Nonlinear Elliptic Equations Research Articles

Solving Nonlinear Elliptic PDEs in 2D and 3D Using Polyharmonic Splines and Low-Degree of Polynomials

In this paper, the improved localized method of approximated particular solutions (ILMAPS) using polyharmonic splines (PHS) together with a low-degree of polynomial basis is used to approximate solutions of various nonlinear elliptic Partial Differential Equations (PDEs). The method is completely meshfree, and it uses a radial basis function (RBF) that has no shape parameters. The discretization process is done through a simple collocation technique on a set of points in the local domain of influence. Resulted system of nonlinear algebraic equations is solved by the Picard method.The performance of the proposed method is tested on various nonlinear elliptical problems, including the Poisson-type problems in 2D and 3D with constant or variable coefficients on rectangular or irregular domains and the Poisson–Boltzmann equation with Dirichlet boundary conditions or mixed boundary conditions. The effect of domain shapes in 2D and 3D, types of boundary conditions, and degrees of PHS, and order of polynomial basis are examined. The performance of the method is compared with other bases such as multiquadrics (MQ) basis functions, and with results reported in the literature (method of particular solutions using polynomials). The numerical experiments suggest that ILMAPS with polyharmonic splines yields considerably superior accuracy than other RBFs as well as other approaches reported in the literature for solving nonlinear elliptic PDEs.

On the optimal Lq-regularity for viscous Hamilton–Jacobi equations with subquadratic growth in the gradient

This paper studies a maximal [Formula: see text]-regularity property for nonlinear elliptic equations of second order with a zeroth order term and gradient nonlinearities having superlinear and subquadratic growth, complemented with Dirichlet boundary conditions. The approach is based on the combination of linear elliptic regularity theory and interpolation inequalities, so that the analysis of the maximal regularity estimates boils down to determine lower order integral bounds. The latter are achieved via a [Formula: see text] duality method, which exploits the regularity properties of solutions to stationary Fokker–Planck equations. For the latter problems, we discuss both global and local estimates. Our main novelties for the regularity properties of this class of nonlinear elliptic boundary-value problems are the analysis of the end-point summability threshold [Formula: see text], [Formula: see text] being the dimension of the ambient space and [Formula: see text] the growth of the first-order term in the gradient variable, along with the treatment of the full integrability range [Formula: see text].

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.

Existence and properties of bubbling solutions for a critical nonlinear elliptic equation

Existence and properties of bubbling solutions for a critical nonlinear elliptic equation

Global regularity for a class of fully nonlinear PDEs with unbalanced variable degeneracy

AbstractWe establish the existence and sharp global regularity results (, and estimates) for a class of fully nonlinear elliptic Partial Differential Equations (PDEs) with unbalanced variable degeneracy. In a precise way, the degeneracy law of the model switches between two different kinds of degenerate elliptic operators of variable order, according to the null set of a modulating function . The model case in question is given by for a bounded, regular, and open set , and appropriate continuous data , , and . Such sharp regularity estimates generalize and improve, to some extent, earlier ones via geometric treatments. Our results are consequences of geometric tangential methods and make use of compactness, localized oscillating, and scaling techniques. In the end, our findings are applied in the study of a wide class of nonlinear models.

Solvability of Iterative Classes of Nonlinear Elliptic Equations on an Exterior Domain

This work explores the possibility that iterative classes of elliptic equations have both single and coupled positive radial solutions. Our approach is based on using the well-known Guo–Krasnoselskii and Avery–Henderson fixed-point theorems in a Banach space. Furthermore, we utilize Rus’ theorem in a metric space, to prove the uniqueness of solutions for the problem. Examples are constructed for the sake of verification.

Combined effects in nonlinear elliptic equations involving fractional operators

Combined effects in nonlinear elliptic equations involving fractional operators

Existence and regularity results for a degenerate elliptic nonlinear equation on weighted Sobolev spaces

This paper deals with the existence of many solutions a degenerate weighted elliptic equation which its the number of solutions depend on degeneracy term a i.e., the number of subdomains of Ω\\a −1(0) whose boundary is made by submanifolds with 1-codimension. Moreover, by the Ljusternik-Schnirelman principle, we would study the corresponding eigenvalue problem and obtained the nonnegative eigenvalues sequence of main problem. Finally, we will discuss of the regularity of the solutions on C 1,α (Ω).

Existence of positive radial solutions for nonlinear elliptic equations with gradient terms in an annulus

Existence of positive radial solutions for nonlinear elliptic equations with gradient terms in an annulus

Local regularity for nonlinear elliptic and parabolic equations with anisotropic weights

AbstractThe main purpose of this paper is to capture the asymptotic behaviour for solutions to a class of nonlinear elliptic and parabolic equations with the anisotropic weights consisting of two power-type weights of different dimensions near the degenerate or singular point, especially covering the weighted p-Laplace equations and weighted fast diffusion equations. As a consequence, we also establish the local Hölder estimates for their solutions in the presence of single power-type weights.

Regularity of Hamiltonian stationary equations in symplectic manifolds

In this paper, we prove that any C1-regular Hamiltonian stationary Lagrangian submanifold in a symplectic manifold is smooth. More broadly, we develop a regularity theory for a class of fourth order nonlinear elliptic equations with two distributional derivatives. Our fourth order regularity theory originates in the geometrically motivated variational problem for the volume functional, but should have applications beyond.

