A Gluing Theorem for the Elliptic Adjoint Operator $L^{\star}$ in the Plane

Orazio Arena

Displaying similar documents to “A Gluing Theorem for the Elliptic Adjoint Operator $L^{\star}$ in the Plane”

Existence and nonexistence of solutions for a quasilinear elliptic system

Qin Li, Zuodong Yang (2015)

Annales Polonici Mathematici

Similarity:

By a sub-super solution argument, we study the existence of positive solutions for the system ⎧ $- Δ_{p} u = a ₁ (x) F ₁ (x, u, v)$ in Ω, ⎪ $- Δ_{q} v = a ₂ (x) F ₂ (x, u, v)$ in Ω, ⎨u,v > 0 in Ω, ⎩u = v = 0 on ∂Ω, where Ω is a bounded domain in $ℝ^{N}$ with smooth boundary or $Ω = ℝ^{N}$ . A nonexistence result is obtained for radially symmetric solutions.

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

Similarity:

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 $\prod_{i = 1}^{N} L^{p^{'} (\cdot)} (Ω)$ .

Partially elliptic differential equations having distributions as their right members

H. Marcinkowska

Similarity:

ContentsIntroduction.............................................................................................................................31. Definitions, notations and some auxiliary lemmas...................................................42. The definition of the spaces $H_{p, q; Y} (Ω, ℬ)$ ..........................................................73. Some properties of the spaces $H_{p, q; Y} (Ω, ℬ)$ ...................................................104. Some examples of the spaces $H_{p, q; Y} (Ω, ℬ)$ ....................................................155....

Multiplicity results for a class of concave-convex elliptic systems involving sign-changing weight functions

Honghui Yin, Zuodong Yang (2011)

Annales Polonici Mathematici

Similarity:

Our main purpose is to establish the existence of weak solutions of second order quasilinear elliptic systems ⎧ $- Δ_{p} u + {| u |}^{p - 2} u = f_{1 λ ₁} {(x) | u |}^{q - 2} u + 2 α / (α + β) g_{μ} {| u |}^{α - 2} u {| v |}^{β}$ , x ∈ Ω, ⎨ $- Δ_{p} v + {| v |}^{p - 2} v = f_{2 λ ₂} {(x) | v |}^{q - 2} v + 2 β / (α + β) g_{μ} {| u |}^{α} {| v |}^{β - 2} v$ , x ∈ Ω, ⎩ u = v = 0, x∈ ∂Ω, where 1 < q < p < N and $Ω \subset ℝ^{N}$ is an open bounded smooth domain. Here λ₁, λ₂, μ ≥ 0 and $f_{i λ_{i}} (x) = λ_{i} f_{i +} (x) + f_{i -} (x)$ (i = 1,2) are sign-changing functions, where $f_{i \pm} (x) = m a x \pm f_{i} (x), 0$ , $g_{μ} (x) = a (x) + μ b (x)$ , and $Δ_{p} u = {d i v (| \nabla u |}^{p - 2} \nabla u)$ denotes the p-Laplace operator. We use variational methods.

On annealed elliptic Green's function estimates

Daniel Marahrens, Felix Otto (2015)

Mathematica Bohemica

Similarity:

We consider a random, uniformly elliptic coefficient field $a$ on the lattice $ℤ^{d}$ . The distribution $〈 \cdot 〉$ of the coefficient field is assumed to be stationary. Delmotte and Deuschel showed that the gradient and second mixed derivative of the parabolic Green’s function $G (t, x, y)$ satisfy optimal annealed estimates which are $L^{2}$ and $L^{1}$ , respectively, in probability, i.e., they obtained bounds on $〈 | \nabla_{x} G (t, x, y) {|^{2} 〉}^{1 / 2}$ and $〈 | \nabla_{x} \nabla_{y} G (t, x, y) | 〉$ . In particular, the elliptic Green’s function $G (x, y)$ satisfies optimal annealed bounds. In their recent work,...

Fonctions biharmoniques adjointes

Emmanuel P. Smyrnelis (2010)

Annales Polonici Mathematici

Similarity:

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.

