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1. RESONANT SOLUTIONS FOR ELLIPTIC SYSTEMS WITH NEUMANN BOUNDARY CONDITIONS

   BRICEYDA, B; DELGADO, ROSA PARDO (January 2023, Electronic journal of differential equations)

   We consider a sublinear perturbation of an elliptic eigenvalue system with homogeneous Neumann boundary conditions. For oscillatory non-linearities and using bifurcation from infinity, we prove the existence of an unbounded sequence of turning points and an unbounded sequence of resonant solutions. 

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2. A Dual Scale Approach to Modeling Sub-Filter Velocities due to Shear-Induced Instabilities

   Goodrich, A.; Herrmann, M. (May 2020, ILASS-Americas 31st Annual Conference on Liquid Atomization and Spray Systems)

   null (Ed.)

   A method to compute sub-filter velocities due to shear induced instabilities on a liquid-gas interface for use in a dual scale LES-DNS model is presented. The method reconstructs the sub-filter velocity field as the sum of a prescribed base velocity profile and a perturbation velocity field determined by the Orr-Sommerfeld equations. The base velocity profile is approximated as an error function appropriately scaled with flow parameters, and the perturbation velocity field is computed by solving the Orr-Sommerfeld equations with appropriate boundary and interface conditions. The perturbation velocities of the Orr-Sommerfeld equations are expanded into Chebyshev polynomials to create a linear eigenvalue problem as outlined by Schmid and Henningson (2001). Finally the eigenvalue problem is solved using a standard linear algebra package and used to evaluate the perturbation velocities. The Chebyshev method is tested under a variety of flow parameters and initial interface disturbances. Results are presented and compared against prior literature and asymptotic solutions. 

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3. Minimizers for the Cahn--Hilliard energy functional under strong anchoring conditions

   https://doi.org/10.1137/19M1309651  

   Dai, S.; Li, B.; Luong, T. (July 2020, SIAM journal on applied mathematics)

   null (Ed.)

   We study analytically and numerically the minimizers for the Cahn-Hilliard energy functional with a symmetric quartic double-well potential and under a strong anchoring condition(i.e., the Dirichlet condition) on the boundary of an underlying bounded domain. We show a bifurcation phenomenon determined by the boundary value and a parameter that describes the thickness of a transition layer separating two phases of an underlying system of binary mixtures. For the case that the boundary value is exactly the average of the two pure phases, if the bifurcation parameter is larger than or equal to a critical value, then the minimizer is unique and is exactly the homogeneous state. Otherwise, there are exactly two symmetric minimizers. The critical bifurcation value is inversely proportional to the first eigenvalue of the negative Laplace operator with the zero Dirichlet boundary condition. For a boundary value that is larger (or smaller) than that of the average of the two pure phases, the symmetry is broken and there is only one minimizer. We also obtain the bounds and morphological properties of the minimizers under additional assumptions on the domain.Our analysis utilizes the notion of the Nehari manifold and connects it to the eigenvalue problem for the negative Laplacian with the homogeneous boundary condition. We numerically minimize the functional E by solving the gradient-flow equation of E, i.e., the Allen-Cahn equation, with the designated boundary conditions, and with random initial values. We present our numerical simulations and discuss them in the context of our analytical results. 

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4. A degenerate bifurcation from simple eigenvalue theorem

   https://doi.org/10.3934/era.2022006  

   Liu, Ping; Shi, Junping (January 2021, Electronic Research Archive)

   A new bifurcation from simple eigenvalue theorem is proved for general nonlinear functional equations. It is shown that in this bifurcation scenario, the bifurcating solutions are on a curve which is tangent to the line of trivial solutions, while in typical bifurcations the curve of bifurcating solutions is transversal to the line of trivial ones. The stability of bifurcating solutions can be determined, and examples from partial differential equations are shown to demonstrate such bifurcations. 

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5. Minimizers for the de Gennes–Cahn–Hilliard energy under strong anchoring conditions

   https://doi.org/10.1002/num.23127  

   Dai, Shibin; Ramadan, Abba (June 2024, Numerical Methods for Partial Differential Equations)

   Abstract In this article, we use the Nehari manifold and the eigenvalue problem for the negative Laplacian with Dirichlet boundary condition to analytically study the minimizers for the de Gennes–Cahn–Hilliard energy with quartic double‐well potential and Dirichlet boundary condition on the bounded domain. Our analysis reveals a bifurcation phenomenon determined by the boundary value and a bifurcation parameter that describes the thickness of the transition layer that segregates the binary mixture's two phases. Specifically, when the boundary value aligns precisely with the average of the pure phases, and the bifurcation parameter surpasses or equals a critical threshold, the minimizer assumes a unique form, representing the homogeneous state. Conversely, when the bifurcation parameter falls below this critical value, two symmetric minimizers emerge. Should the boundary value be larger or smaller from the average of the pure phases, symmetry breaks, resulting in a unique minimizer. Furthermore, we derive bounds of these minimizers, incorporating boundary conditions and features of the de Gennes–Cahn–Hilliard energy. 

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