Liouville type theorems of fully nonlinear elliptic equations in half spaces

In this paper, we investigate the asymptotic behavior of viscosity solutions for a class of fully nonlinear second-order elliptic equations in the half space. Moreover, the result is applied to Monge-Ampère equations and consequently we improve some existent result.

Normalized Wolfe-Powell-type local minimax method for finding multiple unstable solutions of nonlinear elliptic PDEs

The local minimax method (LMM) proposed by Li and Zhou (2001) and Li and Zhou (2002) is an efficient method to solve nonlinear elliptic partial differential equations (PDEs) with certain variational structures for multiple solutions. The steepest descent direction and the Armijo-type step-size search rules are adopted in Li and Zhou (2002) and play a significant role in the performance and convergence analysis of traditional LMMs. In this paper, a new algorithm framework of the LMMs is established based on general descent directions and two normalized (strong) Wolfe-Powell-type step-size search rules. The corresponding algorithm framework named as the normalized Wolfe-Powell-type LMM (NWP-LMM) is introduced with its feasibility and global convergence rigorously justified for general descent directions. As a special case, the global convergence of the NWP-LMM algorithm combined with the preconditioned steepest descent (PSD) directions is also verified. Consequently, it extends the framework of traditional LMMs. In addition, conjugate gradient-type (CG-type) descent directions are utilized to speed up the NWP-LMM algorithm. Finally, extensive numerical results for several semilinear elliptic PDEs are reported to profile their multiple unstable solutions and compared for different algorithms in the LMM’s family to indicate the effectiveness and robustness of our algorithms. In practice, the NWP-LMM combined with the CG-type direction indeed performs much better than its known LMM companions.

𝐿^{∞}-Energy method and its applications to nonlinear partial differential equations

A method called “ L ∞ L^\infty -Energy Method” is introduced. Several basic tools for this method are prepared and some typical ways of usage of this method are exemplified for some nonlinear parabolic equations and nonlinear elliptic equations with nonlinear boundary conditions.

Uniform integrability in periodic homogenization of fully nonlinear elliptic equations

This paper is devoted to the study of uniform W^{1,frac{np}{n-p}}- and W^{2,p}-estimates for periodic homogenization problems of fully nonlinear elliptic equations. We establish sharp, global, large-scale estimates under the Dirichlet boundary conditions. The main novelty of this paper can be found in the characterization of the size of the “effective” Hessian and gradient of viscosity solutions to homogenization problems. Moreover, the large-scale estimates work in a large class of non-convex problems. It should be stressed that our global estimates are new even for the standard problems without homogenization.

Behavior of Entire Solutions of a Nonlinear Elliptic Equation with An Inhomogeneous Singular Term

Behavior of Entire Solutions of a Nonlinear Elliptic Equation with An Inhomogeneous Singular Term

On the Regularity of Non-scattering Anisotropic Inhomogeneities

In this paper we examine the necessary conditions for an anisotropic inhomogeneous medium to be non-scattering at a single wave number and for a single incident field. These conditions are expressed in terms of the regularity of the boundary of the inhomogeneity. We assume that the coefficients, characterizing the constitutive material properties of the medium, are sufficiently smooth, and that the incident wave is appropriately non-degenerate. Our analysis utilizes the Hodograph transform as well as regularity results for nonlinear elliptic partial differential equations. Our approach requires that the boundary a-priori is of class $$C^{1,\alpha }$$ for some $$0<\alpha <1$$ .

Uniform Lipschitz estimates up to the boundary for singular perturbation problem for some nonlinear elliptic PDEs with unbounded ingredients

We prove uniform up to the boundary gradient estimates for one phase nonlinear inhomogeneous singular perturbation problems with unbounded measurable ingredients governed by fully nonlinear elliptic equations. We present similar results for quasilinear PDEs with bounded RHS. Our proof is based on the Lipschitz regularity up to the boundary for free boundary problems (FBP) found in Braga and Moreira (2022).

Energy criteria of global existence for a class of Hartree equations with Coulomb potential

This paper studies a class of Hartree equations with Coulomb potential. Combined with the conservation of mass and energy, we analyze the variational characteristics of the corresponding nonlinear elliptic equation. According to the range of parameters, we construct the evolution invariant flows of the equation in different cases. Then the sharp energy thresholds for global existence and blow-up of solutions are discussed in detail.

Fucik spectrum with weights and existence of solutions for nonlinear elliptic equations with nonlinear boundary conditions

We consider the boundary value problem $$\displaylines{ - \Delta u + c(x) u = \alpha m(x) u^+ - \beta m(x) u^- +f(x,u), \quad x \in \Omega, \cr \frac{\partial u}{\partial \eta} + \sigma (x) u =\alpha \rho (x) u^+- \beta \rho (x) u^- +g(x,u), \quad x \in \partial \Omega, }$$ where \((\alpha, \beta) \in \mathbb{R}^2\), \(c, m \in L^\infty (\Omega)\), \(\sigma, \rho \in L^\infty (\partial\Omega)\), and the nonlinearities f and g are bounded continuous functions. We study the asymmetric (Fucik) spectrum with weights, and prove existence theorems for nonlinear perturbations of this spectrum for both the resonance and non-resonance cases. For the resonance case, we provide a sufficient condition, the so-called generalized Landesman-Lazer condition, for the solvability. The proofs are based on variational methods and rely strongly on the variational characterization of the spectrum. See also https://ejde.math.txstate.edu/special/02/m2/abstr.html