Existence and stability of cylindrical transonic shock solutions under three dimensional perturbations

This study presents a novel strategy to improve solar cell electrical performance by implementing an advanced geometric architecture, specifically comparing the traditional Rectangular Structure Solar Cell (RE SC) and the innovative Half-Octagonal Structure Solar Cell (HO SC). Both solar cell designs incorporate a PIN diode and a Metal-Oxide-Semiconductor (MOS) capacitor to effectively manage the depletion region within the substrate. The HO SC is distinguished by a 38.15° oblique boundary, which is designed to optimize the longitudinal electric field distribution and enhance the collection of photogenerated charge carriers. Three-dimensional numerical simulations using Sentaurus TCAD demonstrated that the HO SC consistently outperforms the RE SC across several key metrics. The HO SC achieved a short circuit current (Isc) of 30 nA (6.4% higher than RE SC's 28.2 nA), a maximum power (PMAX) of 11.1 nW (6.7% higher than RE SC's 10.4 nW), an efficiency ( ) of 18.5% (6.7% higher than RE SC's 17.33%), and a fill factor (FF) of 75.5 (1.8% higher than RE SC's 74.2). The only exception was the open-circuit voltage (VOC), where the RE SC was slightly higher at 0.497 V compared to HO SC's 0.490 V. Therefore, based on these results, the use of PIN diode operating together with the MOS capacitor can be considered a low-cost alternative to enhance the electrical performance of solar cells.

Fully Nonlinear Elliptic PDEs in Thin Domains with Oblique Boundary Condition

We study a good shape property of boundary sections of convex solutions to the oblique boundary value problem for the Monge–Ampère equation \det D^{2}u =f(x) \quad \text{in }\Omega, \quad D_{\beta}u = \phi(x) \quad \text{on }\partial \Omega. In two dimensions, we prove a global C^{2,\alpha} estimate for solutions. For dimensions n \geq 3 , we show that this estimate remains valid provided the solution satisfies a quadratic growth condition in tangential directions. We also prove an existence result for convex solutions to the Monge–Ampère equation with an oblique Robin boundary condition.

Weighted Orlicz-Sobolev and variable exponent Morrey regularity for fully nonlinear parabolic PDEs with oblique boundary conditions and applications

Sharp moduli of continuity for solutions to fully nonlinear elliptic equations with oblique boundary conditions

We provide a sharp C1,α estimate up to the boundary for a viscosity solution of a degenerate fully nonlinear elliptic equation with the oblique boundary condition on a C1 domain. To this end, we first obtain a uniform boundary Hölder estimate with the oblique boundary condition in an “almost C1-flat" domain for the equations which is uniformly elliptic only where the gradient is far from some point, and then we establish a desired C1,α regularity based on perturbation and compactness arguments.

In this article, we prove that on any compact Riemann surface (M,∂M,g) with non-empty smooth boundary ∂M and a Riemannian metric g, (i) any K∈C∞(M) is the Gaussian curvature function of some Riemannian metric on M; (ii) any σ∈C∞(∂M) is the geodesic curvature of some Riemannian metric on M. These geometric results are obtained analytically by solving a semi-linear elliptic equation −Δgu=Ke2u on M with oblique boundary condition ∂u∂ν=σeu . One essential tool is the existence results of Brezis–Merle type equations −Δgu+Au=Ke2uinM and ∂u∂ν+κu=σeuon∂M with given functions K,σ and some constants A,κ . In addition, we rely on the extension of the uniformization theorem given by Osgood, Phillips and Sarnak.

In this article, we mainly study the oblique value problem for the generalized parabolic mean curvature type equations. Using the maximum principle, we prove the uniform gradient estimate of the solution. As applications, the long time existence and asymptic behavior are both obtained.

Weighted Orlicz regularity for fully nonlinear elliptic equations with oblique derivative at the boundary via asymptotic operators

We consider the elliptic diffusion (steady-state heat conduction) equation with spacedependent conductivity and inhomogeneous source subject to a generalized oblique boundary condition on a part of the boundary and Dirichlet or Neumann boundary conditions on the remaining part. The oblique boundary condition represents a linear combination between the dependent variable and its normal and tangential derivatives at the boundary. We first prove the well-posedness of the continuous problems. We then develop new finite volume schemes for these problems and prove rigorously the stability and convergence of these schemes.We also address an application to the inverse corrosion problem concerning the reconstruction of the coefficients present in the generalized oblique boundary condition that is prescribed over a portion Γ0 of the boundary ∂Ω from Cauchy data on the complementary portion Γ1 = ∂Ω\Γ0.

