Jan Kristensen
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
Research Subjects
Jan Kristensen has a total of 41 co-authors. Here are the top co-authors by number of shared publications:
- Mikhail. V. Korobkov
School of Mathematical Sciences, Fudan University, Shanghai, People’s Republic of China
8 shared publications - Giuseppe Mingione
Universitá di Parma
7 shared publications - André Guerra
ETH Zurich, Switzerland
3 shared publications - Antonia Passarelli di Napoli
Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università degli Studi di Napoli “Federico II”, Napoli, Italy
3 shared publications Bernd Kirchheim
Mathematical Institute, University of Oxford, 24–29 St. Gilesʼ, Oxford OX1 3LB, UK
3 shared publications- Daniel Faraco
Departamento de Matemáticas, Universidad Autónoma de Madrid, Instituto de Ciencias Matemáticas, CSIC-UAM-UC3M-UCM, Madrid, Spain
3 shared publications
Latest Publications
- 19 March 2025
- Published by European Mathematical Society - EMS - Publishing House GmbH in Journal of the European Mathematical Society
Distortion of Hausdorff measures under Orlicz–Sobolev maps
A comprehensive theory of the effect of Orlicz–Sobolev maps, between Euclidean spaces, on subsets with zero or finite Hausdorff measure is offered. Arbitrary Orlicz–Sobolev spaces embedded into the space of continuous functions and Hausdorff measures built upon general gauge functions are included in our discussion. An explicit formula for the distortion of the relevant gauge function under the action of these maps is provided in terms of the Young function defining the Orlicz–Sobolev space. Classical results, dealing with standard Sobolev spaces and Hausdorff measures, are recovered, and their optimality is shown to hold in a refined stronger sense. Special instances available in the literature, concerning Young functions and gauge functions of nonpower type, are also recovered and, when not sharp, improved. New phenomena and features, related to the flexibility in the definition of the degree of integrability of weak derivatives of maps and in the notion of measure of sets, are detected. For instance, perhaps surprisingly, Hausdorff measures, other than the Lebesgue measure, which are not distorted by classical Sobolev maps are exhibited. The relevant measures are associated with gauge functions with a logarithmic decay near zero, which cannot be attained by the standard theory.
- 19 March 2025
- Published by European Mathematical Society - EMS - Publishing House GmbH in Journal of the European Mathematical Society
Distortion of Hausdorff measures under Orlicz–Sobolev maps
A comprehensive theory of the effect of Orlicz–Sobolev maps, between Euclidean spaces, on subsets with zero or finite Hausdorff measure is offered. Arbitrary Orlicz–Sobolev spaces embedded into the space of continuous functions and Hausdorff measures built upon general gauge functions are included in our discussion. An explicit formula for the distortion of the relevant gauge function under the action of these maps is provided in terms of the Young function defining the Orlicz–Sobolev space. Classical results, dealing with standard Sobolev spaces and Hausdorff measures, are recovered, and their optimality is shown to hold in a refined stronger sense. Special instances available in the literature, concerning Young functions and gauge functions of nonpower type, are also recovered and, when not sharp, improved. New phenomena and features, related to the flexibility in the definition of the degree of integrability of weak derivatives of maps and in the notion of measure of sets, are detected. For instance, perhaps surprisingly, Hausdorff measures, other than the Lebesgue measure, which are not distorted by classical Sobolev maps are exhibited. The relevant measures are associated with gauge functions with a logarithmic decay near zero, which cannot be attained by the standard theory.
