Nonlinear Elliptic Equations Research Articles

A direct proof of existence of weak solutions to elliptic problems

We provide a direct proof of existence and uniqueness of weak solutions to a broad family of strongly nonlinear elliptic equations with lower-order terms. The leading part of the operator satisfies general growth conditions settling the problem in the framework of fully anisotropic and inhomogeneous Musielak-Orlicz spaces generated by an $N$-function \linebreak $M\colon \Omega\times\mathbb R^d\to [0,\infty)$. Neither $\nabla_2$ nor $\Delta_2$ conditions are imposed on $M$. Our results cover among others problems with anisotropic polynomial, Orlicz, variable exponent, and double phase growth.

Nonlinear elliptic equations by topological degree in Musielak-Orlicz-Sobolev spaces

"We prove by using the topological degree theory the existence of at least one weak solution for the nonlinear elliptic equation − div a1(x, ∇u) + a0(x, u) = f (x, u, ∇u) with homogeneous Dirichlet boundary condition in Musielak-Orlicz-Sobolev spaces. Keywords: Nonlinear elliptic equation, Musielak-Orlicz-Sobolev space, topological degree."

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.

Nonlinear Elliptic Equations on Weighted Sobolev Space

Nonlinear Elliptic Equations on Weighted Sobolev Space

Parallel Computing of Thermo-Fluid Modeling Related to the FSW Process

This paper is devoted to the numerical computation of a steady-state thermo-fluid modeling related to the Friction Stir Welding Process in a two-dimensional cylindrical geometry. It analyzes the efficiency of the implementation on parallel architectures of two finite-difference schemes on a structured grid. The first one applies the Newton-Raphson method to compute a numerical solution of this non-linear elliptic type equation, and uses an iterative sparse solver. The second one is based on a time-marching approach converging to the steady state solution thanks to a time-explicit computation. Their respective performance is presented and discussed. Some numerical simulation results are presented to validate the proposed approach.

A compact high resolution semi-variable mesh exponential finite difference method for non-linear boundary value problems of elliptic nature

In this research an original exponential approximation of second accuracy in y- and third accuracy in x-axis employing full step discretization has been designed for solving 2D non-linear partial differential equation of elliptic nature in a rectangular domain. We adopted non-constant grid spacing in x -axis and constant grid spacing in y -axis in numerical computation of convection-diffusion equation where convection term dominates. An exhaustive error behaviour of the technique has been analysed. Non-linear elliptic equations are computed using this method. Lastly, proposed idea is scrutinized on simulations of physical repute with emphasis on convection-diffusion equation articulating the efficacy of the technique.

Global regularity results for a class of singular/degenerate fully nonlinear elliptic equations

Global regularity results for a class of singular/degenerate fully nonlinear elliptic equations

Concavity principles for nonautonomous elliptic equations and applications

In the study of concavity properties of positive solutions to nonlinear elliptic partial differential equations the diffusion and the nonlinearity are typically independent of the space variable. In this paper we obtain new results aiming to get almost concavity results for a relevant class of anisotropic semilinear elliptic problems with spatially dependent source and diffusion.

Nine-point compact sixth-order approximation for two-dimensional nonlinear elliptic partial differential equations: Application to bi- and tri-harmonic boundary value problems

Nine point sixth order compact numerical approximations are suggested to solve 2D nonlinear elliptic partial differential equations (NLEPDEs) and for the estimation of normal derivatives on a uniform rectangular grid subject to Dirichlet boundary conditions. We deliberate error analysis and reveal that, under specific conditions, our method converges to the sixth order. In addition, we extend our technique to vector form in order to solve the system of NLEPDEs. In application, we discuss nine-point compact sixth order approximations for bi- and tri-harmonic elliptic boundary value problems. Numerical experiments are carried out on several benchmark problems including bi- and tri-harmonic equations, and verified the sixth order convergence of the proposed methods.

On a class of fully nonlinear elliptic equations containing gradient terms on compact almost Hermitian manifolds

On a class of fully nonlinear elliptic equations containing gradient terms on compact almost Hermitian manifolds

Existence of renormalized solutions to fully anisotropic and inhomogeneous elliptic problems

We will present the proof of existence and uniqueness of renormalized solutions to a broad family of strongly non-linear elliptic equations with lower order terms and data of low integrability. The leading part of the operator satisfies general growth conditions settling the problem in the framework of fully anisotropic and inhomogeneous Musielak–Orlicz spaces. The setting considered in this paper generalized known results in the variable exponents, anisotropic polynomial, double phase and classical Orlicz setting.

Constructive proofs for localised radial solutions of semilinear elliptic systems on

Ground state solutions of elliptic problems have been analysed extensively in the theory of partial differential equations, as they represent fundamental spatial patterns in many model equations. While the results for scalar equations, as well as certain specific classes of elliptic systems, are comprehensive, much less is known about these localised solutions in generic systems of nonlinear elliptic equations. In this paper we present a general method to prove constructively the existence of localised radially symmetric solutions of elliptic systems on . Such solutions are essentially described by systems of non-autonomous ordinary differential equations. We study these systems using dynamical systems theory and computer-assisted proof techniques, combining a suitably chosen Lyapunov–Perron operator with a Newton–Kantorovich type theorem. We demonstrate the power of this methodology by proving specific localised radial solutions of the cubic Klein–Gordon equation on , the Swift–Hohenberg equation on , and a three-component FitzHugh–Nagumo system on . These results illustrate that ground state solutions in a wide range of elliptic systems are tractable through constructive proofs.

