Global gradient estimates for divergence-type elliptic problems involving general nonlinear operators

We study nonlinear elliptic problems with nonstandard growth and ellipticity related to an N-function. We establish global Calderón–Zygmund estimates of the weak solutions in the framework of Orlicz spaces over bounded non-smooth domains. Moreover, we prove a global regularity result for asymptotically regular problems which are getting close to the regular problems considered, when the gradient variable goes to infinity.     

Some remarks on elliptic problems with critical growth in the gradient

In this work we analyze existence, nonexistence, multiplicity and regularity of solution to problem

(1)     

Global weighted estimates for the gradient of solutions to nonlinear elliptic equations

We consider nonlinear elliptic equations of p  -Laplacian type that are not necessarily of variation form when the nonlinearity is allowed to be discontinuous and the boundary of the domain can go beyond the Lipschitz category. Under smallness in the BMO nonlinearity and sufficient flatness of the Reifenberg domain, we obtain the global weighted L^q$L^q$ estimates with q∈(p,∞)$q\in (p,\infty )$ for the gradient of weak solutions.     

Elliptic problems with a Hardy potential and critical growth in the gradient: Non-resonance and blow-up results

In this article we analyze existence and nonexistence of positive solutions to problem

     

Smoothness spaces of higher order on lower dimensional subsets of the Euclidean space

We study Sobolev type spaces defined in terms of sharp maximal functions on Ahlfors regular subsets of and the relation between these spaces and traces of classical Sobolev spaces.     

Optimal results for parabolic problems arising in some physical models with critical growth in the gradient respect to a Hardy potential

We deal with the following parabolic problem

     

Widths of embeddings in function spaces

We study the approximation, Gelfand and Kolmogorov numbers of embeddings in function spaces of Besov and Triebel-Lizorkin type. Our aim here is to provide sharp estimates in several cases left open in the literature and give a complete overview of the known results. We also add some historical remarks.     

Criteria for embedding of spaces constructed by the method of means with arbitrary quasi-concave functional parameters

We consider a generalization ?(X₀,X₁)_p0,p1 of the method of means to arbitrary non-degenerate functional parameter. In this case non-trivial embedding ?(X₀,X₁)_p0,p1⊂ψ(X₀,X₁)_q0,q1 take place. We find necessary and sufficient condition for such embedding if 1?q₀?p₀?∞ and 1?q₁?p₁?∞ or 1?p₀?q₀?∞ and 1?p₁?q₁?∞.     

On the centralizer ofK in the universal enveloping algebra of SO(n,1) and SU(n,1)

LetG _o be a non compact real semisimple Lie group with finite center, and letU U(g) ^K denote the centralizer inU U(g) of a maximal compact subgroupK _o ofG _o. To study the algebraU U(g) ^K , B. Kostant suggested to consider the projection mapP:U U(g)→U(k)⊗U(a), associated to an Iwasawa decompositionG _o=K _o A _o N _o ofG _o, adapted toK _o. WhenP is restricted toU U(g) ^K J. Lepowsky showed thatP becomes an injective anti-homomorphism ofU U(g) ^K intoU(k) ^M ⊗U(a). HereU(k) ^M denotes the centralizer ofM _o inU(k),M _o being the centralizer ofA _o inK _o. To pursue this idea further it is necessary to have a good characterization of the image ofU U(g) ^K inU(k)^M×U(a). In this paper we describe such image whenG _o=SO(n,1)_e or SU(n,1). This is acomplished by establishing a (minimal) set of equations satisfied by the elements in the image ofU U(g) ^K , and then proving that they are enough to characterize such image. These equations are derived on one hand from the intertwining relations among the principal series representations ofG _o given by the Kunze-Stein interwining operators, and on the other hand from certain imbeddings among Verma modules. This approach should prove to be useful to attack the general case. Supported in part by Fundación Antorchas     

Dimension of images of subspaces under mappings in Triebel‐Lizorkin spaces

Let and let be a ‐quasicontinuous representative of a mapping in the Triebel‐Lizorkin space . We find an optimal value of such that for a.e. the Hausdorff dimension of is at most α. We construct examples to show that the value of β is optimal and we show that it does not increase once p goes below the critical value α.     

Overview of differential equations with non-standard growth

Differential equations with non-standard growth have been a very active field of investigation in recent years. In this survey we present an overview of the field, as well as several of the most important results. We consider both existence and regularity questions. Finally, we provide a comprehensive list of papers published to date.     

