Abstract We study the first eigenvalue of the Laplace equation in a bounded domain in R (d = 2, 3) with mixed Neumann–Dirichlet (Zaremba) boundary conditions. The Neumann condition is imposed on most of the boundary and the Dirichlet boundary consists of a cluster of small windows. When the windows are well separated the first eigenvalue is asymptotically the sum of eigenvalues of mixed problem...

In this article, we consider the analogue of the Dirichlet problem for second-order elliptic integro-differential equations, which consists in imposing the “boundary conditions” in the whole complementary of the domain. We are looking for conditions on the differential and integral parts of the equation in order to ensure that the Dirichlet boundary condition is satisfied in the classical sense...

Consider a unidimensional, single-phase nonlinear Stefan problem with nonlinear source and permeance terms, and a Dirichlet boundary condition depending on the free-boundary function. This problem is important in groundwater ow. By immobilizing the free-boundary with the help of a Landau type transformation, together with a homogeneous transformation dealing with the nonhomogeneous Dirichlet bo...

We consider the defocusing energy-critical NLS in the exterior of the unit ball in three dimensions. For the initial value problem with Dirichlet boundary condition we prove global well-posedness and scattering with large radial initial data in the Sobolev space Ḣ1 0 . We also point out that the same strategy can be used to treat the energy-supercritical NLS in the exterior of balls with Dirich...

Finite volume renormalization scheme is one of the most fascinating scheme for non-perturbative renormalization on lattice. By using the step scaling function one can follow running of renormalized quantities with reasonable cost. It has been established the Schrödinger functional is very convenient to define a field theory in a finite volume for the renormalization scheme. The Schrödinger func...

In this section, we will introduce the work of Kbabou, Hermi, and Rhonma (2007)[2]. Their main idea is to use the eigenvalues and their ratios of the Dirichlet-Laplacian for various planar shapes as their features for classifying them. Let the sequence 0 < λ 1 < λ 2 ≤ λ 3 ≤ · · · ≤ λ k ≤ · · · → ∞ be the sequence of eigenvalues of Dirichlet-Laplacian problem: −∆u = λu in a given bounded planar ...

We study the effect of a periodic roughness on a Neumann boundary condition. We show that, as in the case of a Dirichlet boundary condition, it is possible to approach this condition by a more complex law on a domain without rugosity, called wall law. This approach is however different from that usually used in Dirichlet case. In particular, we show that this wall law can be explicitly written ...

We consider bosonic noncritical strings as QCD string. We present an basic strategy to construct them, in the context of Liouville theory. We show that Dirichlet boundary condition fills important roles in generalized Liouville theory. Specifically, it is used to stabilize the classical configuration of strings, and also utilized to treat the tachyon condensation, in our model. We point out tha...

The global solvability of the boundary value problem for stationarymagnetohydrodynamic equations under the Dirichlet boundary condition for the velocity and mixed boundary conditions for the magnetic field is proved. © 2014 Elsevier Ltd. All rights reserved.

In the context of the bulk-boundary correspondence we study the correlation functions arising on a boundary for different types of boundary conditions. The most general condition is the mixed one interpolating between the Neumann and Dirichlet conditions. We obtain the general expressions for the correlators on a boundary in terms of Green’s function in the bulk for the Dirichlet, Neumann and m...