Jan Malý - recent publications

Valentino Magnani, Jan Malý, Samuele Mongodi: A low rank property and nonexistence of higher-dimensional horizontal Sobolev sets. J. Geom. Anal. 25 (2015), no. 3, 1444–1458.
We establish a low rank property'' for Sobolev mappings that pointwise solve a first order nonlinear system of PDEs, whose smooth solutions have the so-called contact property''. As a consequence, Sobolev mappings from an open set of the plane, taking values in the first Heisenberg group $$\mathcal H^1$$ and that have almost everywhere maximal rank must have images with positive 3-dimensional Hausdorff measure with respect to the sub-Riemannian distance of $$\mathcal H^1$$. This provides a complete solution to a question raised in a paper by Z. M. Balogh, R. Hoefer-Isenegger and J. T. Tyson. Our approach differs from the previous ones. Its technical aspect consists in performing an exterior differentiation by blow-up'', where the standard distributional exterior differentiation is not possible. This method extends to higher dimensional Sobolev mappings, taking values in higher dimensional Heisenberg groups.

Pekka Koskela, Jan Malý, Thomas Zürcher: Luzin's condition (N) and modulus of continuity. Adv. Calc. Var. 8 (2015), no. 2, 155–171.
In this paper, we establish Luzin's condition (N) for mappings in certain Sobolev-Orlicz spaces with certain moduli of continuity. Further, given a mapping in these Sobolev-Orlicz spaces, we give bounds on the size of the exceptional set where Luzin's condition (N) may fail. If a mapping violates Luzin's condition (N), we show that there is a Cantor set of measure zero that is mapped to a set of positive measure.

Jan Malý, Luděk Zajíček: On Stepanov type differentiability theorems. Acta Math. Hungar. 145 (2015), no. 1, 174–190.
The main result shows that the Rademacher theorem proved by J. Lindenstrauss and D. Preiss in 2003 (which says that, for some pairs $$X$$, $$Y$$ of Banach spaces, each Lipschitz $$f: X \to Y$$ is $$\Gamma$$-a.e. Fréchet differentiable) generalizes to the corresponding Stepanov theorem (which says that, for such $$X$$ and $$Y$$, an arbitrary $$f: X \to Y$$ is Fréchet differentiable at $$\Gamma$$-almost all points at which $$f$$ is Lipschitz). We also present an abstract approach which shows an easy way how (in some cases) a theorem of Stepanov type (for vector functions) can be inferred from the corresponding theorem of Rademacher type. Finally we present Stepanov type differentiability theorems with the assumption of pointwise directional Lipschitzness.

Stanislav Hencl, Zhuomin Liu, Jan Malý: Distributional Jacobian equal to $$\mathcal H^1$$ measure. Ann. Inst. H. Poincaré Anal. Non Linéaire 31 (2014), no. 5, 947–955.
Let $$1\le p\lt 2$$. We construct a Hölder continuous $$W^{1,p}$$ mapping of a square into $$\mathbb R^2$$ such that the distributional Jacobian equals to one dimensional Hausdorff measure on a line segment.

Jan Malý: Non-absolutely convergent integrals with respect to distributions. Ann. Mat. Pura Appl. (4) 193 (2014), no. 5, 1457–1484.
We define an integral of a function with respect to a distribution. In case that the underlying distribution is just the Lebesgue measure, the definition leads to a new non-absolutely convergent integral which is wider than the Denjoy-Perron integral. We present a version of the Gauss-Green theorem where the new integral is used for both interior and boundary terms. As a by-product we characterize the predual Sobolev space $$W^{-1,1}$$.

Petr Honzík and Jan Malý: Non-absolutely convergent integrals and singular integrals. Collect. Math. 65 (2014), no. 3, 367–377.
We define the packing integral (a kind of non-absolutely convergent integral) with respect to distributions of arbitrary order. Then we show that singular integrals can be interpreted as packing integrals with respect to generating distributions. This allows us to consider singular integrals beyond $$L^1$$.

Stanislav Hencl, Luděk Kleprlík and Jan Malý: Composition operator and Sobolev-Lorentz spaces $$WL^{n,q}$$. Studia Math. 221 (2014), no. 3, 197–208.
Let $$\Omega,\Omega'\subset\mathbb R^n$$ be domains and let $$f\colon\Omega\to\Omega'$$ be a homeomorphism. We show that if the composition operator $$T_f\colon u\mapsto u\circ f$$ maps the Sobolev-Lorentz space $$WL^{n,q}(\Omega')$$ to $$WL^{n,q}(\Omega)$$ for some $$q\neq n$$ then $$f$$ must be a locally bilipschitz mapping.

