Nonlinear Elliptic Operators Research Articles

Reiterated Homogenization of nonlinear degenerate elliptic operators with nonstandard growth

Reiterated Homogenization of nonlinear degenerate elliptic operators with nonstandard growth

Dirichlet problem for a class of nonlinear degenerate elliptic operators with critical growth and logarithmic perturbation

Dirichlet problem for a class of nonlinear degenerate elliptic operators with critical growth and logarithmic perturbation

Structure of singularities in the nonlinear nerve conduction problem

We give a characterization of the singular points of the free boundary \partial \{u>0\} for viscosity solutions of the nonlinear equation F(D^{2} u)=-\chi_{\{u>0\}}, where F is a fully nonlinear elliptic operator and \chi is the characteristic function. This equation models the propagation of a nerve impulse along an axon. We analyze the structure of the free boundary \partial\{u>0\} near the singular points where u and \nabla u vanish simultaneously. Our method uses the stratification approach developed in Dipierro and the author’s 2018 paper. In particular, when n=2 we show that near a flat singular free boundary point, \partial\{u>0\} is a union of four C^{1} arcs tangential to a pair of crossing lines.

Sharp estimates for the first Robin eigenvalue of nonlinear elliptic operators

The aim of this paper is to obtain optimal estimates for the first Robin eigenvalue of the anisotropic p-Laplace operator, namely:λ1(β,Ω)=minψ∈W1,p(Ω)∖{0}⁡∫ΩF(∇ψ)pdx+β∫∂Ω|ψ|pF(νΩ)dHN−1∫Ω|ψ|pdx, where p∈]1,+∞[, Ω is a bounded, anisotropic mean convex domain in RN, νΩ is its Euclidean outward normal, β is a real number, and F is a sufficiently smooth norm on RN. The estimates we found are in terms of the first eigenvalue of a one-dimensional nonlinear problem, which depends on β and on geometrical quantities associated to Ω. More precisely, we prove a lower bound of λ1 in the case β>0, and a upper bound in the case β<0. As a consequence, we prove, for β>0, a lower bound for λ1(β,Ω) in terms of the anisotropic inradius of Ω and, for β<0, an upper bound of λ1(β,Ω) in terms of β.

Fully nonlinear free transmission problems

We examine a free transmission problem driven by fully nonlinear elliptic operators. Since the transmission interface is determined endogenously, our analysis regards this object as a free boundary. We start by relating our problem with a pair of viscosity inequalities. Then, approximation methods ensure that strong solutions are of class C ^{1,\operatorname{Log-Lip}} , locally. In addition, under further conditions on the problem, we prove quadratic growth of the solutions away from branch points.

Rate of convergence for singular perturbations of Hamilton-Jacobi equations in unbounded spaces

We prove rate of convergence results for singular perturbations of Hamilton-Jacobi equations in unbounded spaces where the fast operator is linear, uniformly elliptic and has an Ornstein-Uhlenbeck-type drift. The slow operator is a fully nonlinear elliptic operator while the source term is assumed only locally Hölder continuous in both fast and slow variables. We obtain several rates of convergence according to the regularity of the source term.

General framework to construct local-energy solutions of nonlinear diffusion equations for growing initial data

This paper presents an integrated framework to construct local-energy solutions to fairly general nonlinear diffusion equations for initial data growing at infinity under suitable assumptions on local-energy estimates for approximate solutions. A delicate issue for constructing local-energy solutions resides in the identification of weak limits of nonlinear terms for approximate solutions in a limiting procedure. Indeed, such an identification process often needs the maximal monotonicity of nonlinear elliptic operators (involved in the doubly-nonlinear equations) as well as uniform estimates for approximate solutions; however, even the monotonicity is violated due to a localization of the equations, which is also necessary to derive local-energy estimates for approximate solutions. In the present paper, such an inconsistency is systematically overcome by reducing the original equation to a localized one, where a (no longer monotone) localized elliptic operator is decomposed into the sum of a maximal monotone operator and a perturbation, and by integrating all the other relevant processes. Furthermore, the general framework developed in the present paper is also applied to the Finsler porous medium and fast diffusion equations, which are variants of the classical PME and FDE and also classified as a doubly-nonlinear equation.

New regularity estimates for fully nonlinear elliptic equations

We establish new quantitative Hessian integrability estimates for viscosity supersolutions of fully nonlinear elliptic operators. As a corollary, we show that the optimal Hessian power integrability ε=ε(λ,Λ,n) in the celebrated W2,ε–regularity estimate satisfies(1+23(1−λΛ))n−1ln⁡n4⋅(λΛ)n−1≤ε≤nλ(n−1)Λ+λ, where n≥3 is the dimension and 0<λ<Λ are the ellipticity constants. In particular, (Λλ)n−1ε(λ,Λ,n) blows-up, as n→∞; previous results yielded fast decay of such a quantity. The upper estimate improves the one obtained by Armstrong, Silvestre, and Smart in [1].