- Top Cited PapersOpen Access
- 9 September 2024
- Published by Springer Nature in Archive for Rational Mechanics and Analysis
- Vol. 248 (5) , 1-125
Quasiconvex Functionals of (p, q)-Growth and the Partial Regularity of Relaxed Minimizers
We establish $$\textrm{C}^{\infty }$$ -partial regularity results for relaxed minimizers of strongly quasiconvex functionals $$\begin{aligned} \mathscr {F}[u;\Omega ]:=\int _{\Omega }F(\nabla u)\textrm{d}x,\qquad u:\Omega \rightarrow \mathbb {R}^{N}, \end{aligned}$$ subject to a q-growth condition $$|F(z)|\leqq c(1+|z|^{q})$$ , $$z\in \mathbb {R}^{N\times n}$$ , and natural p-mean coercivity conditions on $$F\in \textrm{C}^{\infty }(\mathbb {R}^{N\times n})$$ for the basically optimal exponent range $$1\leqq p\leqq q<\min \{\frac{np}{n-1},p+1\}$$ . With the p-mean coercivity condition being stated in terms of a strong quasiconvexity condition on F, our results include pointwise (p, q)-growth conditions as special cases. Moreover, we directly allow for signed integrands which is natural in view of coercivity considerations and hence the direct method, but is novel in the study of relaxed problems. In the particular case of classical pointwise (p, q)-growth conditions, our results extend the previously known exponent range from Schmidt’s foundational work (Schmidt in Arch Ration Mech Anal 193:311–337, 2009) for non-negative integrands to the maximal range for which relaxations are meaningful, moreover allowing for $$p=1$$ . We also emphasize that our results apply to the canonical class of signed integrands and do not rely in any way on measure representations à la Fonseca and Malý (Ann Inst Henri Poincaré Anal Non Linéaire 14:309–338, 1997).- 9 September 2024
- Published by Springer Nature in Archive for Rational Mechanics and Analysis
- Vol. 248 (5) , 1-125
Top Cited PapersOpen AccessQuasiconvex Functionals of (p, q)-Growth and the Partial Regularity of Relaxed Minimizers
We establish $$\textrm{C}^{\infty }$$ -partial regularity results for relaxed minimizers of strongly quasiconvex functionals $$\begin{aligned} \mathscr {F}[u;\Omega ]:=\int _{\Omega }F(\nabla u)\textrm{d}x,\qquad u:\Omega \rightarrow \mathbb {R}^{N}, \end{aligned}$$ subject to a q-growth condition $$|F(z)|\leqq c(1+|z|^{q})$$ , $$z\in \mathbb {R}^{N\times n}$$ , and natural p-mean coercivity conditions on $$F\in \textrm{C}^{\infty }(\mathbb {R}^{N\times n})$$ for the basically optimal exponent range $$1\leqq p\leqq q<\min \{\frac{np}{n-1},p+1\}$$ . With the p-mean coercivity condition being stated in terms of a strong quasiconvexity condition on F, our results include pointwise (p, q)-growth conditions as special cases. Moreover, we directly allow for signed integrands which is natural in view of coercivity considerations and hence the direct method, but is novel in the study of relaxed problems. In the particular case of classical pointwise (p, q)-growth conditions, our results extend the previously known exponent range from Schmidt’s foundational work (Schmidt in Arch Ration Mech Anal 193:311–337, 2009) for non-negative integrands to the maximal range for which relaxations are meaningful, moreover allowing for $$p=1$$ . We also emphasize that our results apply to the canonical class of signed integrands and do not rely in any way on measure representations à la Fonseca and Malý (Ann Inst Henri Poincaré Anal Non Linéaire 14:309–338, 1997). - Open Access
The local Burkholder functional, quasiconvexity and Geometric Function Theory
We show that the local Burkholder functional $\mathcal B_K$ is quasiconvex. In the limit of $p$ going to 2 we find a class of non-polyconvex functionals which are quasiconvex on the set of matrices with positive determinant. In order to prove the validity of lower semicontinuity arguments in this setting, we show that the Burkholder functionals satisfy a sharp extension of the classical function theoretic area formula. As a corollary, in addition to functionals in geometric function theory, one finds new classes of non-polyconvex functionals, degenerating as the determinant vanishes, for which there is existence of minimizers.Open AccessThe local Burkholder functional, quasiconvexity and Geometric Function Theory
We show that the local Burkholder functional $\mathcal B_K$ is quasiconvex. In the limit of $p$ going to 2 we find a class of non-polyconvex functionals which are quasiconvex on the set of matrices with positive determinant. In order to prove the validity of lower semicontinuity arguments in this setting, we show that the Burkholder functionals satisfy a sharp extension of the classical function theoretic area formula. As a corollary, in addition to functionals in geometric function theory, one finds new classes of non-polyconvex functionals, degenerating as the determinant vanishes, for which there is existence of minimizers. - 1 January 2024
- Published by American Institute of Mathematical Sciences (AIMS) in Communications on Pure and Applied Analysis
- Vol. 23 (10) , 1608-1645
Lower semicontinuity, Stoilow factorization and principal maps
We consider a refinement of the usual quasiconvexity condition of Morrey in two dimensions that allows us to prove lower semicontinuity and existence of minimizers for a class of functionals which are unbounded as the determinant vanishes and are non-polyconvex in general. This notion, that we call principal quasiconvexity, arose from the planar theory of quasiconformal mappings and mappings of finite distortion. We compare it with other quasiconvexity conditions that have appeared in the literature and provide a number of concrete examples of principally quasiconvex functionals that are not polyconvex. We also describe local conditions which combined with quasiconvexity yield principal quasiconvexity. The Stoilow factorization, that in the context of maps of integrable distortion was developed by Iwaniec and Šverák, plays a prominent role in our approach.
