scholarly journals Extrema of Low Eigenvalues of the Dirichlet–Neumann Laplacian on a Disk

2010 ◽  
Vol 62 (4) ◽  
pp. 808-826
Author(s):  
Eveline Legendre
Keyword(s):  
Boundary Conditions ◽  
Mixed Boundary ◽  
Neumann Boundary ◽  
Mixed Boundary Conditions ◽  
First Eigenvalue ◽  
Bounded Number ◽  
Eigenvalues Of The Laplacian ◽  
Neumann Laplacian ◽  
Dirichlet And Neumann Conditions ◽  
Second Eigenvalue

AbstractWe study extrema of the first and the second mixed eigenvalues of the Laplacian on the disk among some families of Dirichlet–Neumann boundary conditions. We show that the minimizer of the second eigenvalue among all mixed boundary conditions lies in a compact 1-parameter family for which an explicit description is given. Moreover, we prove that among all partitions of the boundary with bounded number of parts on which Dirichlet and Neumann conditions are imposed alternately, the first eigenvalue is maximized by the uniformly distributed partition.

scholarly journals Eigenvalues of the Laplacian with Neumann boundary conditions

1985 ◽  
Vol 26 (3) ◽  
pp. 293-309 ◽  
Author(s):  
H. P. W. Gottlieb
Keyword(s):  
Boundary Conditions ◽  
Mixed Boundary ◽  
Three Dimensional ◽  
Neumann Boundary ◽  
Mixed Boundary Conditions ◽  
Summation Formula ◽  
Laplacian Operator ◽  
Eigenvalues Of The Laplacian ◽  
Annular Sector ◽  
Simply Connected

AbstractVarious grometrical properties of a domain may be elicited from the asymptotic expansion of a spectral function of the Laplacian operator for that region with apporpriate boundary conditions. Explicit calculations, using analytical formulae for the eigenvalues, are performed for the cases fo Neumann and mixed boundary conditions, extending earlier work involving Dirichet boundary conditions. Two- and three-dimensional cases are considered. Simply-connected regions dealt with are the rectangle, annular sector, and cuboid. Evaluations are carried out for doubly-connected regions, including the narrow annulus, annular cylinder, and thin concentric spherical cavity. The main summation tool is the Poission summation formula. The calculations utilize asymptotic expansions of the zeros of the eigenvalue equations involving Bessel and related functions, in the cases of curved boundaries with radius ratio near unity. Conjectures concerning the form of the contributions due to corners, edges and vertices in the case of Neumann and mixed boundary conditions are presented.

scholarly journals Nonlinear elliptic equations involving the p-Laplacian with mixed Dirichlet-Neumann boundary conditions

2019 ◽  
Vol 39 (2) ◽  
pp. 159-174 ◽  
Author(s):  
Gabriele Bonanno ◽  
Giuseppina D'Aguì ◽  
Angela Sciammetta
Keyword(s):  
Boundary Conditions ◽  
Elliptic Equations ◽  
Mixed Boundary ◽  
Neumann Boundary ◽  
Nonlinear Elliptic Equations ◽  
Mixed Boundary Conditions ◽  
Neumann Boundary Conditions ◽  
Laplacian Operator ◽  
Differential Problem ◽  
Nonlinear Elliptic

In this paper, a nonlinear differential problem involving the \(p\)-Laplacian operator with mixed boundary conditions is investigated. In particular, the existence of three non-zero solutions is established by requiring suitable behavior on the nonlinearity. Concrete examples illustrate the abstract results.

scholarly journals Optimal Design Problems for the First p-Fractional Eigenvalue with Mixed Boundary Conditions

2018 ◽  
Vol 18 (2) ◽  
pp. 323-335 ◽  
Author(s):  
Julian Fernández Bonder ◽  
Julio D. Rossi ◽  
Juan F. Spedaletti
Keyword(s):  
Optimal Design ◽  
Boundary Conditions ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Dirichlet Condition ◽  
Shape Design ◽  
First Eigenvalue ◽  
Design Problems ◽  
Asymptotic Bounds ◽  
Optimization Variable

AbstractIn this paper, we study an optimal shape design problem for the first eigenvalue of the fractionalp-Laplacian with mixed boundary conditions. The optimization variable is the set where the Dirichlet condition is imposed (that is restricted to have measure equal to a prescribed quantity α). We show existence of an optimal design and analyze the asymptotic behavior when the fractional parametersconverges to 1, and thus obtain asymptotic bounds that are independent of α.

scholarly journals Cost of null controllability for parabolic equations with vanishing diffusivity and a transport term

