Torus-like solutions for the Landau-de Gennes model. Part I: the Lyuksyutov regime

**boundary**

**conditions**and a corresponding physically relevant norm constraint (Lyuksyutov constraint) in the interior, we prove full regularity up to the boundary for the minimizers. Expand abstract.

**boundary**

**conditions**and a corresponding physically relevant norm constraint (Lyuksyutov constraint) in the interior, we prove full regularity up to the

**boundary**for the minimizers. As a consequence, in a relevant range (which we call the Lyuksyutov regime) of parameters of the model we show that even without the norm constraint isotropic melting is anyway avoided in the energy minimizing configurations. Finally, we describe a class of

**boundary**data including radial anchoring which yield in both the previous situations as minimizers smooth configurations whose level sets of the biaxiality carry nontrivial topology. Results in this paper will be largely employed and refined in the next papers of our series. In particular, in [DMP2], we will prove that for smooth minimizers in a restricted class of axially symmetric configurations, the level sets of the biaxiality are generically finite union of tori of revolution.

4/10 relevant

arXiv

Inverse scattering transform for the Kundu-Eckhaus Equation with nonzero
**boundary** **condition**

**boundary**

**conditions**(NZBCs) at infinity by inverse scattering transform method. The solutions of the KE equation with NZBCs can be reconstructed in terms of the solution of an associated $2 \times 2$ matrix Riemann-Hilbert problem (RHP). In our formulation, both the direct and the inverse problems are posed in terms of a suitable uniformization variable which allows us to develop the IST on the standard complex plane instead of a two-sheeted Riemann surface or the cut plane with discontinuities along the cuts. Furthermore, on the one hand, we obtain the N-soliton solutions with simple pole of the defocusing and focusing KE equation with the NZBCs, especially, the explicit one-soliton solutions are given in details. And we prove that the scattering data $a(\zeta)$ of the defocusing KE equation can only have simple zeros. On the other hand, we also obtain the soliton solutions with double pole of the focusing KE equation with NZBCs. And we show that the double pole solutions can be viewed as some proper limit of the two simple pole soliton solutions. Some dynamical behaviors and typical collisions of the soliton solutions of both of the defocusing and focusing KE equation are shown graphically.

8/10 relevant

arXiv

Geometric Perturbation Theory and Acoustic **Boundary** **Condition** Dynamics

**boundary**

**conditions**of the three-dimensional Laplacian belonging to the wave equation in the internal cavity that couples and internally drives the eardrums. In animals with ICE the resulting signal is the superposition of external sound arriving at both eardrums and the internal pressure coupling them. This is also the typical setup for geometric perturbation theory. In the context of ICE it boils down to acoustic

**boundary**-

**condition**dynamics (ABCD) for the coupled dynamical system of eardrums and internal cavity. In acoustics the deviations from equilibrium are extremely small (nm). Perturbation theory is therefore natural and shown to be appropriate. In doing so, we use a time-dependent perturbation theory \`a la Dirac in the context of Duhamel's principle. The relaxation dynamics of the tympanic-membrane system, which neuronal information processing stems from, is explicitly obtained in first order. Furthermore, both the initial and the quasi-stationary asymptotic state are derived and analyzed. Finally, we set the general stage for geometric perturbation theory where (d-1)-dimensional manifolds as subsets of the

**boundary**of a d-dimensional domain are driven by their own dynamics with the domain pressure $p$ and an external source term as input, at the same time constituting time-dependent

**boundary**

**conditions**for $p$.

10/10 relevant

arXiv

Three-dimensional and four-dimensional scalar, vector, tensor cosmological fluctuations and the cosmological decomposition theorem

**boundary**

**conditions**for the higher-derivative equations. While asymptotic

**conditions**suffice for fluctuations around a dS background or a $k=0$ RW background, for fluctuations around a $k\neq 0$ RW background one additionally has to require that the fluctuations be well-behaved at the origin. We show that in certain cases the gauge-invariant combinations themselves involve both scalars and vectors. For such cases there is no decomposition theorem for the individual SVT components themselves, but for the gauge-invariant combinations there still can be. Given the lack of manifest covariance in defining a basis with respect to 3-dimensional rotations, we introduce an alternate SVT basis whose components are defined according to how they transform under 4-dimensional general coordinate transformations. With this basis the fluctuation equations greatly simplify, and while one can again break them up into separate gauge-invariant sectors at the higher-derivative level, in general we find that even with

**boundary**

**conditions**we do not obtain a decomposition theorem in which the fluctuations separate at the level of the fluctuation equations themselves.

5/10 relevant

arXiv

Adapted or Adaptable: How to Manage Entropy Production?

**boundary**

**conditions**play a major role in defining operating points because they define the intensity of the feedback that ultimately characterizes the operation. Expand abstract.

**boundary**

**conditions**are correctly defined. The Novikov-Curzon-Ahlborn approach [1,2] applied to non-endoreversible systems then makes it possible to precisely determine the

**conditions**for obtaining characteristic operating points. As a result, power maximization principle (MPP), entropy minimization principle(mEP), efficiency maximization, or waste minimization states are only specific modalities of system operation. We show that

**boundary**

**conditions**play a major role in defining operating points because they define the intensity of the feedback that ultimately characterizes the operation. Armed with these thermodynamic foundations, we show that the intrinsically most efficient systems are also the most constrained in terms of controlling the entropy and dissipation production. In particular, we show that the best figure of merit necessarily leads to a vanishing production of power. On the other hand, a class of systems emerges which, although they do not offer extreme efficiency or power, have a wide range of use and therefore marked robustness. It therefore appears that the number of degrees of freedom of the system leads to an optimization of the allocation of entropy production.