Kristýna Kuncová, Jan Malý: Non-absolutely convergent integrals in metric spaces. J. Math. Anal. Appl. 401 (2013), no. 2, 578–600.
We develop the theory of Henstock-Kurzweil type integral of functions with respect to metric distributions in the framework of metric spaces. In the setting of metric currents (as originated by E.~De Giorgi, L.~Ambrosio and B.~Kirchheim) we apply the new integral to study a generalization of the Stokes theorem.

Pekka Koskela, Jan Malý, Thomas Zürcher: Luzin's condition (N) and Sobolev mappings. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 23 (2012), no. 4, 455–465.
For the purpose of change of variables in integral, it is important to know how to verify Luzin's condition (N) for Sobolev mappings. It this article we survey some results on this topic sorted according to the method. We discuss the method of absolute continuity, results obtained via degree and results based on the interplay between integrability and modulus of continuity.

Hana Bendová, Jan Malý: An elementary way to introduce a Perron-like integral. Ann. Acad. Sci. Fenn. Math. 36 (2011), no. 1, 153–164.
We give an alternative definition of integral at the generality of the Perron integral and propose an exposition of the foundations of integral theory starting from this new definition. Both definition and proofs needed for the development are unexpectedly simple. We show how to adapt the definition to cover the multidimensional and Stieltjes case and prove that our integral is equivalent to the Henstock-Kurzweil(-Stieltjes) integral.

Stanislav Hencl, Jan Malý, Luboš Pick, Jan Vybíral: Weak estimates cannot be obtained by extrapolation. Expo. Math. 28 (2010), no. 4, 375–377.
We prove that weak-type estimates cannot be obtained via extrapolation.

Marianna Csörnyei, Stanislav Hencl, Jan Malý: Homeomorphisms in the Sobolev space $$W^{1,n-1}$$. J. Reine Angew. Math. 644 (2010), 221–235.
Let $$\Omega\subset\mathbb R^n$$ be open. We show that each homeomorphism $$f\in W^{1,n-1}_{loc}(\Omega,\mathbb R^n)$$ satisfies $$f^{-1}\in BV_{loc}(f(\Omega),\mathbb R^n)$$. If we moreover assume that $$f$$ has finite distortion, then $$f^{-1}\in W^{1,1}_{loc}(f(\Omega),\mathbb R^n)$$ and $$f^{-1}$$ has finite distortion. The main ingredient is a new result on change of variables in integral (area and coarea formula) for such mappings.

Stanislav Hencl and Jan Malý: Jacobians of Sobolev homeomorphisms. Calculus of Variations and Partial Differential Equations Let $$\Omega\subset\mathbb R^n$$ be a domain. We show that each homeomorphism $$f$$ in the Sobolev space $$W^{1,1}_{loc}(\Omega,\mathbb R^n)$$ satisfies either $$J_f\geq 0$$ a.e or $$J_f\leq 0$$ a.e. if $$n=2$$ or $$n=3$$. For $$n>3$$ we prove the same conclusion under the stronger assumption that $$f\in W^{1,s}_{loc}(\Omega,\mathbb R^n)$$ for some $$s>[n/2]$$ (or in the setting of Lorentz spaces).

Piotr Hajlasz; Jan Maly: On approximate differentiability of the maximal function. Proc. Amer. Math. Soc. 138 (2010), 165-174. We prove that if $$f\in L^1(\mathbf R^n)$$ is approximately differentiable a.e., then the Hardy-Littlewood maximal function $$M f$$ is also approximately differentiable a.e. Moreover, if we only assume that $$f\in L^1(\mathbf R^n)$$, then any open set of $$\mathbf R^n$$ contains a subset of positive measure such that $$M f$$ is approximately differentiable on that set. On the other hand we present an example of $$f\in L^1(\mathbf R)$$ such that $$M f$$ is not approximately differentiable a.e.

Jan Maly, David Swanson, William P. Ziemer: Fine behavior of functions whose gradients are in an Orlicz space. Studia Math. 190(1) (2009), 33-71.
For functions whose derivatives belong to an Orlicz space, we develop their fine'' properties that is a generalization of the treatment found in [MZ] for Sobolev functions. Of particular importance is Theorem 8.8 which is used in the development in [MSZ] of the coarea formula for such functions.

