Symmetry results for viscosity solutions of fully nonlinear uniformly elliptic equations

Francesca Da Lio; Boyan Sirakov

Displaying similar documents to “Symmetry results for viscosity solutions of fully nonlinear uniformly elliptic equations”

Uniqueness of solutions for some degenerate nonlinear elliptic equations

Albo Carlos Cavalheiro (2014)

Applicationes Mathematicae

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We investigate the existence and uniqueness of solutions to the Dirichlet problem for a degenerate nonlinear elliptic equation $- \sum_{i, j = 1}^{n} D_{j} (a_{i j} (x) D_{i} u (x)) + b (x) u (x) + d i v (Φ (u (x))) = g (x) - \sum_{j = 1}^{n} f_{j} (x)$ on Ω in the setting of the space H₀(Ω).

Existence of a renormalized solution of nonlinear degenerate elliptic problems

Youssef Akdim, Chakir Allalou (2014)

Applicationes Mathematicae

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We study a general class of nonlinear elliptic problems associated with the differential inclusion $β (u) - d i v (a (x, D u) + F (u)) ∋ f$ in Ω where $f \in L^{\infty} (Ω)$ . The vector field a(·,·) is a Carathéodory function. Using truncation techniques and the generalized monotonicity method in function spaces we prove existence of renormalized solutions for general $L^{\infty}$ -data.

The restriction theorem for fully nonlinear subequations

F. Reese Harvey, H. Blaine Lawson (2014)

Annales de l’institut Fourier

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Let $X$ be a submanifold of a manifold $Z$ . We address the question: When do viscosity subsolutions of a fully nonlinear PDE on $Z$ , restrict to be viscosity subsolutions of the restricted subequation on $X$ ? This is not always true, and conditions are required. We first prove a basic result which, in theory, can be applied to any subequation. Then two definitive results are obtained. The first applies to any “geometrically defined” subequation, and the second to any subequation which can be...

Some results on strongly nonlinear anisotropic differential equations

L. Bougoffa, A. El Khalil, S. El Manouni (2010)

Applicationes Mathematicae

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The paper concerns the existence of weak solutions to nonlinear elliptic equations of the form A(u) + g(x,u,∇u) = f, where A is an operator from an appropriate anisotropic function space to its dual and the right hand side term is in $L^{1 + m}$ with 0 < m < 1. We assume a sign condition on the nonlinear term g, but no growth restrictions on u.

T-p(x)-solutions for nonlinear elliptic equations with an L¹-dual datum

El Houssine Azroul, Abdelkrim Barbara, Meryem El Lekhlifi, Mohamed Rhoudaf (2012)

Applicationes Mathematicae

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We establish the existence of a T-p(x)-solution for the p(x)-elliptic problem $- d i v (a (x, u, \nabla u)) + g (x, u) = f - d i v F$ in Ω, where Ω is a bounded open domain of $ℝ^{N}$ , N ≥ 2 and $a : Ω \times ℝ \times ℝ^{N} \to ℝ^{N}$ is a Carathéodory function satisfying the natural growth condition and the coercivity condition, but with only a weak monotonicity condition. The right hand side f lies in L¹(Ω) and F belongs to $\prod_{i = 1}^{N} L^{p^{'} (\cdot)} (Ω)$ .

A nonlinear elliptic equation with singular potential and applications to nonlinear field equations

Marino Badiale, Vieri Benci, Sergio Rolando (2007)

Journal of the European Mathematical Society

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We prove the existence of cylindrical solutions to the semilinear elliptic problem $- Δ u + \frac{u}{{| y |}^{2}} = f (u)$ , $u \in H^{1} (ℝ^{N})$ , $u \geq 0$ , where $(y, z) \in ℝ^{k} \times ℝ^{N - k}$ , $N > k \geq 2$ and $f$ has a double-power behaviour, subcritical at infinity and supercritical near the origin. This result also implies the existence of solitary waves with nonvanishing angular momentum for nonlinear Schr¨odinger and Klein–Gordon equations.

On a semilinear elliptic eigenvalue problem

Mario Michele Coclite (1997)

Annales Polonici Mathematici

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We obtain a description of the spectrum and estimates for generalized positive solutions of -Δu = λ(f(x) + h(u)) in Ω, ${u |}_{\partial Ω} = 0$ , where f(x) and h(u) satisfy minimal regularity assumptions.

A nonlocal elliptic equation in a bounded domain

Piotr Fijałkowski, Bogdan Przeradzki, Robert Stańczy (2004)

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The existence of a positive solution to the Dirichlet boundary value problem for the second order elliptic equation in divergence form $- \sum_{i, j = 1}^{n} D_{i} (a_{i j} D_{j} u) = f (u, \int_{Ω} g (u^{p}))$ , in a bounded domain Ω in ℝⁿ with some growth assumptions on the nonlinear terms f and g is proved. The method based on the Krasnosel’skiĭ Fixed Point Theorem enables us to find many solutions as well.

Optimal $L^{p}$ -properties of Green’s functions for non-divergence elliptic equations in two dimensions

Gioconda Moscariello, Carlo Sbordone (2005)

Studia Mathematica

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A sharp integrability result for non-negative adjoint solutions to planar non-divergence elliptic equations is proved. A uniform estimate is also given for the Green's function.

Space-time adaptive $h p$ -FEM: Methodology overview

Šolín, Pavel, Segeth, Karel, Doležel, Ivo

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We present a new class of self-adaptive higher-order finite element methods ( $h p$ -FEM) which are free of analytical error estimates and thus work equally well for virtually all PDE problems ranging from simple linear elliptic equations to complex time-dependent nonlinear multiphysics coupled problems. The methods do not contain any tuning parameters and work reliably with both low- and high-order finite elements. The methodology was used to solve various types of problems including thermoelasticity,...