5/10 relevant

Preprints.org

Convergence of simultaneous distributed-**boundary** parabolic optimal
control problems

**boundary**

**conditions**in a n-dimensional domain $\Omega$ with regular

**boundary**$\Gamma$ and a family of problems $S_{\alpha}$, where the parameter $\alpha>0$ is the heat transfer coefficient on the portion of the

**boundary**$\Gamma_{1}$. In relation to these state systems, we formulate simultaneous \emph{distributed-boundary} optimal control problems on the internal energy $g$ and the heat flux $q$ on the complementary portion of the

**boundary**$\Gamma_{2}$. We obtain existence and uniqueness of the optimal controls, the first order optimality

**conditions**in terms of the adjoint state and the convergence of the optimal controls, the system and the adjoint states when the heat transfer coefficient $\alpha$ goes to infinity. Finally, we prove estimations between the simultaneous distributed-

**boundary**optimal control and the distributed optimal control problem studied in a previous paper of the first author.

4/10 relevant

arXiv

Generalized Elastodynamic Model for Nanophotonics

**boundary**

**conditions**in the low spatial dispersion limit for insulators and conductors, quantified by means of a parameter defined as the "characteristic length". Expand abstract.

**Boundary**

**Conditions**. In this paper, we derive an approach where non-local effects are studied in a precise and uniquely defined way, thus allowing the treatment of all solid-solid interfaces (metals, semiconductors or insulators), as well as solid-vacuum interfaces in the same framework. The theory is based on the derivation of a potential energy for an ensemble of electrons in a given potential, where the deformation of the ensemble is treated as in a solid, including both shear and compressional deformations, instead of a fluid described only by a bulk compressibility like in the hydrodynamical approach. The derived classical equation of motion for the ensemble describes the deformation vector and the corresponding polarization vector as an elastodynamic field, including viscous forces, from which a generalized non-local constitutive equation for the dielectric constant is derived.

**Boundary**

**conditions**are identical to that of elastodynamics and they emerge in a natural way, without any physical hypothesis outside the current description, as it is commonly required in other non-local approaches. This description does not require the discontinuity of any component of the electric, magnetic or polarization fields and, consequently, no bounded currents or charges are present at the interface, which is a more suitable description from the microscopic point of view. It is shown that the method converges to the local

**boundary**

**conditions**in the low spatial dispersion limit for insulators and conductors, quantified by means of a parameter defined as the "characteristic length".

6/10 relevant

arXiv

Eliminating turbulent self-interaction through the parallel **boundary**
**condition** in local gyrokinetic simulations

**boundary**

**condition**. Expand abstract.

**boundary**

**condition**. Given a sufficiently long parallel correlation length, individual turbulent eddies can span the full domain and "bite their own tails," thereby altering their statistical properties. Such self-interaction is only modeled accurately when the simulation domain corresponds to a full flux surface, otherwise it is artificially strong. For Cyclone Base Case parameters and typical domain sizes, we find that this mechanism modifies the heat flux by roughly 40% and it can be even more important. The effect is largest when using kinetic electrons, low magnetic shear, and strong turbulence drive (i.e. steep background gradients). It is found that parallel self-interaction can be eliminated by increasing the parallel length and/or the binormal width of the simulation domain until convergence is achieved.

7/10 relevant

arXiv

Deformed Heisenberg charges in three-dimensional gravity

**boundary**

**conditions**define so-called Bryant surfaces, which can be classified completely in terms of holomorphic maps from Riemann surfaces into the spinor bundle. Expand abstract.

**boundary**phase space for three-dimensional gravity with negative cosmological constant for a particular choice of conformal

**boundary**conditions: the conformal class of the induced metric at the

**boundary**is kept fixed and the mean extrinsic curvature is constrained to be one. Such specific conformal

**boundary**

**conditions**define so-called Bryant surfaces, which can be classified completely in terms of holomorphic maps from Riemann surfaces into the spinor bundle. To study the observables and gauge symmetries of the resulting bulk plus

**boundary**system, we introduce an extended phase space, where these holomorphic maps are now part of the gravitational bulk plus

**boundary**phase space. The physical phase space is obtained by introducing two sets of Kac-Moody currents, which are constrained to vanish. The constraints are second-class and the corresponding Dirac bracket yields an infinite-dimensional deformation of the Heisenberg algebra for the spinor-valued surface charges. Finally, we compute the Poisson algebra among the generators of conformal diffeomorphisms and demonstrate that there is no central charge. Although the central charge vanishes and the

**boundary**CFT is likely non-unitary, we will argue that a version of the Cardy formula still applies in this context, such that the entropy of the BTZ black hole can be derived from the degeneracy of the eigenstates of quasi-local energy.

4/10 relevant

arXiv

On C^{1/2,1}, C^{1,2}, and C^{0,0} estimates for linear parabolic operators

**boundary**

**conditions**are continuously differentiable up to the boundary when the leading coefficients have Dini mean oscillation and the lower order coefficients verify certain

**condition**s. Expand abstract.

**boundary**

**conditions**are continuously differentiable up to the

**boundary**when the leading coefficients have Dini mean oscillation and the lower order coefficients verify certain

**conditions**. Similar results are obtained for non-divergence form parabolic operators and their adjoint operators.

4/10 relevant

arXiv