Piotr Hajlasz, Tadeusz Iwaniec, Jan Maly and Jani Onninen: Weakly Differentiable Mappings Between Manifolds. Mem. Amer. Math. Soc. 192 (2008), no. 899.
We study Sobolev classes of weakly differentiable mappings $$f:X\rightarrow Y$$ between compact Riemannian manifolds without boundary. These mappings need not be continuous. They actually possess less regularity than the mappings in $$W^{1,n}(X, Y)$$, $$n=\dim X$$. The central themes being discussed are:

1. smooth approximation of those mappings
2. integrability of the Jacobian determinant
The approximation problem in the category of Sobolev spaces between manifolds $$W^{1,p}(X, Y)$$, $$1\le p \le n$$, has been recently settled by Bethuel, Bethuel/Zhong, and Hang/Lin. However, the point of our results is that we make no topological restrictions on manifolds $$X$$ and $$Y$$. We characterize, essentially all, classes of weakly differentiable mappings which satisfy the approximation property. The novelty of our approach is that we were able to detect tiny sets on which the mappings are continuous. These sets give rise to the so-called web like structure of $$X$$ associated with the given mapping $$f: X\rightarrow Y$$.
The integrability theory of Jacobians in a manifold setting is really different than one might a priori expect based on the results in the Euclidean space. To our surprise, the case when the target manifold $$Y$$ admits only trivial cohomology groups $$H^\ell(Y)$$, $$1\le \ell \le n= \dim Y$$, like $$n$$-sphere, is more difficult than the nontrivial case in which $$Y$$ has at least one non-zero $$\ell$$-cohomology. The necessity of topological constraints on the target manifold is a new phenomenon in the theory of Jacobians.

Luigi Ambrosio, Camillo De Lellis, Jan Maly: On the chain rule for the divergence of BV like vector fields: applications, partial results, open problems. In: Perspectives in Nonlinear Partial Differential Equations (In honor of Haim Brezis). Contemporary Math. 446, AMS 2007, ISBN 978-0-8218-4190-7, pp.31--67.
We discuss the problem of computing the distributional divergence of a vector field of the form h(w)B, where h is a smooth scalar function and B is a BV vector field, knowing the distributional divergence of all vector fields w_iB, where w_i are the components of w. We present partial results on this problem, conjectures, and links with other problems related to the SBV regularity of solutions of Hamilton-Jacobi equations and systems of conservation laws, and a conjecture recently made by Bressan.

Luigi Ambrosio, Jan Maly: Very weak notions of differentiability. Proc. Roy. Soc. Edinburgh Sect. A 137 (2007), no. 3, 447--455.
In this paper we study a measure-theoretic notion of differentiability introduced in a paper by Le Bris and Lions, in connection with the differentiability properties of the flow associated to a Sobolev vector field. We characterize in various ways this differentiability property, showing through an example that it is strictly weaker than the classical approximate differentiability.

Stanislav Hencl, Pekka Koskela, Jan Maly: Regularity of the inverse of a Sobolev homeomorphism in space. Proc. Roy. Soc. Edinburgh Sect. A 136 (2006), no. 6, 1267-1285.
Let $$\Omega\subset\mathbb R^n$$ be open. Given a homeomorphism $$f\in W^{1,1}_{loc}(\Omega,\mathbb R^n)$$ of finite distortion with $$|Df|$$ in the Lorentz space $$L^{n-1,1}(\Omega)$$, we show that $$f^{-1}\in W^{1,1}_{loc}(f(\Omega),\mathbb R^n)$$ and that $$f^{-1}$$ has finite distortion. A class of counterexamples demonstrating sharpness of the results is constructed.

Jan Maly and Miroslav Zeleny: A note on Buczolich's solution of the Weil gradient problem: a construction based on an infinite game. Acta Mathematica Hungarica 113 (1-2), 145-158.
Using an infinite game approach we reprove Buczolich's result that there exists a differentiable function $f$ such that $\nabla f(0) = 0$ and $|\nabla f| \geq 1$ a.e.

Irene Fonseca and Jan Maly: From Jacobian to Hessian: distributional form and relaxation. Riv. Mat. Univ. Parma (7) 4* (2005), 45-74.
A weak formulation of the determinant of the matrix of second order derivatives is introduced and several of its properties are explored in analogy with the theory developed for the weak Jacobian.

Irene Fonseca, Giovanni Leoni, Jan Maly: Weak Continuity and Lower Semicontinuity Results for Determinants Arch Rational Mech. Anal. 178, no. 3 (2005), 411 - 448.
Weak continuity properties of minors and lower semicontinuity properties of functionals with polyconvex integrands are addressed in this paper. In particular, it is shown that if $\left\{ u_{n}\right\}$ is bounded in $W^{1,N-1}\left( \Omega;\mathbb{R}^{N}\right)$, $\left\{ \mathrm{adj}\nabla u_{n}\right\} \subset L^{\frac{N}{N-1}}\left( \Omega;\mathbb{R}^{{N}\times{N}}\right) ,$ and if $u\in BV\left( \Omega;\mathbb{R}^{N}\right)$ are such that $u_{n}\rightarrow u$ in $L^{1}\left( \Omega;\mathbb{R}^{N}\right)$ and $\det\nabla u_{n}\overset{\ast}{\rightharpoonup}\mu$ in the sense of measures, then for $\mathcal{L}^{N}$ a.e. $x\in\Omega$ $\det\nabla u\left( x\right) =\frac{d\mu}{d\mathcal{L}^{N}}\left( x\right).$ The result is sharp and counterexamples are provided in the cases where regularity of $\left\{ u_{n}\right\}$ or the type of weak convergence are weakened.

