On W^{2,p}-estimates for solutions of obstacle problems for fully nonlinear elliptic equations with oblique boundary conditions

<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>

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

W2,p-estimates for 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.

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

Much has been written about various obstacle problems in the context of variational inequalities. In particular, if the obstacle is smooth enough and if the coefficients of associated elliptic operator satisfy appropriateconditions, then the solution of the obstacle problem has continuous first derivatives. For a general class of obstacle problems, we show here that this regularity is attained under minimal smoothness hypotheses on the data and with a one-sided analog of the usual modulus of continuity assumption for the gradient of the obstacle. Our results apply to linear elliptic operators with Holder continuous coefficients and, more generally, to a large class of fully nonlinear operators and boundary conditions.

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

We study nonlinear elliptic double obstacles problems with an asymptotically regular nonlinearity in a non-smooth domain. Here, we obtain a global Calderón–Zygmund type estimate in the setting of Lorentz spaces by making use of the perturbation method. More precisely, we approximate the solutions of asymptotically regular double obstacles problems to the solutions of regular single obstacle problems when the gradients of solutions are close to infinity.

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

This paper is concerned with the nonlinear boundary value problem (1) $\beta u''-u'+f(u)=0$, (2) $u'(0)-au(0)=0,u'(1)=0$, where $f(u)=b(c-u)\exp(-k/(1+u))$ and $\beta,a,b,c,k$ are constants. First a formal singular perturbation procedure is applied to reveal the possibility of multiple solutions of (1) and (2). Then an iteration procedure is introduced which yields sequences converging to the maximal solution from above and the minimal solution from below. A criterion for a unique solution of (1), (2) is given. It is mentioned that for certain values of the parameters multiple solutions have been found numerically. Finally, the stability of solutions of (1), (2) is discussed for certain values of the parameters. A solution $u(x)$ of (1), (2) is said to be stable if the first eigenvalue $\sigma$ of the variational equations $(1)' \beta v''-v'+[\sigma\beta+f'(u)]v=0$ and $(2)' v'(0)-av(0)=0, v'(1)=0$, is positive.

Elliptic obstacle problems are formulated to find either superharmonic solutions or minimal surfaces that lie on or over the obstacles, by incorporating inequality constraints. In order to solve such problems effectively using finite difference (FD) methods, the article investigates simple iterative algorithms based on the successive over-relaxation (SOR) method. It introduces subgrid FD methods to reduce the accuracy deterioration occurring near the free boundary when the mesh grid does not match with the free boundary. For nonlinear obstacle problems, a method of gradient-weighting is introduced to solve the problem more conveniently and efficiently. The iterative algorithm is analyzed for convergence for both linear and nonlinear obstacle problems. An effective strategy is also suggested to find the optimal relaxation parameter. It has been numerically verified that the resulting obstacle SOR iteration with the optimal parameter converges about one order faster than state-of-the-art methods and the subgrid FD methods reduce numerical errors by one order of magnitude, for most cases. Various numerical examples are given to verify the claim.

Superconvergence of a class of expanded discontinuous Galerkin methods for fully nonlinear elliptic problems in divergence form

The theory of nonlinear elliptic equations is currently one of the most actively developing branches of the theory of partial differential equations. This book investigates boundary value problems for nonlinear elliptic equations of arbitrary order. In addition to monotone operator methods, a broad range of applications of topological methods to nonlinear differential equations is presented: solvability, estimation of the number of solutions, and the branching of solutions of nonlinear equations. Skrypnik establishes, by various procedures, a priori estimates and the regularity of solutions of nonlinear elliptic equations of arbitrary order. Also covered are methods of homogenization of nonlinear elliptic problems in perforated domains. The book is suitable for use in graduate courses in differential equations and nonlinear functional analysis.

New implicit iterative methods are presented for the efficient numerical solution of non-linear elliptic boundary-value problems. Isomorphic iterative schemes in conjunction with preconditioning techniques are used for solving non-linear elliptic equations in two and three-space dimensions. The application of the derived methods on characteristic 2D and 3D non-linear boundary-value problems is discussed and numerical results are given.

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.