Compact embedding results of Sobolev spaces and existence of positive solutions to quasilinear equations

In this paper, we study mainly the existence of multiple positive solutions for a quasilinear elliptic equation of the following form on ${\mathbf{R}}^N$, when $N\ge 2$,

(0.1)$?{\mathrm{\Delta }}_{N}u+V(x){\left|u\right|}^{N?2}u=\lambda {\left|u\right|}^{r?2}u+f(x,u).$

Here, $V(x)>0:{\mathbf{R}}^N\to \mathbf{R}$ is a suitable potential function, $r\in (1,N)$, $f(x,u)$ is a continuous function of N-superlinear and subcritical exponential growth without having the Ambrosetti–Rabinowitz condition, while $\lambda >0$ is a constant. A suitable Moser–Trudinger inequality and the compact embedding $W_{\mathrm{V}}^{1,N}\left({\mathbf{R}}^N\right)?L^r\left({\mathbf{R}}^N\right)$ are proved to study problem (0.1). Moreover, the compact embedding $H_{\mathrm{V}}^1\left({\mathbf{R}}^N\right)?L_{\mathrm{K}}^t\left({\mathbf{R}}^N\right)$ is also analyzed to investigate the existence of a positive ground state to the following nonlinear Schrödinger equation

(0.2)$?\mathrm{\Delta }u+V(x)u=K(x)g(u)$

with potentials vanishing at infinity in a measure-theoretic sense when $N\ge 3$.     

Sharp gradient estimates for quasilinear elliptic equations with p(x) growth on nonsmooth domains

In this paper, we study quasilinear elliptic equations with the nonlinearity modelled after the $p(x)$-Laplacian on nonsmooth domains and obtain sharp Calderón–Zygmund type estimates in the variable exponent setting. In a recent work of [12], the estimates obtained were strictly above the natural exponent and hence there was a gap between the natural energy estimates and estimates above $p(x)$, see (1.3) and (1.4). Here, we bridge this gap to obtain the end point case of the estimates obtained in [12], see (1.5). In order to do this, we have to obtain significantly improved a priori estimates below $p(x)$, which is the main contribution of this paper. We also improve upon the previous results by obtaining the estimates for a larger class of domains than what was considered in the literature.     

Lower bound of measure and embeddings of Sobolev,Besov and Triebel–Lizorkin spaces

In this article, we study the relation between Sobolev-type embeddings for Sobolev spaces or Hajłasz–Besov spaces or Hajłasz–Triebel–Lizorkin spaces defined on a doubling and geodesic metric measure space and lower bound for measure of balls either in the whole space or in a domain inside the space.     

The property (DN) and the exponential representation of holomorphic functions

It is shown that a nuclear Fréchet spaceE has the property (DN) if and only if every holomorphic function onE ^*, the strongly dual space ofE, with values in the strongly dual space of a Fréchet spaceF having the property ( ) can be represented in the exponential form. Moreover, it is shown that the space of holomorphic functions onC , the space of all complex number sequences, has a linearly absolutely exponential representation system. But the space of holomorphic functions onE ^* does not have such a system whenE is a nuclear Fréchet space that does not have the property (DN).Supported by the State Program for Fundamental Researches in Natural Sciences     

On approximate differentiability of functions with bounded deformation

We prove that functions with bounded deformationu: Ω → ℝⁿ, Ω ⊂ ℝⁿ, i.e., such mappings that the symmetric part of the gradient is a measure, are approximately differentiable a.e. Then we generalize the result to a more general class of functions. This research was carried out while the author stayed in the ICTP in Trieste in 1995. He wishes to thank the ICTP for the hospitality. The author was partially supported by KBN grant no. 2-PO3A-034-08.     

Topological-algebraic properties of function spaces with set-open topologies

For a Tychonoff space X, we denote by Cλ(X) the space of all real-valued continuous functions on X with set-open topology. In this paper, we study the topological-algebraic properties of Cλ(X). Our main results state that (1) Cλ(X) is a topological vector space (a topological group) iff λ is a family of C-compact sets and Cλ(X)=Cλ^′(X), where λ^′ consists of all C-compact subsets of every set of λ. In particular, if Cλ(X) is a topological group, then the set-open topology coincides with the topology of uniform convergence on a family λ; (2) a topological group Cλ(X) is ω-narrow iff λ is a family of metrizable compact subsets of X.     

Some function spaces on spaces of homogeneous type

We introduce Campanato, Morrey, BMO and Sobolev-type spaces for mappings from a space of homogeneous type into a complete metric space which possess properties comparable to their classical analogues. In particular we show integral characterizations, the validity of the John–Nirenberg theorem, Poincarè and Sobolev inequalities, Sobolev's embedding theorem and estimates on the pointwise behavior of Sobolev-type mappings. Received: 4 December 2000 / Revised version: 5 July 2001