Irene Fonseca, Jan Maly, Giuseppe Mingione: Scalar minimizers with fractal singular sets. Arch Rational Mech. Anal. 172 (2004), 295-307.
Lack of regularity of local minimizers for convex functionals with non-standard growth conditions is considered. It is shown that for every $\varepsilon > 0$ there exists a function $a \in C^{\alpha}(\Omega)$ such that the functional $$\mathcal{F}:u \mapsto \int_{\Omega} (|Du|^p + a(x)|Du|^q)\, dx$$ admits a local minimizer $u \in W^{1,p}(\Omega)$ whose set of non-Lebesque points is a closed set $\Sigma$ with $\operatorname{dim}_{\mathcal H}(\Sigma)>N-p-\varepsilon$, and where $1< p Janne Kauhanen, Pekka Koskela, Jan Maly, Jani Onninen and Xiao Zhong: Mappings of finite distortion: Sharp Orlicz-conditions. Revista Matematica Iberoamericana 19,3 (2003), 857-872. Let$f$be a mapping of finite distortion$K$. Under sharp Orlicz conditions on integrability of the gradient or of$K$we show that$f$is continuous and either constant or both open and discrete. Moreover,$f$maps sets of Lebesgue measure zero to sets of measure zero. The sharpness is demonstrated by counterexamples. Pekka Koskela and Jan Maly: Mappings of finite distortion: The zero set of the Jacobian. J. Eur. Math. Soc. (JEMS) 5 (2003), no. 2, 95--105. Both the zero set of the Jacobian and the branch set of a non-constant mapping of finite, (sub)exponentially integrable distortion have volume zero. If a mapping$f$of finite distortion$K$has essentially bounded multiplicity, and$K^{1/(n-1)}$and the Jacobian$J_f$are integrable, then$J_f>0$a.e. Stanislav Hencl, Jan Maly: Absolutely continuous functions of several variables and diffeomorphisms. Central European J. Math. 1,4 (2003), 690-705. In a previous paper of the second author, a class of absolutely continuous functions of$d$-variables, motivated by applications to change of variables in integral, has been introduced. The main result of this paper states that such absolutely continuous functions are not stable under diffeomorphisms. We also show an example of a function which is absolutely continuous with respect cubes but not with respect to balls. Jan Maly: Coarea integration in metric spaces. In: Nonlinear Analysis, Function Spaces and Applications Vol. 7. Proceedings of the Spring School held in Prague July 17-22, 2002. Eds. B. Opic and J. Rakosnik. Math. Inst. of the Academy of Sciences of the Czech Republic, Praha 2003, pp. 142-192. Coarea properties and Eilenberg-type inequalities are proved in the framework of metric spaces with a doubling measure, for mappings with an upper gradient in the Lorentz space$L_{m,1}$. This relies on estimates of Hausdorff content of level sets of mappings between measure spaces and analysis of their Lebesgue points. Adapted versions of the Frostman lemma and of the Muckenhoupt-Wheeden inequality appear as essential tools. Jaroslav Lukes, Jan Maly, Ivan Netuka, Michal Smrcka and Jiri Spurny: On approximation of affine Baire-one functions. Israel J. Math. 134 (2003), 255-287. It is known (G.~Choquet, G.~Mokobodzki) that a Baire--one affine function on a compact convex set satisfies the barycentric formula and can be expressed as a pointwise limit of a sequence of continuous affine functions. Moreover, the space of Baire--one affine functions is uniformly closed. The aim of this paper is to discuss to what extent analogous properties are true in the context of general function spaces. In particular, we investigate the function space$H(U)$, consisting of the functions continuous on the closure of a bounded open set$U \subset \mathbb R ^m$and harmonic on$U$, which has been extensively studied in potential theory. We demonstrate that the barycentric formula does not hold for the space$\B _1^b(H(U))$of bounded functions which are pointwise limits of functions from the space$H(U)$and that$\B _1^b(H(U))$is not uniformly closed. On the other hand, every Baire--one$H(U)$--affine function (in particular a solution of the generalized Dirichlet problem for continuous boundary data) is a pointwise limit of a bounded sequence of functions belonging to$H(U)$. It turns out that such a situation always occurs for simplicial spaces whereas it is not the case of general function spaces. The paper provides