As computer vision become more widespread and deeper, the requirements for accuracy and clarity of an image gradually increase, and more accurate image reconstruction algorithms urgently need to be developed and researched. Existing image reconstruction algorithms mainly use the conversion of images into RGB or use the difference to predict certain regions to build mathematical models and then revert to images. The existing bilinear interpolation algorithm may reduce the resolution and damage the high frequency part, thus blurring the edges and making it difficult to achieve the expected results. The colorimetric constant method is greatly affected by noise, and the boundary problem is still unsolved. The gradient edge interpolation method only considers the horizontal and vertical directions, which is a single idea and does not meet the needs of oblique edges, and there will be problems such as color overflow. And based on some features near the oblique boundary, this paper focus on the case of large or small edge slope, and select the horizontal difference and vertical difference for algorithm improvement respectively by comparing the color difference along all four directions. Experimental results show that the PSNR value increases slightlyby 0.0013. It effectively optimizes the image and lays the foundation for feature perception in the image.

This paper describes the reflection and refraction of a pressure wave as it passes through a bubble medium—pure liquid boundary in the case of an oblique incidence of a wave on the interface. In this study, the gas inside the bubbles is explosive. A significant decrease in the amplitude of an initial wave capable of initiating detonation in a bubbly liquid due to wave interference at an oblique boundary is established.

Sharp Hessian estimates for fully nonlinear elliptic equations under relaxed convexity assumptions, oblique boundary conditions and applications

This paper concerns the generalized div‐curl‐gradient inequalities which control certain norms of the gradient of a vector field by using its divergence and curl in the domain and either the oblique trace or the oblique component on the domain boundary, where is a nontangential vector field. These inequalities provide a priori estimates of the solutions of the div‐curl system with the boundary condition of prescribing either the oblique trace or the oblique component. We shall prove the inequalities and derive existence of the solutions under some geometric condition on the domain and on the vector field .

Abstract We consider a nonlinear Neumann problem, with periodic oscillation in the elliptic operator and on the boundary condition. Our focus is on problems posed in half-spaces, but with general normal directions that may not be parallel to the directions of periodicity. As the frequency of the oscillation grows, quantitative homogenization results are derived. When the homogenized operator is rotation-invariant, we prove the Hölder continuity of the homogenized boundary data. While we follow the outline of Choi and Kim (Homogenization for nonlinear PDEs in general domains with oscillatory Neumann boundary data, Journal de Mathématiques Pures et Appliquées 102 (2014), no. 2, 419–448), new challenges arise due to the presence of tangential derivatives on the boundary condition in our problem. In addition, we improve and optimize the rate of convergence within our approach. Our results appear to be new even for the linear oblique problem.

A [formula omitted] finite element approximation of planar oblique derivative problems in non-divergence form

We study the existence of classical solutions to a broad class of local, first order, forward-backward extended mean field games systems, that includes standard mean field games, mean field games with congestion, and mean field type control problems. We work with a strictly monotone cost that may be fully coupled with the Hamiltonian, which is assumed to have superlinear growth. Following previous work on the standard first order mean field games system, we prove the existence of smooth solutions under a coercivity condition that ensures a positive density of players, assuming a strict form of the uniqueness condition for the system. Our work relies on transforming the problem into a partial differential equation with oblique boundary conditions, which is elliptic precisely under the uniqueness condition.

Regularity refers to the properties of the solution, including the smoothness, symmetry, and asymptotic of the solution. It is an important part of the theoretical study of partial differential equations. It plays a key role in the existence, uniqueness, stability, and smoothness of the theoretical solutions of partial differential equations. It is an important basis for understanding the nature of partial differential equations and their corresponding physical reality. This paper studies the boundary regularity of elliptic partial differential equations, including the problem of the oblique boundary of completely nonlinear equations. It is well known that the regularity of the solution at the region boundary depends not only on the equation, but also on the geometric properties of the region boundary. This is why boundary regularity is complicated. This paper is to obtain the regularity of the solution at the region boundary under different boundary conditions.

The paper is devoted to the study of positive solutions of a second-order linear elliptic equation in divergence form in a domain [Formula: see text] that satisfy an oblique boundary condition on a portion of [Formula: see text]. First, we study weak solutions for the degenerate mixed boundary value problem [Formula: see text] where [Formula: see text] is a bounded Lipschitz domain, [Formula: see text] is a relatively open portion of [Formula: see text], and [Formula: see text] is an oblique (Robin) boundary operator defined on [Formula: see text] in a weak sense. In particular, we discuss the unique solvability of the above problem, the existence of a principal eigenvalue, and the existence of a minimal positive Green function. Then we establish a criticality theory for positive weak solutions of the operator [Formula: see text] in a general domain [Formula: see text] with no boundary condition on [Formula: see text] and no growth condition at infinity. The paper extends results obtained by Pinchover and Saadon for classical solutions of such a problem, where stronger regularity assumptions on the coefficients of [Formula: see text], and the boundary [Formula: see text] are assumed.