A Littlewood-Paley type inequality with applications to the elliptic Dirichlet problem

Caroline Sweezy (2007)

Annales Polonici Mathematici

Similarity:

Let L be a strictly elliptic second order operator on a bounded domain Ω ⊂ ℝⁿ. Let u be a solution to $L u = d i v \vec{f}$ in Ω, u = 0 on ∂Ω. Sufficient conditions on two measures, μ and ν defined on Ω, are established which imply that the $L^{q} (Ω, d μ)$ norm of |∇u| is dominated by the $L^{p} (Ω, d v)$ norms of $d i v \vec{f}$ and $| \vec{f} |$ . If we replace |∇u| by a local Hölder norm of u, the conditions on μ and ν can be significantly weaker.

Solutions to a class of singular quasilinear elliptic equations

Lin Wei, Zuodong Yang (2010)

Annales Polonici Mathematici

Similarity:

We study the existence of positive solutions to ⎧ ${d i v (| \nabla u |}^{p - 2} \nabla u) + q (x) u^{- γ} = 0$ on Ω, ⎨ ⎩ u = 0 on ∂Ω, where Ω is $ℝ^{N}$ or an unbounded domain, q(x) is locally Hölder continuous on Ω and p > 1, γ > -(p-1).

A bound for the solutions of a basic elliptic system with non-linearity $q\geq 2$

Sergio Campanato (1986)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Similarity:

In questa Nota si dimostra un risultato enunciato nel § 5 della pubblicazione [4]. Per le soluzioni di un sistema ellittico base, con non-linearità $q\geq 2$ , vale un principio di massimo analogo a quello dimostrato in [3] nel caso di non-linearità $q=2$ .

On some $L^{p}$ -estimates for solutions of elliptic equations in unbounded domains

Sara Monsurrò, Maria Transirico (2015)

Mathematica Bohemica

Similarity:

In this review article we present an overview on some a priori estimates in $L^{p}$ , $p > 1$ , recently obtained in the framework of the study of a certain kind of Dirichlet problem in unbounded domains. More precisely, we consider a linear uniformly elliptic second order differential operator in divergence form with bounded leading coeffcients and with lower order terms coefficients belonging to certain Morrey type spaces. Under suitable assumptions on the data, we first show two $L^{p}$ -bounds, $p > 2$ , for...

Perturbed nonlinear degenerate problems in $ℝ^{N}$

A. El Khalil, S. El Manouni, M. Ouanan (2009)

Applicationes Mathematicae

Similarity:

Via critical point theory we establish the existence and regularity of solutions for the quasilinear elliptic problem ⎧ $d i v (x, \nabla u) + {a (x) | u |}^{p - 2} u = {g (x) | u |}^{p - 2} u + h (x) {| u |}^{s - 1} u$ in $ℝ^{N}$ ⎨ ⎩ u > 0, $l i m_{| x | \to \infty} u (x) = 0$ , where 1 < p < N; a(x) is assumed to satisfy a coercivity condition; h(x) and g(x) are not necessarily bounded but satisfy some integrability restrictions.

Convex integration and the $L^{p}$ theory of elliptic equations

Kari Astala, Daniel Faraco, László Székelyhidi Jr. (2008)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Similarity:

This paper deals with the $L^{p}$ theory of linear elliptic partial differential equations with bounded measurable coefficients. We construct in two dimensions examples of weak and so-called very weak solutions, with critical integrability properties, both to isotropic equations and to equations in non-divergence form. These examples show that the general $L^{p}$ theory, developed in [1, 24] and [2], cannot be extended under any restriction on the essential range of the coefficients. Our constructions...

Separable solutions of quasilinear Lane–Emden equations

Alessio Porretta, Laurent Véron (2013)

Journal of the European Mathematical Society

Similarity:

For $0 < p - 1 < q$ and either $ϵ = 1$ or $ϵ = - 1$ , we prove the existence of solutions of $- Δ_{p} u = ϵ u^{q}$ in a cone $C_{S}$ , with vertex 0 and opening $S$ , vanishing on $\partial C_{S}$ , of the form $u (x) = {|x|}^{- β} ω (x / |x|)$ . The problem reduces to a quasilinear elliptic equation on $S$ and the existence proof is based upon degree theory and homotopy methods. We also obtain a nonexistence result in some critical case by making use of an integral type identity.

Existence of a renormalized solution of nonlinear degenerate elliptic problems

Youssef Akdim, Chakir Allalou (2014)

Applicationes Mathematicae

Similarity:

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.

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

Yongxiang Li, Huanhuan Zhang (2016)

Annales Polonici Mathematici

Similarity:

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...