several characterizations of those Baire--one functions which can be approximated pointwise by bounded sequences of elements of a given function space. Jan Maly: Wolff potential estimates of superminimizers of Orlicz type Dirichlet integrals. Manuscripta Math. 110 (2003), 513--525. If$u$is a minimizer of$\int_{\Omega}\ef(|\nabla u|)\,dx -\int_{\Omega}u\,d\mu$, then the pointwise estimate $$u(x)\le K+ C\int_0^R[\ef']^{-1}\Bigl(r^{1-n}\mu(B(x,r))\Bigr)\,dr$$ can be reached. This results is obtained for a Young function$\ef$with the global$\Delta_2\;\&\;\nabla_2$property. Links to applications to real analysis are given. Jan Maly: Coarea properties of Sobolev functions. In: Function Spaces, Differential Operators and Nonlinear Analysis. The Hans Triebel Aniversary Volume. D. Haroske, T. Runst, H.-J. Schmeisser (eds.). Birhhaeuser, Basel 2003. The Lusin N-property is well known as a criterion for validity of theorems on change of variables in integral. Here we consider related properties motivated by the coarea formula. They also imply a generalization of Eilenberg's intequality. We prove them for functions with gradient in the Lorentz space$L_{m,1}$. This relies on estimates of Hausdorff content of level sets for Sobolev functions and analysis of their Lebesgue points. A significant part of presented results has its origin in a joint work with David Swanson and William P.~Ziemer. Jan Maly, David Swanson and William P. Ziemer: Coarea formula for Sobolev mappings. Trans. Amer. Math. Soc. 355 (2003), 477-492. We extend Federer's coarea formula to mappings$f$belonging to the Sobolev class$W^{1,p}(\R^n;\R^m)$,$1 \le m < n$,$p>m$, and more generally, to mappings with gradient in the Lorentz space$L^{m,1}(\R^n)$. This is accomplished by showing that the graph of$f$in$\R^{n+m}$is a Hausdorff$n$-rectifiable set. Stanislav Hencl and Jan Maly: Mapping of bounded distortion: Hausdorff measure of zero sets. Math. Ann. 324 (2002), 451-464. We prove that a for a mapping$f$of finite distortion$K\in L^{p/(n-p)}$, the$(n-p)$-Hausdorff measure of any point preimage is zero provided$J_f$is integrable,$Df\in L^s$with$s>p$, and the multiplicity function of$f$is essentially bounded. As a consequence for$p=n-1$we obtain that the mapping is then open and discrete. 30C65, 26B10, 74B20. Piotr Hajlasz and Jan Maly: Approximation of nonlinear expressions involving gradient in Sobolev spaces. Ark. Mat. 40,2 (2002), 245-274. Approximation of Sobolev functions by functions which are locally monotone in the sense of Lebesgue. The approximation is based on non-convolution methods which allow to handle nonlinear expressions like some functions of gradient minors. Irene Fonseca, Giovanni Leoni, Jan Maly and Roberto Paroni: A note on Meyers' Theorem in$W^{k,1}$Trans. Amer. Math. Soc. 354,9 (2002), 3723-3741 Lower semicontinuity properties of multiple-integrals $$u\in W^{k,1}(\Omega;\mathbb{R}^{d})\mapsto\int_{\Omega}f(x,u(x),\cdots ,\nabla^{k}u(x))\,dx$$ are studied when$f$grows at most linearly with respect to the highest order derivative,$\nabla^{k}u,$and admissible$W^{k,1}(\Omega;\mathbb{R}^{d})$sequences converge strongly in$W^{k-1,1}(\Omega;\mathbb{R}^{d}).$It is shown that under certain continuity assumptions on$f,$convexity,$1$-quasiconvexity or$k$-polyconvexity of$\xi\longmapsto f(x_{0},u(x_{0}),\cdots,\nabla^{k-1}u(x_{0}),\xi)$ensure lower semicontinuity. The case where$f(x_{0},u(x_{0}),\cdots,\nabla^{k-1}u(x_{0}),\cdot)$is$k$-quasiconvex remains open except in some very particular cases, such as when$f(x,u(x),\cdots,\nabla^{k}u(x))=h(x)g(\nabla^{k}u(x)).$Robert Cerny and Jan Maly: Another counterexample to lower semicontinuity in Calculus of Variations. Journal of Convex Analysis 9 (2002), No. 1, 295-299 An example is shown of a functional $$F(u)=\int_{I}f(u,u')\,dt$$ which is not lower semicontinuous with respect to$L^1$-con\-ver\-gen\-ce. The function$f$is nonnegative, continuous and strictly convex in the second variable for each$u \in \er^n$. This solves a problem related to Serin's lower semocontinuity theorem. Jan Maly and Lubos Pick: The sharp Riesz potential estimates in metric spaces. Indiana. Univ. Math. J. 51,2 (2002), 251-268. We introduce the Riesz potential operator for