- 1 January 2024
- Published by American Institute of Mathematical Sciences (AIMS) in Communications on Pure and Applied Analysis
- Vol. 23 (10) , 1608-1645
Lower semicontinuity, Stoilow factorization and principal maps
We consider a refinement of the usual quasiconvexity condition of Morrey in two dimensions that allows us to prove lower semicontinuity and existence of minimizers for a class of functionals which are unbounded as the determinant vanishes and are non-polyconvex in general. This notion, that we call principal quasiconvexity, arose from the planar theory of quasiconformal mappings and mappings of finite distortion. We compare it with other quasiconvexity conditions that have appeared in the literature and provide a number of concrete examples of principally quasiconvex functionals that are not polyconvex. We also describe local conditions which combined with quasiconvexity yield principal quasiconvexity. The Stoilow factorization, that in the context of maps of integrable distortion was developed by Iwaniec and Šverák, plays a prominent role in our approach.
- Open Access
Distortion of Hausdorff measures under Orlicz--Sobolev maps
A comprehensive theory of the effect of Orlicz-Sobolev maps, between Euclidean spaces, on subsets with zero or finite Hausdorff measure is offered. Arbitrary Orlicz-Sobolev spaces embedded into the space of continuous function and Hausdorff measures built upon general gauge functions are included in our discussion. An explicit formula for the distortion of the relevant gauge function under the action of these maps is exhibited in terms of the Young function defining the Orlicz-Sobolev space. New phenomena and features, related to the flexibility in the definition of the degree of integrability of weak derivatives of maps and in the notion of measure of sets, are detected. Classical results, dealing with standard Sobolev spaces and Hausdorff measures, are recovered, and their optimality is shown to hold in a refined stronger sense. Special instances available in the literature, concerning Young functions and gauge functions of non-power type, are also reproduced and, when not sharp, improved.
Open AccessDistortion of Hausdorff measures under Orlicz--Sobolev maps
A comprehensive theory of the effect of Orlicz-Sobolev maps, between Euclidean spaces, on subsets with zero or finite Hausdorff measure is offered. Arbitrary Orlicz-Sobolev spaces embedded into the space of continuous function and Hausdorff measures built upon general gauge functions are included in our discussion. An explicit formula for the distortion of the relevant gauge function under the action of these maps is exhibited in terms of the Young function defining the Orlicz-Sobolev space. New phenomena and features, related to the flexibility in the definition of the degree of integrability of weak derivatives of maps and in the notion of measure of sets, are detected. Classical results, dealing with standard Sobolev spaces and Hausdorff measures, are recovered, and their optimality is shown to hold in a refined stronger sense. Special instances available in the literature, concerning Young functions and gauge functions of non-power type, are also reproduced and, when not sharp, improved.
