

Author(s):

Blaise Kone ◽

Stanislas Ouaro

Keyword(s):

Boundary Condition ◽

Boundary Conditions ◽

Boundary Value Problems ◽

Existence And Uniqueness ◽

Boundary Value ◽

Neumann Boundary ◽

Neumann Boundary Conditions ◽

Point Boundary ◽

Uniqueness Of Solutions ◽

Wirtinger Inequality

We prove the existence and uniqueness of solutions for a family of discrete boundary value problems by using discrete's Wirtinger inequality. The boundary condition is a combination of Dirichlet and Neumann boundary conditions.

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