functions defined on a quasi-metric space endowed with a nonnegative doubling measure satisfying certain lower bound for the~measure of a~ball. For this operator we prove sharp Lorentz-norm estimates in the spirit of the results due to O'Neil and Peetre (in the non-limiting case) and to Hansson and Brezis-Wainger (in the limiting case). J. Maly and L. Pick: An elementary proof of sharp Sobolev embeddings. Proc. Amer. Math. Soc. 130,2 (2002), 555-563 We present an elementary unified and self-contained proof of sharp Sobolev embedding theorems. We introduce a~new function space and use it to improve the limiting Sobolev embedding theorem due to Br\'ezis and Wainger. Janne Kauhanen, Pekka Koskela and Jan Maly: Mappings of finite distortion: discreteness and openness. Arch. Rat. Mech. Anal. 160 (2001), 135-151. We study mappings$f\in W^{1,1}(\Omega,R^n)$satisfying the distortion inequality$|Df(x)|^n\le K(x) J(x,f)$. If$K\ge 1$,$\exp(\lambda K)$is integrable for some$\lambda>0$and the Jacobian$J(x,f)$is integrable, then$f$is continuous and either constant or discrete and open. If$K\in L^p(\Omega)$for some$p>n-1$and$\lim_{\epsilon\to0} \epsilon\int_{\Omega}|Df(x)|^{n-\epsilon}\,dx=0$, then the same conclusion holds. Conversely, there is a non-constant continuous mapping$f$with a.e. positive and integrable Jacobian determinant and with finite distortion$K$such that$\exp(\lambda K/\log^2(1+K))$is integrable,$\limsup _{\epsilon\to0} \epsilon\int_{\Omega}|Df(x)|^{n-\epsilon}\,dx<\infty$, and$f$is neither open, nor discrete, nor sense preserving. Robert Cerny and Jan Maly: Counterexample to lower semicontinuity in Calculus of Variations. Math. Z. 238 (2001) 689-694. An example is shown of a functional $$F(u)=\int_{I}f(u,u')\,dt$$ which is not lower semicontinuous with respect to$L^1$-con\-ver\-gen\-ce. The function$f$is lower semicontinuous, convex in the second variable and linearly coercive. Application to nonexistence of minimizers in$BV$-setting is also given. Janne Kauhanen, Pekka Koskela and Jan Maly: Mappings of finite distortion: Condition N. Michigan Math. J. 49 (2001), 169-181. If$f\in W^{1,1}(\Omega,R^n)$is a sense-preserving continuous mapping satisfying$\lim_{\epsilon\to0} \epsilon \int_{\Omega} |Df(x)|^{n-\epsilon} \,dx = 0$, then$f$satisfies the Lusin condition N, i.e. maps null sets to null sets. Conversely, there is a homeomorphism such that$\limsup _{\epsilon\to0} \epsilon \int_{\Omega} |Df(x)|^{n-\epsilon} \,dx < \infty$which does not satisfy N. The positive result is a consequence of more general technical theorems and applies to mapping of finite distortion. J. Maly: Sufficient Conditions for Change of Variables in Integral. In: Proceedings on Analysis and Geometry, Sobolev Institute Press, Novosibirsk 2000, 370--386. It is well known that Luzin's property (N) of a mapping$f$is essential for justifying the change of variables through$f$in integral. Let us highlight the deep and fundamental contribution of Professor Yurij G. {\sc Reshetnyak} in investigation of Luzin's property for Sobolev mappings. In this paper we discuss further conditions. One of them is the following {\em$n$-absolute continuity} property of a function$f$: for any$\varepsilon>0$there exists$\delta>0$such that for every disjoint collection$\{B_j\}$of balls we have $$\sum ({\rm diam\,}B_j)^n<\delta \Rightarrow\sum \Bigl({\rm diam\,}f(B_j)\Bigr)^n<\varepsilon.$$ This condition is satisfied if the gradient of$f$belongs to the Lorentz space$\lor$. If$\nabla f$belongs only to$L^n$, then the Luzin property holds except a singular set, which is the set of all points, where$f$is not approximately H\"older continuous. Another modification of the result applies to some$BV$function with nontrivial singular part of the gradient. A large part of referred results stems from joint works with Olli {\sc Martio}, Pekka {\sc Koskela}, Janne {\sc Kauhanen} and Irene {\sc Fonseca}. J. Maly and S. Ponomarev: On the Sobolev class$W^{1,p}_{loc}$and quasiregularity. Siberian Math. J. 41,6 (2000), 1381-1389. If a function satisfies a special Morera-type inequality with respect to a diferentiation basis, then it is quasiregular. The results is based on a Morera-type criterion for belonging to the Sobolev space. J. Maly and U. Mosco: Remarks on measure-valued Lagrangians on homogeneous spaces. Ricerche Mat. XLVII - Supplemento 1999, 217-231 The aim of this note is to present a proof of Sobolev inequalities on spaces of homogeneous type, for measure-valued Lagrangians that - as in the fractal case - may not have pointwise defined densities. We use an approach based on Riesz potential estimates. T. Kilpelainen and J. Maly: Sobolev inequalities on sets with irregular boundaries. Z. Anal. Angew. 