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 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 Systems with Local and Nonlocal Diffusions, Mixed Boundary Conditions, and Reaction Terms

2018 ◽  
Vol 2018 ◽  
pp. 1-10 ◽  
Author(s):  
Mauricio Bogoya ◽  
Julio D. Rossi
Keyword(s):  
Global Existence ◽  
Boundary Conditions ◽  
Existence And Uniqueness ◽  
Blow Up ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Asymptotic Bounds ◽  
Blowing Up ◽  
Blow Up Rate ◽  
Blowing Up Solutions

We study systems with different diffusions (local and nonlocal), mixed boundary conditions, and reaction terms. We prove existence and uniqueness of the solutions and then analyze global existence vs blow up in finite time. For blowing up solutions, we find asymptotic bounds for the blow-up rate.

Diffraction of a plane wave by a half plane under mixed boundary conditions

1981 ◽  
Vol 59 (8) ◽  
pp. 974-984 ◽  
Author(s):  
T. C. Kaladhar Rao
Keyword(s):  
Plane Wave ◽  
Boundary Conditions ◽  
Half Plane ◽  
Infinite Series ◽  
Series Solution ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Dirichlet Condition ◽  
Neumann Condition ◽  
Transform Method

The problem of diffraction of a plane wave by a semi-infinite half plane with mixed boundary conditions (Dirichlet condition on one face and Neumann condition on the other) is solved by a direct and rather straightforward method. The infinite series solution and the far field are in agreement with the previous solutions obtained by the Lebedev–Kontorovich transform method as expected, as the two methods are basically equivalent. An alternate representation of the infinite series solution is presented which is valid for any type of incident field including cylindrical and spherical fields. This representation facilitates easy analysis of transient problems and the special case of an incident plane unit step function on the half plane is given as an example.

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

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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