On invariants of elliptic curves on average

Amir Akbary, Adam Tyler Felix (2015)

Acta Arithmetica

Similarity:

We prove several results regarding some invariants of elliptic curves on average over the family of all elliptic curves inside a box of sides A and B. As an example, let E be an elliptic curve defined over ℚ and p be a prime of good reduction for E. Let $e_{E} (p)$ be the exponent of the group of rational points of the reduction modulo p of E over the finite field $_{p}$ . Let be the family of elliptic curves $E_{a, b} : y^{2} = x^{3} + a x + b$ , where |a| ≤ A and |b| ≤ B. We prove that, for any c > 1 and k∈ ℕ, $1 / | | \sum_{E \in} \sum_{p \leq x} e_{E}^{k} (p) = C_{k} l i (x^{k + 1}) + O ((x^{k + 1}) / {(l o g x)}^{c}$ )as x → ∞, as long...

On Relative $\gamma_{k}$ -Sets

Maddalena Bonanzinga, Filippo Cammaroto, Bruno A. Pansera (2007)

Bollettino dell'Unione Matematica Italiana

Similarity:

In this note we show a relative version of $\gamma_{k}$ -set introduced and studied in [12]. We give several characterizations of this property; in particular one of the characterizations is Ramsey theoretical. Also we give a result involving a property of the corresponding mapping between function spaces.

A bound for the solutions of a basic elliptic system with non-linearity $q\geq 2$

Sergio Campanato (1986)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

Similarity:

Nonlinear Elliptic Equations with Lower Order Terms and Symmetrization Methods

Angelo Alvino, Anna Mercaldo (2008)

Bollettino dell'Unione Matematica Italiana

Similarity:

We consider the homogeneous Dirichlet problem for nonlinear elliptic equations as $-\operatorname{div}a(x,\nabla u)=b(x,\nabla u)+\mu$ where $\mu$ is a measure with bounded total variation. We fix structural conditions on functions $a$ , $b$ which ensure existence of solutions. Moreover, if $\mu$ is an $L^{1}$ function, we prove a uniqueness result under more restrictive hypotheses on the operator.

On the potential theory of some systems of coupled PDEs

Abderrahim Aslimani, Imad El Ghazi, Mohamed El Kadiri, Sabah Haddad (2016)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

In this paper we study some potential theoretical properties of solutions and super-solutions of some PDE systems (S) of type $L_{1} u = - μ_{1} v$ , $L_{2} v = - μ_{2} u$ , on a domain $D$ of $ℝ^{d}$ , where $μ_{1}$ and $μ_{2}$ are suitable measures on $D$ , and $L_{1}$ , $L_{2}$ are two second order linear differential elliptic operators on $D$ with coefficients of class $𝒞^{\infty}$ . We also obtain the integral representation of the nonnegative solutions and supersolutions of the system (S) by means of the Green kernels and Martin boundaries associated with $L_{1}$ and $L_{2}$ , and...

Isomorphisms of Royden Type Algebras Over $\mathbb{S}^{1}$

Teresa Radice, Eero Saksman, Gabriella Zecca (2009)

Bollettino dell'Unione Matematica Italiana

Similarity:

Let $\mathbb{S}^{1}$ and $\mathbb{D}$ be the unit circle and the unit disc in the plane and let us denote by $\mathcal{A}(\mathbb{S}^{1})$ the algebra of the complex-valued continuous functions on $\mathbb{S}^{1}$ which are traces of functions in the Sobolev class $W^{1,2}(\mathbb{D})$ . On $\mathcal{A}(\mathbb{S}^{1})$ we define the following norm $\|u\|=\|u\|_{L^{\infty}(\mathbb{S}^{1})}+\left(\iint_{\mathbb{D}}|\nabla\tilde% {u}|^{2}\right)^{1/2}$ where is the harmonic extension of $u$ to $\mathbb{D}$ . We prove that every isomorphism of the functional algebra $\mathcal{A}(\mathbb{S}^{1})$ is a quasitsymmetric change of variables on $\mathbb{S}^{1}$ .

Displaying similar documents to “A Gluing Theorem for the Elliptic Adjoint Operator L ⋆ in the Plane”

Displaying similar documents to “A Gluing Theorem for the Elliptic Adjoint Operator $L^{\star}$ in the Plane”