Estimates for the first eigenvalue for p-Laplacian with mixed boundary conditions

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.

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The buckling eigenvalue problem in the annulus

Communications in Contemporary Mathematics ◽

10.1142/s0219199720500443 ◽

2020 ◽

pp. 2050044

Author(s):

Davide Buoso ◽

Enea Parini

Keyword(s):

Asymptotic Behavior ◽

Eigenvalue Problem ◽

Analytical Study ◽

Mixed Boundary ◽

Mixed Boundary Conditions ◽

First Eigenvalue ◽

Eigenvalues And Eigenfunctions ◽

Nodal Domains ◽

The First Eigenvalue ◽

Inner Radius

We consider the buckling eigenvalue problem for a clamped plate in the annulus. We identify the first eigenvalue in dependence of the inner radius, and study the number of nodal domains of the corresponding eigenfunctions. Moreover, in order to investigate the asymptotic behavior of eigenvalues and eigenfunctions as the inner radius approaches the outer one, we provide an analytical study of the buckling problem in rectangles with mixed boundary conditions.

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

Advanced Nonlinear Studies ◽

10.1515/ans-2018-0001 ◽

2018 ◽

Vol 18 (2) ◽

pp. 323-335 ◽

Cited By ~ 2

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.

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POSITIVE SOLUTIONS OF NONLINEAR ELLIPTIC EQUATIONS WITH MIXED BOUNDARY CONDITIONS

Acta Mathematica Scientia ◽

10.1016/s0252-9602(18)30162-0 ◽

1992 ◽

Vol 12 (3) ◽

pp. 292-303

Author(s):

Ruying Xue

Keyword(s):

Boundary Conditions ◽

Positive Solutions ◽

Elliptic Equations ◽

Mixed Boundary ◽

Nonlinear Elliptic Equations ◽

Mixed Boundary Conditions ◽

Nonlinear Elliptic

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Effects of transverse shear on large deflection random response of symmetric composite laminates with mixed boundary conditions

10.2514/6.1989-1356 ◽

1989 ◽

Cited By ~ 3

Author(s):

C. PRASAD ◽

CHUH MEI

Keyword(s):

Boundary Conditions ◽

Transverse Shear ◽

Composite Laminates ◽

Large Deflection ◽

Mixed Boundary ◽

Mixed Boundary Conditions ◽

Random Response

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General solutions in Chern-Simons gravity and $$ T\overline{T} $$-deformations

Journal of High Energy Physics ◽

10.1007/jhep01(2021)039 ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Eva Llabrés

Keyword(s):

Variational Principle ◽

Boundary Conditions ◽

Dirichlet Boundary ◽

Mixed Boundary ◽

Mixed Boundary Conditions ◽

Dirichlet Boundary Conditions ◽

Spin Connection ◽

Departure Point ◽

Boundary Stress ◽

Chern Simons

Abstract We find the most general solution to Chern-Simons AdS3 gravity in Fefferman-Graham gauge. The connections are equivalent to geometries that have a non-trivial curved boundary, characterized by a 2-dimensional vielbein and a spin connection. We define a variational principle for Dirichlet boundary conditions and find the boundary stress tensor in the Chern-Simons formalism. Using this variational principle as the departure point, we show how to treat other choices of boundary conditions in this formalism, such as, including the mixed boundary conditions corresponding to a $$ T\overline{T} $$ T T ¯ -deformation.

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