Fully Nonlinear Elliptic PDEs in Thin Domains with Oblique Boundary Condition

Navier–Stokes equations in 3D thin domains with Navier friction boundary condition

The behaviour of Newtonian and non-Newtonian flows through a thin three-dimensional domain are widely studied in the literature. Usually, authors deal with special models related to particular concrete fluids. In this work, our aim is to present a general model, governing the behaviour of a large class of Newtonian and non-Newtonian fluids. Moreover, we deal with mixed boundary conditions, which are not often studied in the literature related to flows in thin domains. We consider a nonlinear model of a flow in a thin three-dimensional domain, and we study its behaviour when the thickness in one direction tends to zero. At the limit, we obtain a quasilinear two-dimensional problem for the pressure, a nonlinear Reynolds's law for the velocity and a nonlinear Darcy's law for the averaged velocity. Finally, we check that our results hold for a large class of non-Newtonian fluids by producing concrete examples.

This paper concerns fully nonlinear elliptic obstacle problems with oblique boundary conditions. We investigate the existence, uniqueness and W^{2,p}-regularity results by finding approximate non-obstacle problems with the same oblique boundary condition and then making a suitable limiting process.

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.

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

<p style='text-indent:20px;'>We prove a global <inline-formula><tex-math id="M1">\begin{document}$ W^{2, p} $\end{document}</tex-math></inline-formula>-estimate for the viscosity solution to fully nonlinear elliptic equations <inline-formula><tex-math id="M2">\begin{document}$ F(x, u, Du, D^{2}u) = f(x) $\end{document}</tex-math></inline-formula> with oblique boundary condition in a bounded <inline-formula><tex-math id="M3">\begin{document}$ C^{2, \alpha} $\end{document}</tex-math></inline-formula>-domain for every <inline-formula><tex-math id="M4">\begin{document}$ \alpha\in (0, 1) $\end{document}</tex-math></inline-formula>. Here, the nonlinearities <inline-formula><tex-math id="M5">\begin{document}$ F $\end{document}</tex-math></inline-formula> is assumed to be asymptotically <inline-formula><tex-math id="M6">\begin{document}$ \delta $\end{document}</tex-math></inline-formula>-regular to an operator <inline-formula><tex-math id="M7">\begin{document}$ G $\end{document}</tex-math></inline-formula> that is <inline-formula><tex-math id="M8">\begin{document}$ (\delta, R) $\end{document}</tex-math></inline-formula>-vanishing with respect to <inline-formula><tex-math id="M9">\begin{document}$ x $\end{document}</tex-math></inline-formula>. We employ the approach of constructing a regular problem by an appropriate transformation. With a similar argument, we also obtain a global <inline-formula><tex-math id="M10">\begin{document}$ W^{2, p} $\end{document}</tex-math></inline-formula>-estimate for the viscosity solution to fully nonlinear parabolic equations <inline-formula><tex-math id="M11">\begin{document}$ F(x, t, u, Du, D^{2}u)-u_{t} = f(x, t) $\end{document}</tex-math></inline-formula> with oblique boundary condition in a bounded <inline-formula><tex-math id="M12">\begin{document}$ C^{3} $\end{document}</tex-math></inline-formula>-domain.</p>

Lp-estimates for the Hessians of solutions to fully nonlinear parabolic equations with oblique boundary conditions

A linear approximation for the regular reflection of a weak shock at a wedge satisfying sonic condition

Some Results on the Navier–Stokes Equations in Thin 3D Domains

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

W2,p-estimates for fully nonlinear elliptic equations with oblique boundary conditions

Electrohydrodynamic instabilities in the nematic phase of Merck ’’Phase V’’ with oblique boundary conditions were optically observed with a polarizing microscope in 25–100 μm ’’sandwich’’ cells. Oblique anchoring of the nematic was achieved by oblique evaporation of SiO on the plates. Two types of cells were used having the respective in-plane projection of the direction of evaporation on the two plates either parallel (p-type cells), or antiparallel (a-type cells). The low voltage dc instability observed for the p-type cells forms in an almost regular hexagonal pattern. By gradually increasing the voltage, the dc instability observed for the a-type cells forms at first as flows which originate at order disturbances created at imperfections in the SiO coating. Voltage increase causes these flows to detach themselves from the places of the imperfections and move solitarily. The moving flows are associated with what appears to be moving tilt inversion deformations (of splay-bend type) extending from the central part of the flow to some distance from it. When the voltage is further increased, a repeated process of replication of the flows, occurring on the associated tilt inversion deformations, leads to the creation of a periodic grid of moving flows. Other observed types of static and dynamic patterns under ac and dc excitation are reported, in particular: different types of cross rolls (ac conduction regime); variations of a pattern of what appears to be walls associated with flows, exhibiting an approximate wave number dependence on the electric field k∼E; a striped pattern associated with what appears to be twist walls and the propagating interference patterns associated with their oscillations; a toroidal flow (sometimes associated with closed inversion walls) which creates and caries along closed nematic threads (dc regime); a polygonal grid of turbulent flows (dc regime); a flow pattern correlated with the movement of the moving chevron pattern; a cellular fast turn-off pattern related to the chevron pattern. This cellular pattern appears at first as moving snakelike regions in the chevron pattern which are bordered by disclination lines. Some features of dark, spotlike figures appearing on the chevron pattern are described. Preliminary interpretations of some of the observations are offered.

Oblique boundary conditions are introduced in the Fourier modal method at each slice of the staircase decomposition of an arbitrary profile of a dielectric corrugation grating. The precision and convergence improvement are demonstrated by comparison with reference methods.

Least-squares for second-order elliptic problems

Consider the scattering of electromagnetic waves by a penetrable homogeneous cylinder at oblique incident. The Maxwell equations are then reduced to a system of a pair of the two-dimensional Helmholtz equations for 𝑧-components of the electric and magnetic field through coupled oblique boundary conditions. This paper studies an inverse problem of recovering the penetrable obstacle from the far-field pattern of the electric field. The well-known linear sampling method is used to solve this problem. Compared with the usual inverse scattering problem, the coupled system and the oblique derivative boundary condition bring difficulties in theoretical analysis. Some numerical examples are presented to illustrate the validity and feasibility of the proposed method.