New estimates for elliptic equations and Hodge type systems

Jean Bourgain; Haïm Brezis

Displaying similar documents to “New estimates for elliptic equations and Hodge type systems”

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.

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

Caroline Sweezy (2007)

Annales Polonici Mathematici

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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.

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

Solutions to a class of singular quasilinear elliptic equations

Lin Wei, Zuodong Yang (2010)

Annales Polonici Mathematici

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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).

Limiting Sobolev inequalities for vector fields and canceling linear differential operators

Jean Van Schaftingen (2013)

Journal of the European Mathematical Society

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The estimate ${∥D^{k - 1} u∥}_{L^{n / (n - 1)}} \leq {∥A (D) u∥}_{L^{1}}$ is shown to hold if and only if $A (D)$ is elliptic and canceling. Here $A (D)$ is a homogeneous linear differential operator $A (D)$ of order $k$ on $ℝ^{n}$ from a vector space $V$ to a vector space $E$ . The operator $A (D)$ is defined to be canceling if $⋂_{ξ \in ℝ^{n} ∖ {0}} A (ξ) [V] = {0}$ . This result implies in particular the classical Gagliardo–Nirenberg–Sobolev inequality, the Korn–Sobolev inequality and Hodge–Sobolev estimates for differential forms due to J. Bourgain and H. Brezis. In the proof, the class of cocanceling homogeneous...

On the existence of exactly one solution of integral equations in the space $L^{P}$ with a mixed norm

Bogdan Rzepecki (1975)

Annales Polonici Mathematici

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On annealed elliptic Green's function estimates

Daniel Marahrens, Felix Otto (2015)

Mathematica Bohemica

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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,...

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

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

Applicationes Mathematicae

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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.

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

Sara Monsurrò, Maria Transirico (2015)

Mathematica Bohemica

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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...

Partially elliptic differential equations having distributions as their right members

H. Marcinkowska

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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

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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.

Existence and nonexistence of solutions for a quasilinear elliptic system

Qin Li, Zuodong Yang (2015)

Annales Polonici Mathematici

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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.

Hodge type decomposition

Wojciech Kozłowski (2007)

Annales Polonici Mathematici

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In the space $Λ^{p}$ of polynomial p-forms in ℝⁿ we introduce some special inner product. Let $H^{p}$ be the space of polynomial p-forms which are both closed and co-closed. We prove in a purely algebraic way that $Λ^{p}$ splits as the direct sum $d * (Λ^{p + 1}) \oplus δ * (Λ^{p - 1}) \oplus H^{p}$ , where d* (resp. δ*) denotes the adjoint operator to d (resp. δ) with respect to that inner product.

Nonlinear Elliptic Equations with Lower Order Terms and Symmetrization Methods

Angelo Alvino, Anna Mercaldo (2008)

Bollettino dell'Unione Matematica Italiana

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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.

Remarks on the Bourgain-Brezis-Mironescu Approach to Sobolev Spaces

B. Bojarski (2011)

Bulletin of the Polish Academy of Sciences. Mathematics

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For a function $f \in L_{l o c}^{p} (ℝ ⁿ)$ the notion of p-mean variation of order 1, $₁^{p} (f, ℝ ⁿ)$ is defined. It generalizes the concept of F. Riesz variation of functions on the real line ℝ¹ to ℝⁿ, n > 1. The characterisation of the Sobolev space $W^{1, p} (ℝ ⁿ)$ in terms of $₁^{p} (f, ℝ ⁿ)$ is directly related to the characterisation of $W^{1, p} (ℝ ⁿ)$ by Lipschitz type pointwise inequalities of Bojarski, Hajłasz and Strzelecki and to the Bourgain-Brezis-Mironescu approach.

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

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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...

An example of the equation $u_{t} = u_{x x} + f (x, t, u)$ with distinct maximum and minimum solutions of a mixed problem

W. Mlak (1963)

Annales Polonici Mathematici

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Functions with prescribed singularities

Giovanni Alberti, S. Baldo, G. Orlandi (2003)

Journal of the European Mathematical Society

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The distributional $k$ -dimensional Jacobian of a map $u$ in the Sobolev space $W^{1, k - 1}$ which takes values in the sphere $S^{k - 1}$ can be viewed as the boundary of a rectifiable current of codimension $k$ carried by (part of) the singularity of $u$ which is topologically relevant. The main purpose of this paper is to investigate the range of the Jacobian operator; in particular, we show that any boundary $M$ of codimension $k$ can be realized as Jacobian of a Sobolev map valued in $S^{k - 1}$ . In case $M$ is polyhedral, the...

Energy and Morse index of solutions of Yamabe type problems on thin annuli

Mohammed Ben Ayed, Khalil El Mehdi, Mohameden Ould Ahmedou, Filomena Pacella (2005)

Journal of the European Mathematical Society

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We consider the Yamabe type family of problems $(P_{ε}) : - Δ u_{ε} = u_{ε}^{(n + 2) / (n - 2)}$ , $u_{ε} > 0$ in $A_{ε}$ , $u_{ε} = 0$ on $\partial A_{ε}$ , where $A_{ε}$ is an annulus-shaped domain of $ℝ^{n}$ , $n \geq 3$ , which becomes thinner as $ε \to 0$ . We show that for every solution $u_{ε}$ , the energy $\int_{A_{ε}} | \nabla u_{|}^{2}$ as well as the Morse index tend to infinity as $ε \to 0$ . This is proved through a fine blow up analysis of appropriate scalings of solutions whose limiting profiles are regular, as well as of singular solutions of some elliptic problem on $ℝ^{n}$ , a half-space or an infinite strip. Our argument also involves a Liouville...

Arbitrary number of positive solutions for an elliptic problem with critical nonlinearity

Olivier Rey, Juncheng Wei (2005)

Journal of the European Mathematical Society

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We show that the critical nonlinear elliptic Neumann problem $Δ u - μ u + u^{7 / 3} = 0$ in $Ω$ , $u > 0$ in $Ω$ , $\frac{\partial u}{\partial ν} = 0$ on $\partial Ω$ , where $Ω$ is a bounded and smooth domain in $ℝ^{5}$ , has arbitrarily many solutions, provided that $μ > 0$ is small enough. More precisely, for any positive integer $K$ , there exists $μ_{K} > 0$ such that for $0 < μ < μ_{K}$ , the above problem has a nontrivial solution which blows up at $K$ interior points in $Ω$ , as $μ \to 0$ . The location of the blow-up points is related to the domain geometry. The solutions are obtained as critical points of some finite-dimensional...

Fourth-order nonlinear elliptic equations with critical growth

David E. Edmunds, Donato Fortunato, Enrico Jannelli (1989)

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

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In this paper we consider a nonlinear elliptic equation with critical growth for the operator $\Delta^{2}$ in a bounded domain $\Omega\subset\mathbb{R}^{n}$ . We state some existence results when $n\geq 8$ . Moreover, we consider $5\leq n\leq 7$ , expecially when $\Omega$ is a ball in $\mathbb{R}^{n}$ .