Solvability for semilinear PDE with multiple characteristics

Alessandro Oliaro, Luigi Rodino (2003)

Banach Center Publications

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We prove local solvability in Gevrey spaces for a class of semilinear partial differential equations. The linear part admits characteristics of multiplicity k ≥ 2 and data are fixed in $G^{σ}$ , 1 < σ < k/(k-1). The nonlinearity, containing derivatives of lower order, is assumed of class $G^{σ}$ with respect to all variables.

Existence of positive radial solutions for the elliptic equations on an exterior domain

Yongxiang Li, Huanhuan Zhang (2016)

Annales Polonici Mathematici

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We discuss the existence of positive radial solutions of the semilinear elliptic equation ⎧-Δu = K(|x|)f(u), x ∈ Ω ⎨αu + β ∂u/∂n = 0, x ∈ ∂Ω, ⎩ $l i m_{| x | \to \infty} u (x) = 0$ , where $Ω = x \in ℝ^{N} : | x | > r ₀$ , N ≥ 3, K: [r₀,∞) → ℝ⁺ is continuous and $0 < \int_{r ₀}^{\infty} r K (r) d r < \infty$ , f ∈ C(ℝ⁺,ℝ⁺), f(0) = 0. Under the conditions related to the asymptotic behaviour of f(u)/u at 0 and infinity, the existence of positive radial solutions is obtained. Our conditions are more precise and weaker than the superlinear or sublinear growth conditions. Our discussion is based on the...

Entire solutions to a class of fully nonlinear elliptic equations

Ovidiu Savin (2008)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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We study nonlinear elliptic equations of the form $F (D^{2} u) = f (u)$ where the main assumption on $F$ and $f$ is that there exists a one dimensional solution which solves the equation in all the directions $ξ \in ℝ^{n}$ . We show that entire monotone solutions $u$ are one dimensional if their $0$ level set is assumed to be Lipschitz, flat or bounded from one side by a hyperplane.

Homogenization in polygonal domains

David Gérard-Varet, Nader Masmoudi (2011)

Journal of the European Mathematical Society

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We consider the homogenization of elliptic systems with $ε$ -periodic coefficients. Classical two-scale approximation yields an $O (ε)$ error inside the domain. We discuss here the existence of higher order corrections, in the case of general polygonal domains. The corrector depends in a non-trivial way on the boundary. Our analysis substantially extends previous results obtained for polygonal domains with sides of rational slopes.

Some Remarks on Nonlinear Composition Operators in Spaces of Differentiable Functions

J. Appell, Z. Jesús, O. Mejía (2011)

Bollettino dell'Unione Matematica Italiana

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In this note we study the nonlinear composition operator $f\mapsto g\circ f$ in various spaces of differentiable functions over an interval. It turns out that this operator is always bounded in the corresponding norm, whenever it maps such a space into itself, but continuous only in exceptional cases.

Fonctions biharmoniques adjointes

Emmanuel P. Smyrnelis (2010)

Annales Polonici Mathematici

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The study of the equation (L₂L₁)*h = 0 or of the equivalent system L*₂h₂ = -h₁, L*₁h₁ = 0, where $L_{j} (j = 1, 2)$ is a second order elliptic differential operator, leads us to the following general framework: Starting from a biharmonic space, for example the space of solutions (u₁,u₂) of the system L₁u₁ = -u₂, L₂u₂ = 0, $L_{j} (j = 1, 2)$ being elliptic or parabolic, and by means of its Green pairs, we construct the associated adjoint biharmonic space which is in duality with the initial one.

Existence of positive solutions for a nonlinear fourth order boundary value problem

Ruyun Ma (2003)

Annales Polonici Mathematici

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We study the existence of positive solutions of the nonlinear fourth order problem $u^{(4)} (x) = λ a (x) f (u (x))$ , u(0) = u’(0) = u”(1) = u”’(1) = 0, where a: [0,1] → ℝ may change sign, f(0) < 0, and λ < 0 is sufficiently small. Our approach is based on the Leray-Schauder fixed point theorem.

On the average value of the canonical height in higher dimensional families of elliptic curves

Wei Pin Wong (2014)

Acta Arithmetica

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Given an elliptic curve E over a function field K = ℚ(T₁,...,Tₙ), we study the behavior of the canonical height $h ̂_{E_{ω}}$ of the specialized elliptic curve $E_{ω}$ with respect to the height of ω ∈ ℚⁿ. We prove that there exists a uniform nonzero lower bound for the average of the quotient $(h ̂_{E_{ω}} (P_{ω})) / h (ω)$ over all nontorsion P ∈ E(K).

The gradient superconvergence of the finite volume method for a nonlinear elliptic problem of nonmonotone type

Tie Zhu Zhang, Shu Hua Zhang (2015)

Applications of Mathematics

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We study the superconvergence of the finite volume method for a nonlinear elliptic problem using linear trial functions. Under the condition of $C$ -uniform meshes, we first establish a superclose weak estimate for the bilinear form of the finite volume method. Then, we prove that on the mesh point set $S$ , the gradient approximation possesses the superconvergence: ${max}_{P \in S} | (\nabla u - \bar{\nabla} u_{h}) (P) | = O (h^{2}) {| ln h |}^{3 / 2}$ , where $\bar{\nabla}$ denotes the average gradient on elements containing vertex $P$ . Furthermore, by using the interpolation post-processing...