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

<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>

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.

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

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

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.

Design and analysis of finite volume methods for elliptic equations with oblique derivatives; application to Earth gravity field modelling

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

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

We obtain, up to a flat boundary, regularity results in parabolic Hölder spaces for viscosity solutions of fully nonlinear parabolic equations with oblique boundary conditions.

In this paper, we obtain a series of regularity results for viscosity solutions of fully nonlinear elliptic equations with oblique derivative boundary conditions. In particular, we derive the pointwise $C^{\alpha}$, $C^{1,\alpha}$ and $C^{2,\alpha}$ regularity. As byproducts, we also prove the A-B-P maximum principle, Harnack inequality, uniqueness and solvability of the equations.

We show that the Strang splitting method applied to a diffusion-reaction equation with inhomogeneous general oblique boundary conditions is of order two when the diffusion equation is solved with the Crank-Nicolson method, while order reduction occurs in general if using other Runge-Kutta schemes or even the exact flow itself for the diffusion part. We prove these results when the source term only depends on the space variable, an assumption which makes the splitting scheme equivalent to the Crank-Nicolson method itself applied to the whole problem. Numerical experiments suggest that the second order convergence persists with general nonlinearities.

In gas-solid reactions, one of the most important factors that determine the overall reaction rate is the solid morphology, which can be characterized by a combination of smooth, convex and concave structures. Generally, the solid surface structure varies in the course of reactions, which is classically noted as being attributed to one or more of the following three mechanisms: mechanical interaction, molar volume change, and sintering. Here we show that if a gas-solid reaction involves the outward ionic diffusion of a solid-phase reactant then this outward ionic diffusion could eventually smooth the surface with an initial concave and/or convex structure. Specifically, the concave surface is filled via a larger outward diffusing surface pointing to the concave valley, whereas the height of the convex surface decreases via a lower outward diffusion flux in the vertical direction. A quantitative 2-D continuum diffusion model is established to analyze these two morphological variation processes, which shows consistent results with the experiments. This surface morphology variation by solid-phase ionic diffusion serves to provide a fourth mechanism that supplements the traditionally acknowledged solid morphology variation or, in general, porosity variation mechanisms in gas-solid reactions.

Properties of generalized solutions of model nonlinear elliptic systems of second order are studied in the semiball $$B_1^ + = B_1 (0) \cap \{ x_n > 0\} \subset $$ ℝ n , with the oblique derivative type boundary condition on $$\Gamma _1 = B_1 (0) \cap \{ x_n = 0\} $$ . For solutionsu∈H 1(B 1 + ) of systems of the form $$\frac{d}{{dx_\alpha }}a_\alpha ^k (u_x ) = 0, k \leqslant {\rm N}$$ , it is proved that the derivatives ux are Holder in $$B_1^ + \cup \Gamma _1 )\backslash \Sigma $$ , where Hn−p(σ)=0,p>2. It is shown for continuous solutions u from H1(B1/+) of systems $$\frac{d}{{dx_\alpha }}a_\alpha ^k (u,u_x ) = 0$$ that the derivatives ux are Holder on the set $$(B_1^ + \cup \Gamma _1 )\backslash \Sigma , dim_\kappa \Sigma \leqslant n - 2$$ . Bibliography: 13 titles.

Sharp moduli of continuity for solutions to 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.