Displaying similar documents to “ C 1 , α regularity for elliptic equations with the general nonstandard growth conditions”

Hölder continuity of bounded generalized solutions for some degenerated quasilinear elliptic equations with natural growth terms

Salvatore Bonafede (2018)

Commentationes Mathematicae Universitatis Carolinae

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We prove the local Hölder continuity of bounded generalized solutions of the Dirichlet problem associated to the equation i = 1 m x i a i ( x , u , u ) - c 0 | u | p - 2 u = f ( x , u , u ) , assuming that the principal part of the equation satisfies the following degenerate ellipticity condition λ ( | u | ) i = 1 m a i ( x , u , η ) η i ν ( x ) | η | p , and the lower-order term f has a natural growth with respect to u .

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...

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 , u ) ) + g ( x , u ) = f - d i v F in Ω, where Ω is a bounded open domain of N , N ≥ 2 and a : Ω × × N 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 i = 1 N L p ' ( · ) ( Ω ) .

A regularity theory for scalar local minimizers of splitting-type variational integrals

Michael Bildhauer, Martin Fuchs, Xiao Zhong (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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Starting from Giaquinta’s counterexample [12] we introduce the class of splitting functionals being of ( p , q ) -growth with exponents p q < and show for the scalar case that locally bounded local minimizers are of class C 1 , μ . Note that to our knowledge the only C 1 , μ -results without imposing a relation between p and q concern the case of two independent variables as it is outlined in Marcellini’s paper [15], Theorem A, and later on in the work of Fusco and Sbordone [10], Theorem 4.2.

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 ( 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 ( 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 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 ± ( x ) = m a x ± f i ( x ) , 0 , g μ ( x ) = a ( x ) + μ b ( x ) , and Δ p u = d i v ( | u | p - 2 u ) denotes the p-Laplace operator. We use variational methods.

Perturbed nonlinear degenerate problems in 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 , u ) + a ( x ) | u | p - 2 u = g ( x ) | u | p - 2 u + h ( x ) | u | s - 1 u in N ⎨ ⎩ u > 0, l i m | x | 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.

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 ( x ) F ( x , u , v ) in Ω, ⎪ - Δ q v = a ( x ) F ( x , u , v ) in Ω, ⎨u,v > 0 in Ω, ⎩u = v = 0 on ∂Ω, where Ω is a bounded domain in N with smooth boundary or Ω = N . A nonexistence result is obtained for radially symmetric solutions.

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 , ) ..........................................................73. Some properties of the spaces H p , q ; Y ( Ω , ) ...................................................104. Some examples of the spaces H p , q ; Y ( Ω , ) ....................................................155....

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...

Existence of renormalized solutions for some degenerate and non-coercive elliptic equations

Youssef Akdim, Mohammed Belayachi, Hassane Hjiaj (2023)

Mathematica Bohemica

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This paper is devoted to the study of some nonlinear degenerated elliptic equations, whose prototype is given by t 2 - div ( b ( | u | ) | u | p - 2 u ) + d ( | u | ) | u | p = f - div ( c ( x ) | u | α ) in Ω , u = 0 on Ω , t where Ω is a bounded open set of N ( N 2 ) with 1 < p < N and f L 1 ( Ω ) , under some growth conditions on the function b ( · ) and d ( · ) , where c ( · ) is assumed to be in L N ( p - 1 ) ( Ω ) . We show the existence of renormalized solutions for this non-coercive elliptic equation, also, some regularity results will be concluded.

Pointwise regularity associated with function spaces and multifractal analysis

Stéphane Jaffard (2006)

Banach Center Publications

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The purpose of multifractal analysis of functions is to determine the Hausdorff dimensions of the sets of points where a function (or a distribution) f has a given pointwise regularity exponent H. This notion has many variants depending on the global hypotheses made on f; if f locally belongs to a Banach space E, then a family of pointwise regularity spaces C E α ( x ) are constructed, leading to a notion of pointwise regularity with respect to E; the case E = L corresponds to the usual Hölder regularity,...

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 d . The distribution · 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 | x G ( t , x , y ) | 2 1 / 2 and | x y G ( t , x , y ) | . In particular, the elliptic Green’s function G ( x , y ) satisfies optimal annealed bounds. In their recent work,...

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 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 f and | 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

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

Separable solutions of quasilinear Lane–Emden equations

Alessio Porretta, Laurent Véron (2013)

Journal of the European Mathematical Society

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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 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.

An improved regularity criteria for the MHD system based on two components of the solution

Zujin Zhang, Yali Zhang (2021)

Applications of Mathematics

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As observed by Yamazaki, the third component b 3 of the magnetic field can be estimated by the corresponding component u 3 of the velocity field in L λ ( 2 λ 6 ) norm. This leads him to establish regularity criterion involving u 3 , j 3 or u 3 , ω 3 . Noticing that λ can be greater than 6 in this paper, we can improve previous results.