- Open Access
$\mathcal{A}$-harmonic approximation and partial regularity, revisited
We give a direct harmonic approximation lemma for local minima of quasiconvex multiple integrals that entails their $\mathrm{C}^{1,\alpha}$ or $\mathrm{C}^{\infty}$-partial regularity. Different from previous contributions, the method is fully direct and elementary, only hinging on the $\mathrm{L}^{p}$-theory for strongly elliptic linear systems and Sobolev's embedding theorem. Especially, no heavier tools such as Lipschitz truncations are required.Open Access$\mathcal{A}$-harmonic approximation and partial regularity, revisited
We give a direct harmonic approximation lemma for local minima of quasiconvex multiple integrals that entails their $\mathrm{C}^{1,\alpha}$ or $\mathrm{C}^{\infty}$-partial regularity. Different from previous contributions, the method is fully direct and elementary, only hinging on the $\mathrm{L}^{p}$-theory for strongly elliptic linear systems and Sobolev's embedding theorem. Especially, no heavier tools such as Lipschitz truncations are required. - Open Access
Quasiconvex functionals of $(p,q)$-growth and the partial regularity of relaxed minimizers
We establish $\mathrm{C}^{\infty}$-partial regularity results for relaxed minimizers of strongly quasiconvex functionals \begin{align*} \mathscr{F}[u;\Omega]:=\int_{\Omega}F(\nabla u)\,\mathrm{d} x,\qquad u\colon\Omega\to\mathbb{R}^{N}, \end{align*} subject to a $q$-growth condition $|F(z)|\leq c(1+|z|^{q})$, $z\in\mathbb{R}^{N\times n}$, and natural $p$-mean coercivity conditions on $F\in\mathrm{C}^{\infty}(\mathbb{R}^{N\times n})$ for the basically optimal exponent range $1\leq p\leq q<\min\{\frac{np}{n-1},p+1\}$. With the $p$-mean coercivity condition being stated in terms of a strong quasiconvexity condition on $F$, our results include pointwise $(p,q)$-growth conditions as special cases. Moreover, we directly allow for signed integrands which is natural in view of coercivity considerations and hence the direct method, but is novel in the study of relaxed problems. In the particular case of classical pointwise $(p,q)$-growth conditions, our results extend the previously known exponent range from Schmidt's foundational work (Arch. Ration. Mech. Anal. 193, 311-337 (2009)) for non-negative integrands to the maximal range for which relaxations are meaningful, moreover allowing for $p=1$. As further key novelties, our results apply to the canonical class of signed integrands and do not rely in any way on measure representations \`{a} la Fonseca & Mal\'{y} (Ann. Inst. Henri Poincar\'{e}, Anal. Non Lin\'{e}aire 14, 309-338 (1997)).Open AccessQuasiconvex functionals of $(p,q)$-growth and the partial regularity of relaxed minimizers
We establish $\mathrm{C}^{\infty}$-partial regularity results for relaxed minimizers of strongly quasiconvex functionals \begin{align*} \mathscr{F}[u;\Omega]:=\int_{\Omega}F(\nabla u)\,\mathrm{d} x,\qquad u\colon\Omega\to\mathbb{R}^{N}, \end{align*} subject to a $q$-growth condition $|F(z)|\leq c(1+|z|^{q})$, $z\in\mathbb{R}^{N\times n}$, and natural $p$-mean coercivity conditions on $F\in\mathrm{C}^{\infty}(\mathbb{R}^{N\times n})$ for the basically optimal exponent range $1\leq p\leq q<\min\{\frac{np}{n-1},p+1\}$. With the $p$-mean coercivity condition being stated in terms of a strong quasiconvexity condition on $F$, our results include pointwise $(p,q)$-growth conditions as special cases. Moreover, we directly allow for signed integrands which is natural in view of coercivity considerations and hence the direct method, but is novel in the study of relaxed problems. In the particular case of classical pointwise $(p,q)$-growth conditions, our results extend the previously known exponent range from Schmidt's foundational work (Arch. Ration. Mech. Anal. 193, 311-337 (2009)) for non-negative integrands to the maximal range for which relaxations are meaningful, moreover allowing for $p=1$. As further key novelties, our results apply to the canonical class of signed integrands and do not rely in any way on measure representations \`{a} la Fonseca & Mal\'{y} (Ann. Inst. Henri Poincar\'{e}, Anal. Non Lin\'{e}aire 14, 309-338 (1997)). - 22 May 2022
- Published by Springer Nature in Archive for Rational Mechanics and Analysis
- Vol. 245 (1) , 479-500
Automatic Quasiconvexity of Homogeneous Isotropic Rank-One Convex Integrands
- 22 May 2022