Fully nonlinear Hamilton–Jacobi equations of degenerate type

We examine Hamilton–Jacobi equations driven by fully nonlinear degenerate elliptic operators in the presence of superlinear Hamiltonians. By exploring the Ishii–Jensen inequality, we prove that viscosity solutions are locally Lipschitz-continuous, with estimates depending on the structural conditions of the problem. We close the paper with an application of our findings to a two-phase free boundary problem.

On Numerical Approximation of Diffusion Problems Governed by Variable-Exponent Nonlinear Elliptic Operators

We highlight the interest and the limitations of the L1-based Young measure technique for studying convergence of numerical approximations for diffusion problems of the variable-exponent p(x)- and p(u)- laplacian kind. CVFE (Control Volume Finite Element) and DDFV (Discrete Duality Finite Volume) schemes are analyzed and tested. In the situation where the variable exponent is log-Hölder continuous, convergence is proved along the guidelines elaborated in [Andreianov et al. Nonlinear Anal. 72, 4649–4660, 2010 & Nonlinear Anal. 73, 2–24, 2010] while investigating the structural stability of weak solutions for this class of PDEs. In general, the lack of density of the smooth functions in the energy space, related to the Lavrentiev phenomenon for the associated variational problems, makes it necessary to distinguish two notions of solutions, the narrow ones (the H-solutions) and the broad ones (the W-solutions). Some situations where approximation methods “select” the one or the other of these two solution notions are described and illustrated by numerical tests.

On the σ2-Nirenberg problem on [formula omitted

We establish theorems on the existence and compactness of solutions to the σ2-Nirenberg problem on the standard sphere S2. A first significant ingredient, a Liouville type theorem for the associated fully nonlinear Möbius invariant elliptic equations, was established in an earlier paper of ours. Our proof of the existence and compactness results requires a number of additional crucial ingredients which we prove in this paper: A Liouville type theorem for the associated fully nonlinear Möbius invariant degenerate elliptic equations, a priori estimates of first and second order derivatives of solutions to the σ2-Nirenberg problem, and a Bôcher type theorem for the associated fully nonlinear Möbius invariant elliptic equations. Given these results, we are able to complete a fine analysis of a sequence of blow-up solutions to the σ2-Nirenberg problem. In particular, we prove that there can be at most one blow-up point for such a blow-up sequence of solutions. This, together with a Kazdan-Warner type identity, allows us to prove L∞ a priori estimates for solutions of the σ2-Nirenberg problem under some simple generic hypothesis. The higher derivative estimates then follow from classical estimates of Nirenberg and Schauder. In turn, the existence of solutions to the σ2-Nirenberg problem is obtained by an application of the by now standard degree theory for second order fully nonlinear elliptic operators.

Regularity of the singular set in the fully nonlinear obstacle problem

For the obstacle problem involving a convex fully nonlinear elliptic operator, we show that the singular set in the free boundary stratifies. The top stratum is locally covered by a C^{1,\alpha} manifold, and the lower strata are covered by C^{1,\log^\varepsilon} manifolds. This recovers some of the recent regularity results due to Colombo–Spolaor–Velichkov (2018) and Figalli–Serra (2019) when the operator is the Laplacian.

Convergent Finite Difference Methods for Fully Nonlinear Elliptic Equations in Three Dimensions

We introduce a generalized finite difference method for solving a large range of fully nonlinear elliptic partial differential equations in three dimensions. Methods are based on Cartesian grids, augmented by additional points carefully placed along the boundary at high resolution. We introduce and analyze a least-squares approach to building consistent, monotone approximations of second directional derivatives on these grids. We then show how to efficiently approximate functions of the eigenvalues of the Hessian through a multi-level discretization of orthogonal coordinate frames in \(\mathbb {R}^3\). The resulting schemes are monotone and fit within many recently developed convergence frameworks for fully nonlinear elliptic equations including non-classical Dirichlet problems that admit discontinuous solutions, Monge–Ampère type equations in optimal transport, and eigenvalue problems involving nonlinear elliptic operators. Computational examples demonstrate the success of this method on a wide range of challenging examples.

On the lower bound of the principal eigenvalue of a nonlinear operator

We prove sharp lower bound estimates for the first nonzero eigenvalue of the non-linear elliptic diffusion operator L_p on a smooth metric measure space, without boundary or with a convex boundary and Neumann boundary condition, satisfying BE(kappa ,N) for kappa ne 0. Our results extends the work of Koerber Valtorta (Calc Vari Partial Differ Equ. 57(2), 49 2018) for case kappa =0 and Naber–Valtorta (Math Z 277(3–4):867–891, 2014) for the p-Laplacian.

