Oblique Boundary Conditions Research Articles

Kazdan–Warner problem on compact Riemann surfaces with boundary

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.

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

We prove weighted Orlicz-Sobolev regularity for fully nonlinear elliptic equations with oblique boundary condition under asymptotic conditions of the following problem{F(D2u,Du,u,x)=f(x)inΩβ⋅Du+γu=gon∂Ω, where Ω is a bounded domain in Rn (n≥2), under suitable assumptions on the source term f, data β,γ and g. Our approach guarantees such estimates under conditions where the governing operator F does not require a convex (or concave) structure. We also deal with weighted Orlicz-type estimates for the obstacle problem with oblique derivative condition on the boundary. As a final application, the developed methods provide weighted Orlicz-BMO regularity for the Hessian, provided that the source lies in that space and in weighted Orlicz space associated.

Steady-state inhomogeneous diffusion with generalized oblique boundary conditions

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.

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

We derive global W2,p estimates (with n≤p<∞) for viscosity solutions to fully nonlinear elliptic equations under relaxed structural assumptions on the governing operator that are weaker than convexity and oblique boundary conditions as follows:{F(D2u,Du,u,x)=f(x)inΩβ(x)⋅Du(x)+γ(x)u(x)=g(x)on∂Ω, for f∈Lp(Ω) and under appropriate assumptions on the data β,γ, g and Ω⊂Rn. Our approach makes use of geometric tangential methods, which consist of importing “fine regularity estimates” from a limiting profile, i.e., the Recession operator, associated with the original second-order one via compactness and stability procedures. As a result, we pay special attention to the borderline scenario, i.e., f∈BMOp⊋L∞. In such a setting, we prove that solutions enjoy BMOp type estimates for their second derivatives. Finally, as another application of our findings, we obtain Hessian estimates to obstacle-type problems under oblique boundary conditions and no convexity assumptions, which may have their own mathematical interest. A density result for a suitable class of viscosity solutions will also be addressed.

Div‐curl‐gradient inequalities and div‐curl system with oblique boundary condition

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 .

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

This paper proposes a C0 (non-Lagrange) primal finite element approximation of the linear elliptic equations in non-divergence form with oblique boundary conditions in planar, curved domains. As an extension of [Calcolo, 58 (2022), No. 9], the Miranda–Talenti estimate for oblique boundary conditions at a discrete level is established by enhancing the regularity on the vertices. Consequently, the coercivity constant for the proposed scheme is exactly the same as that from PDE theory. The quasi-optimal order error estimates are established by carefully studying the approximation property of the finite element spaces. Numerical experiments are provided to verify the convergence theory and to demonstrate the accuracy and efficiency of the proposed methods.

Classical solutions to local first-order extended mean field games

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.

On criticality theory for elliptic mixed boundary value problems in divergence form

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.

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

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.

The linear sampling method for penetrable cylinder with inclusions for obliquely incident polarized electromagnetic waves

Abstract 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.

Classical and weak solutions to local first-order mean field games through elliptic regularity

We study the regularity and well-posedness of the local, first-order forward–backward mean field games system, assuming a polynomially growing cost function and a Hamiltonian of quadratic growth. We consider systems and terminal data that are strictly monotone in density and study two different regimes depending on whether there exists a lower bound for the running cost function. The work relies on a transformation due to P.-L. Lions, which gives rise to an elliptic partial differential equation with oblique boundary conditions, that is strictly elliptic when the coupling is unbounded from below. In this case, we prove that the solution is smooth. When the problem is degenerate elliptic, we obtain existence and uniqueness of weak solutions analogous to those obtained by P. Cardaliaguet and P. J. Graber for the case of a terminal condition that is independent of the density. The weak solutions are shown to arise as viscous limits of classical solutions to strictly elliptic problems.

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

We study fully nonlinear parabolic equations in nondivergence form with oblique boundary conditions. An optimal and global Calderón-Zygmund estimate is obtained by proving that the Hessian of the viscosity solution to the oblique boundary problem is as integrable as the nonhomogeneous term in Lp spaces under minimal regularity requirement on the nonlinear operator, the boundary data and the boundary of the domain.

Superconvergence of the Strang splitting when using the Crank-Nicolson scheme for parabolic PDEs with Dirichlet and oblique boundary conditions

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.

Regularity for fully nonlinear parabolic equations with oblique boundary data

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.

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

We study fully nonlinear elliptic equations with oblique boundary conditions. We obtain a global W2,p-estimate, n−τ0<p<∞, for viscosity solutions of such problems when the boundary of the domain is in C2,α for every 0<α<1.

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

We develop and analyse finite volume methods for the Poisson problem with boundary conditions involving oblique derivatives. We design a generic framework, for finite volume discretisations of such models, in which internal fluxes are not assumed to have a specific form, but only to satisfy some (usual) coercivity and consistency properties. The oblique boundary conditions are split into a normal component, which directly appears in the flux balance on control volumes touching the domain boundary, and a tangential component which is managed as an advection term on the boundary. This advection term is discretised using a finite volume method based on a centred discretisation (to ensure optimal rates of convergence) and stabilised using a vanishing boundary viscosity. A convergence analysis, based on the 3rd Strang Lemma [9], is conducted in this generic finite volume framework, and yields the expected O(h) optimal convergence rate in discrete energy norm.We then describe a specific choice of numerical fluxes, based on a generalised hexahedral meshing of the computational domain. These fluxes are a corrected version of fluxes originally introduced in [29]. We identify mesh regularity parameters that ensure that these fluxes satisfy the required coercivity and consistency properties. The theoretical rates of convergence are illustrated by an extensive set of 3D numerical tests, including some conducted with two variants of the proposed scheme. A test involving real-world data measuring the disturbing potential in Earth gravity modelling over Slovakia is also presented.

A Discontinuous Galerkin Finite Element Method fOR Uniformly Elliptic Two Dimensional Oblique Boundary-Value Problems

In this paper we present and analyze a discontinuous Galerkin finite element method (DGFEM) for the approximation of solutions to elliptic partial differential equations in nondivergence form, with oblique boundary conditions, on curved domains. In Kawecki [A DGFEM for Nondivergence Form Elliptic Equations with Cordes Coefficients on Curved Domains], the author introduced a DGFEM for the approximation of solutions to elliptic partial differential equations in nondivergence form, with Dirichlet boundary conditions. In this paper, we extend the framework further, allowing for the oblique boundary condition. The method also provides an approximation for the constant occurring in the compatibility condition for the elliptic problems under consideration.

Regularity for Fully Nonlinear Elliptic 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.

A degree theory for second order nonlinear elliptic operators with nonlinear oblique boundary conditions

In this paper, we introduce an integer-valued degree for second order fully nonlinear elliptic operators with nonlinear oblique boundary conditions. We also give some applications to the existence of solutions of certain nonlinear elliptic equations arising from a Yamabe problem with boundary and reflector problems.

A new proof of gradient estimates for mean curvature equations with oblique boundary conditions

In this paper, we will use the maximum principle to give a new proof of the gradient estimates for mean curvature equations with some oblique derivative problems. In particular, we shall give a new proof for the capillary problem with zero gravity.