- Published by Springer Nature in Archive for Rational Mechanics and Analysis
- Vol. 245 (1) , 479-500
Automatic Quasiconvexity of Homogeneous Isotropic Rank-One Convex Integrands
- 1 September 2021
- Published by Elsevier in Journal of Mathematical Analysis and Applications
- Vol. 501 (1)
A trace preserving operator and applications
- 1 September 2021
- Published by Elsevier in Journal of Mathematical Analysis and Applications
- Vol. 501 (1)
A trace preserving operator and applications
- Open Access
A bridge between Dubovitskii - Federer theorems and the coarea formula
The Morse-Sard theorem requires that a mapping $v:R^n \to R^m$ is of class $C^k$, $k>n-m$. In 1957 Dubovitski\u{\i} generalized this result by proving that almost all level sets for a $C^k$ mapping have $H^s$-negligible intersection with its critical set, where $s=\max(n-m-k+1,0)$. Here the critical set, or $m$-critical set is defined as $Z_{v,m} = \{ x \in R^n : {\rm rank} \nabla v(x) < m \}$. Another generalization was obtained independently by Dubovitski\u{\i} and Federer in 1966, namely for $C^k$ mappings $v:R^n\to R^d$ and integers $m\le d$ they proved that the set of $m$-critical values $v(Z_{v,m})$ is $H^{b}$-negligible for $b= m-1+\frac{n-m+1}{k}$. They also established the sharpness of these results within the $C^k$ category. Here we prove that Dubovitski\u{\i}'s theorem can be generalized to the case of continuous mappings of the Sobolev-Lorentz class $W^{k}_{p,1}(R^n,R^d )$, $p=\frac{n}k$ (this is the minimal integrability assumption that guarantees the continuity of mappings). In this situation the mappings need not be everywhere differentiable and in order to handle the set of nondifferentiability points, we establish for such mappings an analog of the Luzin $N$-property with respect to lower dimensional Hausdorff content. Finally, we formulate and prove a~${\rm bridge\ theorem}$ that includes all the above results as particular cases. This result is new also for smooth mappings but is presented here in the general Sobolev context. The proofs of the results are based on our previous joint papers with J.~Bourgain (2013, 2015). Note, that in this paper some result concerning the Coarea formula was not formulated accurately. Now we put an Addendum consisting of three parts: first, we describe the accurate formulation of this result, then we give some historical remarks, and finally its relation to other results of the paper.Open AccessA bridge between Dubovitskii - Federer theorems and the coarea formula
The Morse-Sard theorem requires that a mapping $v:R^n \to R^m$ is of class $C^k$, $k>n-m$. In 1957 Dubovitski\u{\i} generalized this result by proving that almost all level sets for a $C^k$ mapping have $H^s$-negligible intersection with its critical set, where $s=\max(n-m-k+1,0)$. Here the critical set, or $m$-critical set is defined as $Z_{v,m} = \{ x \in R^n : {\rm rank} \nabla v(x) < m \}$. Another generalization was obtained independently by Dubovitski\u{\i} and Federer in 1966, namely for $C^k$ mappings $v:R^n\to R^d$ and integers $m\le d$ they proved that the set of $m$-critical values $v(Z_{v,m})$ is $H^{b}$-negligible for $b= m-1+\frac{n-m+1}{k}$. They also established the sharpness of these results within the $C^k$ category. Here we prove that Dubovitski\u{\i}'s theorem can be generalized to the case of continuous mappings of the Sobolev-Lorentz class $W^{k}_{p,1}(R^n,R^d )$, $p=\frac{n}k$ (this is the minimal integrability assumption that guarantees the continuity of mappings). In this situation the mappings need not be everywhere differentiable and in order to handle the set of nondifferentiability points, we establish for such mappings an analog of the Luzin $N$-property with respect to lower dimensional Hausdorff content. Finally, we formulate and prove a~${\rm bridge\ theorem}$ that includes all the above results as particular cases. This result is new also for smooth mappings but is presented here in the general Sobolev context. The proofs of the results are based on our previous joint papers with J.~Bourgain (2013, 2015). Note, that in this paper some result concerning the Coarea formula was not formulated accurately. Now we put an Addendum consisting of three parts: first, we describe the accurate formulation of this result, then we give some historical remarks, and finally its relation to other results of the paper.