19,2 (2000), 369-380. Optimal exponent for the (weighted) Sobolev-Poincare inequality on s-John domains. I. Fonseca and J. Maly: Remarks on the Determinant in Nonlinear Elasticity and Fracture Mechanics. Applied Nonlinear Analysis. Eds. A. Sequiera, H. B. da Veiga, J. H. Videman. Kluwer Academic / Plenum Publishers, New York 1999, 117-132. The role of the determinant in ensuring local invertibility of Sobolev functions in$W^{1,N}(\Omega;\mathbb R^N)$is studied. Weak continuity of minors of gradients of functions in$W^{1,p}(\Omega;\mathbb R^N)$for$p<N$is fully characterized. Properties of the determinant are addressed within the framework of functions of bounded variation, and a change of variables formula is obtained. These results are relevant in the study of equilibria, cavitation, and fracture of nonlinear elastic materials. B. Dacorogna, I. Fonseca, J. Maly and K. Trivisa: Manifold Constrained Variational Problems. Calc. Var. 9.3 (1999), 185-206. The integral representation for the relaxation of a class of energy functionals where the admissible fields are constrained to remain on a$C^1m$-dimensional manifold$\mathcal M \subset {\mathbb R}^d$is obtained. J. Kauhanen, P. Koskela and J. Maly: On functions with derivatives in a Lorentz space. Manuscripta Math. 100,1 (1999), 87-101. Functions of$n$-variables with weak derivatives in the Lorentz space$L(n,1)$are$n$-absolutely continuous in the sense of [1]. In particular, are a.e. differentiable and if vector valued, they have the Lusin's$N$property and are almost open''. Conditions of Orlicz type and counterexamples showing sharpness of the condition are also presented. J. Maly: A Simple Proof of the Stepanov Theorem of Differentiability Almost Everywhere. Expo. Math. 17 (1999), 059-062. If$f$is a function of several variables whose pointwise Lipschitz constant is a. e. finite, then$f$is a. e. differentiable. This generalization of Rademacher's theorem is due to Stepanov. We give here a simple proof. J. Maly: Absolutely continuous functions of several variables. J. Math. Anal. Appl. 231 (1999) 492-508. http://www.idealibrary.com A class of absolutely continuous functions'' of$n$variables is introduced. The absolute continuity implies continuity, weak differentiability with gradient in$L^n$, differentiability almost everywhere, area, coarea and degree formulae. This gives a unified approach to some previously known conditions for the mentioned results based on monotonicity, finite dilatation or higher integrability of the gradient. G. Bouchitte, I. Fonseca and J. Maly: The effective bulk energy of the relaxed energy of multiple integrals below the growth exponent. Proceedings of the Royal Society of Edinburgh, 128A (1998), 463-479. The characterization of the bulk energy density of the relaxation in$W^{1,p}(\Omega;\er^d)$of a functional $$F(u,\Omega) := \int_{\Omega} f(\nabla u)\,dx$$ is obtained for$p > q- q/N$, where$u \in W^{1,p}(\Omega;\er^d)$, and$f$is a continuous function on the set of$d\times N$matrices verifying $$0\le f(\xi)\le C(1+|\xi|^q)$$ for some constant$C>0$and$1\leq q <+\infty$. Typical examples may be found in cavitation and related theories. Standard techniques cannot be used due to the gap between the exponent$q$of the growth condition and the exponent$p$of integrability of the macroscopic strain$\nabla u. A recently introduced global method for relaxation and fine Sobolev trace and extension theorems are applied. I. Fonseca and J. Maly: Relaxation of multiple integrals below the growth exponent. Ann. Inst. Henri Poincare (Analyse Non Lineaire), 14,3 (1997), 309-338. The integral representation of the relaxed energies \eqalign{ {\cal F}^{q,p}(u,\Omega)=\inf_{u_n}\Bigl\{&\liminf_{n\to \infty} \int_{\Omega}F(x,u_n,\nabla