Perturbations of Nonlinear Elliptic Operators by Potentials in the Space of Multiplicators

Perturbations of Nonlinear Elliptic Operators by Potentials in the Space of Multiplicators

Global gradient estimates for a general class of quasilinear elliptic equations with Orlicz growth

We provide an optimal global Calderón-Zygmund theory for quasilinear elliptic equations of a very general form with Orlicz growth on bounded nonsmooth domains under minimal regularity assumptions of the nonlinearity A = A ( x , u , D u ) A=A(x,u,Du) in the first and second variables ( x , z ) (x,z) as well as on the boundary of the domain. Our result improves known regularity results in the literature regarding nonlinear elliptic operators depending on a given bounded weak solution.

Non-existence of dead cores in fully nonlinear elliptic models

We investigate non-existence of non-negative dead core solutions for the problem [Formula: see text] Here, [Formula: see text] is a bounded smooth domain, [Formula: see text] is a fully nonlinear elliptic operator, [Formula: see text] is a sign-changing weight, [Formula: see text], and [Formula: see text]. We show that this problem has no nontrivial dead core solutions if either [Formula: see text] is close enough to [Formula: see text] or the negative part of [Formula: see text] is sufficiently small. In addition, we obtain the existence and uniqueness of a positive solution under these conditions on [Formula: see text] and [Formula: see text]. Our results extend previous ones established in the semilinear case, and are new even for the simple model [Formula: see text], where [Formula: see text] is a uniformly elliptic and non-negative matrix.

The obstacle problem for a class of degenerate fully nonlinear operators

We study the obstacle problem for fully nonlinear elliptic operators with an anisotropic degeneracy on the gradient: \left\{\begin{array}{rll} \min\left\{f-|Du|^\gamma F(D^2u),u-\phi\right\} &amp;= 0 &amp; \textrm{ in } \Omega,\\ u &amp; = g &amp; \textrm{ on } \partial \Omega, \end{array}\right. for some degeneracy parameter \gamma\geq 0 , uniformly elliptic operator F , bounded source term f , and suitably smooth obstacle \phi and boundary datum g . We obtain existence/uniqueness of solutions and prove sharp regularity estimates at the free boundary points, namely \partial\{u&gt;\phi\} \cap \Omega . In particular, for the homogeneous case ( f\equiv0 ) we get that solutions are C^{1,1} at free boundary points, in the sense that they detach from the obstacle in a quadratic fashion, thus beating the optimal regularity allowed for such degenerate operators. We also prove several non-degeneracy properties of solutions and partial results regarding the free boundary. These are the first results for obstacle problems driven by degenerate type operators in non-divergence form and they are a novelty even for the simpler prototype given by an operator of the form \mathcal{G}[u] = |Du|^\gamma\Delta u , with \gamma &gt;0 and f \equiv 1 .

Geometric regularity estimates for fully nonlinear elliptic equations with free boundaries

AbstractIn this manuscript we study geometric regularity estimates for problems driven by fully nonlinear elliptic operators (which can be either degenerate or singular when “the gradient is small”) under strong absorption conditions of the general form: where the mapping fails to decrease fast enough at the origin, so allowing that nonnegative solutions may create plateau regions, that is, a priori unknown subsets where a given solution vanishes identically. We establish improved geometric regularity along the set (the free boundary of the model), for a sharp value of (obtained explicitly) depending only on structural parameters. Non‐degeneracy among others measure theoretical properties are also obtained. A sharp Liouville result for entire solutions with controlled growth at infinity is proved. We also present a number of applications consequential of our findings.

Analysis of a Composition Operator’s Eigenvalue Equation on Unitary Spaces by the Krein-Rutman Theorem

The Krein-Rutman theorem is vital in partial differential equations that are non-linear and provides evidence of the presence of several significant eigenvalues useful in topological degree calculations, stability analysis, and bifurcation theory. Schr?der’s equation which has been used extensively in studies of turbulence is an equation with a single independent variable suitable for encoding self-similarity. The concept of Hilbert spaces has been an inner product space frequently used due to its convenience in countless dimensional vector analysis. This paper is aimed at proving a number of solutions through the Krein-Rutman theorem in unitary spaces especially in Hilbert spaces. It has been certainly observed that the whole Krein-Rutman theorem system has a fairly stable scope, and has strong regular features, and many non-linear elliptic operators need the most ethical principles to satisfy the comparison policy.