In this paper, on Riemannian manifolds with boundary, we establish a Yau type gradient estimate and Liouville theorem for harmonic functions under Dirichlet boundary condition. Under similar setting, also formulate Souplet-Zhang ancient solutions to the heat equation.

We give a necessary and sufficient condition, of geometrical type, for the uniform decay of energy of solutions of the linear system of magnetoelasticity in a bounded domain with smooth boundary. A Dirichlet-type boundary condition is assumed. When the geometrical condition is not fulfilled, we show polynomial decay of the energy, for smooth initial conditions. Our strategy is to use micro-loca...

We establish the saddle point property of the focusing selfsimilar solution of a free boundary problem for the heat equation with free boundary conditions given by u = 0 and ∂u ∂ν = 1.

the purpose of this paper is to study the higher order asymptotic distributions of the eigenvalues associated with a class of sturm-liouville problem with equation of the form w??=(?2f(x)?r(x)) (1), on [a,b, where ? is a real parameter and f(x) is a real valued function in c2(a,b which has a single zero (so called turning point) at point 0x=x and r(x) is a continuously differentiable function. ...

In this paper, we introduce the Navier-Stokes equations with a new boundary condition. In this context, we show the existence and uniqueness of the solution of the weak formulation associated with the proposed problem. To solve this latter, we use the discretization by mixed finite element method. In addition, two types of a posteriori error indicator are introduced and are shown to give global...

The diffraction by a finite sinusoidal grating is analyzed for the H-polarized plane wave incidence using the Wiener-Hopf technique combined with the perturbation method. Assuming the depth of the grating to be small compared with the wavelength and approximating the boundary condition on the grating surface, the problem is reduced to the diffraction problem involving a flat strip with a certai...

Abstract. In this paper we consider, in a bounded and smooth domain, the wave equation with a delay term in the boundary condition. We also consider the wave equation with a delayed velocity term and mixed Dirichlet–Neumann boundary condition. In both cases, under suitable assumptions, we prove exponential stability of the solution. These results are obtained by introducing suitable energies an...

We study the quantum integrability of the O(N) Nonlinear σ (nlσ) model and the O(N) Gross-Neveu (GN) model on the half-line. We show that the nlσ model is integrable with Neumann, Dirichlet and a mixed boundary condition and that the GN model is integrable if ψ a + | x=0 = ±ψ a − | x=0. We also comment on the boundary condition found by Corrigan and Sheng for the O(3) nlσ model. 1 Work done und...

In [7, 8], they proposed a new singular function(NSF) method to compute singular solutions of Poisson equations on a polygonal domain with re-entrant angles. Singularities are eliminated and only the regular part of the solution that is in H is computed. The stress intensity factor and the solution can be computed as a post processing step. This method was extended to the interface problem and ...

We consider the mixed boundary value problem for the Stokes and steady NavierStokes equations in two-dimensional domain with corner points on boundary, where the boundary conditions change their type. Our goal is to prove some regularity results near corner points, where the boundary conditions change.