Smooth approximations for fully nonlinear nonlocal elliptic equations

We show that any viscosity solution to a general fully nonlinear nonlocal elliptic equation can be approximated by smooth ( C ∞ C^\infty ) solutions.

The deformed Hermitian–Yang–Mills equation, the Positivstellensatz, and the solvability

Let (M,ω) be a compact connected Kähler manifold of complex dimension four and let [χ]∈H1,1(M;R). We confirm the conjecture by Collins–Jacob–Yau [8] of the solvability of the deformed Hermitian–Yang–Mills equation, which is given by the following nonlinear elliptic equation ∑iarctan⁡(λi)=θˆ, where λi are the eigenvalues of χ with respect to ω and θˆ is a topological constant. This conjecture was stated in [8], wherein they proved that the existence of a supercritical C-subsolution or the existence of a C-subsolution when θˆ∈[((n−2)+2/n)π/2,nπ/2) will give the solvability of the deformed Hermitian–Yang–Mills equation. Collins–Jacob–Yau conjectured that their existence theorem can be improved to θˆ>(n−2)π/2, where n is the complex dimension of the manifold. In this paper, we confirm their conjecture that when the complex dimension equals four and θˆ is close to the supercritical phase π from the right, then the existence of a C-subsolution implies the solvability of the deformed Hermitian–Yang–Mills equation.

Solvability of the two-dimensional stationary incompressible inhomogeneous Navier–Stokes equations with variable viscosity coefficient

We show the existence and the regularity properties of (a class of) weak solutions to the two-dimensional stationary incompressible inhomogeneous Navier–Stokes equations with density-dependent viscosity coefficients, by analyzing a fourth-order nonlinear elliptic equation for the stream function. For some stationary symmetric flows, we reformulate the Navier–Stokes equations as ordinary differential equations and give explicit examples of weak solutions. We present some further (ir-)regularity results in the case of piecewise-constant viscosity coefficients.

Optical soliton solutions of the generalized sine-Gordon equation

In this study, the extended trial equation method based on the general form of the nonlinear elliptic ordinary differential equation isemployed to solve the nonlinear generalized sine-Gordon equations. By using this method, we achieve, unlike new types of exact wave solutions such as Elliptic-F, Elliptic-E, and Elliptic-Π functions that are known as elliptic integrals.

Complete conformal metrics of negative Ricci curvature in open manifolds

In this paper, we study the problem of finding complete conformal metrics determined by some symmetric function of the modified Schouten tensor in open manifolds. We prove the existence of such metrics by solving a class of fully nonlinear elliptic equations with the infinity Dirichlet boundary condition.

Monotone operators and a class of nonlinear elliptic equations on the Sierpiński gasket

Abstract We study the existence of solutions for the nonlinear elliptic problem - Δ ⁢ v - λ ⁢ g 1 ⁢ v + h 1 ⁢ ( v ) = f 1 \displaystyle-\Delta v-\lambda g_{1}v+h_{1}(v)=f_{1} in ⁢ V ∖ V 0 , \displaystyle\phantom{}\text{in }V\setminus V_{0}, v = 0 \displaystyle v=0 on ⁢ V 0 , \displaystyle\phantom{}\text{on }V_{0}, where V is the Sierpiński gasket in ℝ N - 1 {\mathbb{R}^{N-1}} ( N ≥ 2 {N\geq 2} ), V 0 {V_{0}} is its boundary (consisting of its N corners) and λ ∈ ℝ {\lambda\in\mathbb{R}} . Here f 1 , g 1 : V → ℝ {f_{1},g_{1}:V\to\mathbb{R}} , h 1 : ℝ → ℝ {h_{1}:\mathbb{R}\to\mathbb{R}} are the maps satisfying suitable hypotheses.

Differential inclusions for the Schouten tensor and nonlinear eigenvalue problems in conformal geometry

Let g0 be a smooth Riemannian metric on a closed manifold Mn of dimension n≥3. We study the existence of a smooth metric g conformal to g0 whose Schouten tensor Ag satisfies the differential inclusion λ(g−1Ag)∈Γ on Mn, where Γ⊂Rn is a cone satisfying standard assumptions. Inclusions of this type are often assumed in the existence theory for fully nonlinear elliptic equations in conformal geometry. We assume the existence of a continuous metric g1 conformal to g0 satisfying λ(g1−1Ag1)∈Γ′‾ in the viscosity sense on Mn, together with a nondegenerate ellipticity condition, where Γ′=Γ or Γ′ is a cone slightly smaller than Γ. In fact, we prove not only the existence of metrics satisfying such differential inclusions, but also existence and uniqueness results for fully nonlinear eigenvalue problems for the Schouten tensor. We also give a number of geometric applications of our results. We show that the sign of a nonlinear eigenvalue for the σ2 operator is a conformal invariant in three dimensions. We also give a generalisation of a theorem of Aubin & Ehrlick on pinching of the Ricci curvature, and an application in the study of Green's functions for fully nonlinear Yamabe problems.

Nonlinear elliptic p-Laplacian equations in the whole space

In this paper the existence and multiplicity of non-zero solutions for nonlinear Dirichlet problems involving the p-Laplacian operator and which are defined in the whole space is established. In particular, the existence of two non-zero solutions, one with negative energy and other with positive one for equations having combined effects of concave and convex nonlinearities is obtained. The approach is based on variational methods.