An extension of the lower-bound lemma of Boggio is given for the weak forms of certain elliptic operators, which are in general nonlinear and have partially Dirichlet and partially Neumann boundary conditions. Its consequences and those of an adapted Hardy inequality for the location of the bottom of the spectrum are explored in corollaries wherein a variety of assumptions are placed on the sha...

This paper continues the study of the mixed problem for the Laplacian. We consider a bounded Lipschitz domain Ω ⊂ Rn, n ≥ 2, with boundary that is decomposed as ∂Ω = D ∪ N , D and N disjoint. We let Λ denote the boundary of D (relative to ∂Ω) and impose conditions on the dimension and shape of Λ and the sets N and D. Under these geometric criteria, we show that there exists p0 > 1 depending on ...

The isospectrality of a well-known pair of shapes constructed from two arrangements of seven congruent right isosceles triangles with the Neumann boundary condition is verified numerically to high precision. Equally strong numerical evidence for isospectrality is presented for the eigenvalues of this standard pair in new boundary configurations with alternating Dirichlet and Neumann boundary co...

We introduce Monte Carlo methods to compute the solution of elliptic equations with pure Neumann boundary conditions. We first prove that the solution obtained by the stochastic representation has a zero mean value with respect to the invariant measure of the stochastic process associated to the equation. Pointwise approximations are computed by means of standard and new simulation schemes espe...

This work considers properties of the Neumann-to-Dirichlet map for the conductivity equation under the assumption that the conductivity is identically one close to the boundary of the examined smooth, bounded, and simply connected domain. It is demonstrated that the socalled bisweep data, i.e., the (relative) potential differences between two boundary points when delta currents of opposite sign...

Solvability issues of four boundary value problems for a nonlocal biharmonic equation in the unit ball are investigated. Dirichlet, Neumann, Navier and Riquier–Neumann studied. For under consideration, existence uniqueness theorems proved. Necessary sufficient conditions solvability all obtained an integral representations solutions given terms corresponding Green’s functions.

We present results of a renormalization-group calculation of the thermal conductivity of confined He in a L ×∞ geometry above and at Tλ within model F with Dirichlet boundary conditions for the order parameter. We assume a heat flow parallel to the boundaries which implies Neumann boundary conditions for the entropy density. No adjustable parameters other than those known from bulk theory and s...

We evaluate the Casimir energy and force for a massive scalar field with general curvature coupling parameter, subject to Robin boundary conditions on two codimension-one parallel plates, located on a (D + 1)-dimensional background spacetime with an arbitrary internal space. The most general case of different Robin coefficients on the two separate plates is considered. With independence of the ...

Dispersive instability appears in time-domain solutions of classical cochlear models. In this letter, a derivation of optimal initial data is presented to minimize the effect of instability. A second-order accurate implicit boundary integral method is introduced. Numerical solutions of two-dimensional models show that the optimal initial conditions work successfully in time-domain steady-state ...

We discuss the Casimir effect for a massive bosonic field with mixed (Dirichlet-Neumann) boundary conditions. We use the ζ-function regularization prescription to obtain our physical results. Particularly, we analyse how the Casimir energy varies with the mass of the field and compare this mass dependence with those obtained for other boundary conditions. This is done graphically. Some other gr...