2021 ◽  
Author(s):  
Jon Asier Bárcena-Petisco
Keyword(s):  
Boundary Conditions ◽  
Parabolic Equations ◽  
Space Variable ◽  
Mixed Boundary ◽  
Neumann Boundary ◽  
Mixed Boundary Conditions ◽  
Null Controllability ◽  
Dirichlet Boundary Conditions ◽  
The Cost ◽  
Transport Term

In this paper we consider the heat equation with Neumann, Robin and mixed boundary conditions (with coefficients on the boundary which depend on the space variable). The main results concern the behaviour of the cost of the null controllability with respect to the diffusivity when the control acts in the interior. First, we prove that if we almost have Dirichlet boundary conditions in the part of the boundary in which the flux of the transport enters, the cost of the controllability decays for a time $T$ sufficiently large. Next, we show some examples of Neumann and mixed boundary conditions in which for any time $T>0$ the cost explodes exponentially as the diffusivity vanishes. Finally, we study the cost of the problem with Neumann boundary conditions when the control is localized in the whole domain.

scholarly journals One-loop correction to the Casimir energy in Lifshitz-like theory

2021 ◽  
pp. 2150204
Author(s):  
M. A. Valuyan
Keyword(s):  
Scalar Field ◽  
Boundary Conditions ◽  
Mixed Boundary ◽  
Perturbation Expansion ◽  
Neumann Boundary ◽  
Mixed Boundary Conditions ◽  
Casimir Energy ◽  
Dependent Manner ◽  
Loop Correction ◽  
Systematic Perturbation

In this paper, Radiative Correction (RC) to the Casimir energy was computed for the self-interacting massive/massless Lifshitz-like scalar field, confined between a pair of plates with Dirichlet and Mixed boundary conditions in 3 + 1 dimensions. Moreover, using the results obtained for the Dirichlet Casimir energy, the RC to the Casimir energy for Periodic and Neumann boundary conditions were also draw outed. To renormalize the bare parameters of the Lagrangian, a systematic perturbation expansion was used in which the counterterms were automatically obtained in a position-dependent manner. In our view, the position dependency of the counterterm was allowed, since it reflected the effects of the boundary condition imposed or the background space in the problem. All the answers obtained for the Casimir energy were consistent with well-known physical expects. In a language of graphs, the Casimir energy for the massive Lifshitz-like scalar field confined with four boundary conditions (Dirichlet, Neumann, Mixed, and Periodic) was also compared to each other, and as a concluding remark, the sign and magnitude of their values were discussed.

scholarly journals Exact solution of problems of diffraction of unsteady waves on a wedge under mixed boundary conditions by the Smirnov-Sobolev method

2019 ◽  
Vol 485 (4) ◽  
pp. 434-437
Author(s):  
M. Sh. Israilov
Keyword(s):  
Exact Solution ◽  
Boundary Conditions ◽  
Mixed Problem ◽  
Mixed Boundary ◽  
Neumann Boundary ◽  
Mixed Boundary Conditions ◽  
Blast Waves ◽  
Neumann Boundary Conditions

Antil now, the Smirnov-Sobolev method has been applied only to diffraction problems with the Dirichlet and Neumann boundary conditions. In this study, it is shown that the method also leads to an exact solution of the mixed problem of diffraction on a wedge, which is very important, for example, for estimating the possibility of protection from blast waves by wedge-shaped barriers with different reflecting properties of the sides.

Existence theorems for the quantum drift–diffusion equations with mixed boundary conditions

2016 ◽  
Vol 18 (04) ◽  
pp. 1550048 ◽  
Author(s):  
Xiangsheng Xu
Keyword(s):  
Boundary Conditions ◽  
Mixed Boundary ◽  
Existence Theorems ◽  
Neumann Boundary ◽  
Mixed Boundary Conditions ◽  
Approximate Solutions ◽  
Diffusion Equations ◽  
Drift Diffusion ◽  
High Regularity ◽  
Recent Theorem

In this paper, we construct a weak solution to the unipolar quantum drift–diffusion equations coupled with initial and mixed boundary conditions in up to four space dimensions. We only assume that the boundary of the domain is Lipschitz and the interface between the Dirichlet boundary and the Neumann boundary is a Lipschitzian hypersurface, and thus the lack of high regularity for solutions of such problems is an issue. The convergence of a sequence of approximate solutions is established via an application of a recent theorem by the author on the logarithmic upper bound for solutions of elliptic equations with the same type of boundary conditions [Logarithmic up bounds for solutions of elliptic partial differential equations, Proc. Amer. Math. Soc. 139 (2011) 3485–3490].

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