u_n)\,dx:u_n\in W^{1,q}(\Omega,{\bf R}^d)), \cr & u_n\to u \hbox{ weakly in }W^{1,p}(\Omega,{\bf R}^d)\Bigr\}, }\relax \eqalign{ {\cal F}_{\rm loc}^{q,p}(u,\Omega)=\inf_{u_n}\Bigl\{&\liminf_{n\to \infty} \int_{\Omega}F(x,u_n,\nabla u_n)\,dx:u_n\in W_{\rm loc}^{1,q}(\Omega,{\bf R}^d)), \cr & u_n\to u \hbox{ weakly in }W^{1,p}(\Omega,{\bf R}^d)\Bigr\}, } of a functional $$E:u\mapsto \int_{\Omega}F(x,u,\nabla u)\,dx,\qquad W^{1,q}(\Omega,{\bf R}^d),$$ where0\le F(x,\zeta,\xi)\le C(1+|\zeta|^r+|\xi|^q)$and$\max\Bigl\{1,r{N-1\over N+r},q{N-1\over N}\Bigr\}<p\le q$, is studied. In particular,$W^{1,p}$-sequential weak lower semicontinuity of$E(\cdot)$is obtained in the case where$F=F(\xi)$is a quasiconvex function and$p>q(N-1)/N$. J. Maly and W.P. Ziemer: Fine Regularity of Solutions of Elliptic Differential Equations''. Mathematical Surveys and Monographs, AMS 1997. For more information The primary objective of this book is to give a comprehensive exposition of results surrounding the work of the authors concerning boundary regularity of weak solutions of second order elliptic quasilinear equations in divergence form. The structure of these equations allow coefficients in certain$L^{p}$spaces and thus, it is known from classical results that weak solutions are locally H\"older continuous in the interior. Here, it is shown that weak solutions are continuous at the boundary if and only if a Wiener-type condition is satisfied. This condition reduces to the celebrated Wiener criterion in the case of harmonic functions. The work that accompanies this analysis includes the fine'' analysis of Sobolev spaces and a development of the associated nonlinear potential theory. The term fine'' refers to a topology on$R^{n}$which is induced by the Wiener condition. The book also contains a complete development of regularity of solutions of variational inequalities, including the double obstacle problem, where the obstacles are allowed to be discontinuous. The regularity of the solution is given in terms involving the Wiener-type condition and the fine topology. The case of differential operators with a differentaible structure and$C^{1,\alpha }$obstacles is also developed. The book concludes with a chapter devoted to the existence theory thus providing the reader with a complete treatment of the subject ranging from regularity of weak solutions to the existence of weak solutions. J. Maly: Darboux property of Frechet derivatives. Real Analysis Exchange 22,1 (1996-7), 167-173. P.Holicky, J. Maly, C.E. Weil and L. Zajicek: A note on the gradient problem. Real Analysis Exchange 22,1 (1996-7), 225-235. J. Maly: Nonlinear potentials and quasilinear PDE's. In: Potential Theory - ICPT 94''. Kral, J., Lukes, J., Netuka, I. and Vesely, J. (Eds.) Walter de Gruyter, Berlin-New York, 1996, 103-128. J. Maly: Potential estimates and Wiener criteria for quasilinear elliptic equations. In: XVIth Rolf Nevanlinna Colloquium''. Laine, I. and Martio, O. (Eds.) Walter de Gruyter, Berlin, 1996, 119-133. J. Maly: Pointwise estimates of nonnegative subsolutions of quasilinear elliptic equations at irregular boundary points. Comment Math. Univ. Carolinae 37,1 (1996) 23-42. J. Lukes and J. Maly: Measure and integral''. Matfyzpress, Matematicko-fyzikalni fakulta Univerzity Karlovy, Praha 1995. J. Maly and O. Martio: Lusin's condition (N) and mappings of the class$W^{1,n}$. J. Reine Angew. Math. 458 (1995) 19-36. J. Maly: Examples of weak minimizers with continuous singularities. Expositiones Math. 13 (1995), 446-454. J. Maly: Lower semicontinuity of quasiconvex integrals. Manuscripta Math. 85 (1994), 419-428. T. Kilpelainen and J. Maly: The Wiener test and potential estimates for quasilinear elliptic equations. Acta Math. 172 (1994), 137-161. I. Hlavacek, M. Krizek and J. Maly: On Galerkin approximation of a quasilinear nonpotential elliptic problem of a nonmonotone type. J. Math. Anal. Appl. 184,1 (1994), 168-189. J. Maly: The area formula for$W^{1,n}\$-mappings. Comment. Math. Univ. Carolinae 35,2 (1994) 291-298.

J. Maly: Weak lower semicontinuity of polyconvex integrals. Proc. Roy. Soc. Edinburgh 123A (1993), 681-691.

J. Maly: Holder type quasicontinuity. Potential Analysis 2 (1993), 249-254.