The Singular Set of Minima of Multiple Integrals
The Hausdorff dimension of the singular set of local minimizers of quasiconvex integrals is strictly less than the ambient dimension.
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arXiv subjects
Jan Kristensen
Publications and source records attributed to Jan Kristensen.
Lower semicontinuity, Stoilow factorization and principal maps
We consider a strengthening of the usual quasiconvexity condition of Morrey in two dimensions, which allows us to prove lower semicontinuity for functionals which are unbounded as the determinant vanishes. 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. The Stoilow factorization, that in the context of maps of integrable distortion was developed by Iwaniec and \v{S}ver\'ak, plays a prominent role in our approach.
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Quantified Legendreness and the regularity of minima
We introduce a new quantification of nonuniform ellipticity in variational problems via convex duality, and prove higher differentiability and $2d$-smoothness results for vector valued minimizers of possibly degenerate functionals. Our framework covers convex, anisotropic polynomials as prototypical model examples - in particular, we improve in an essentially optimal fashion Marcellini's original results \cite{ma1}.
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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.
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$\mathcal{A}$-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.
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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)).
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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.
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On the validity of the Euler-Lagrange system without growth assumptions
The constrained minimisers of convex integral functionals of the form $\mathscr F(v)=\int_\Omega F(\nabla^k v(x))\mathrm d x $ defined on Sobolev mappings $v\in \mathrm W^{k,1}_g(\Omega , \mathbb R^N )\cap K$, where $K$ is a closed convex subset of the Dirichlet class $\mathrm W^{k,1}_{g}(\Omega , \mathbb R^N ),$ are characterised as the energy solutions to the Euler-Lagrange inequality for $\mathscr F$. We assume that the essentially smooth integrand $F\colon \mathbb R^{N} \otimes \odot^{k}\mathbb R^{n} \to \mathbb R\cup\{+\infty\}$ is convex, lower semi-continuous, proper and at least super-linear at infinity. In the unconstrained case $K=\mathrm W^{k,1}_{g}(\Omega , \mathbb R^N )$, if the integrand $F$ is convex, real-valued, and satisfies a demi-coercivity condition, then $$ \int_{\Omega} \! F^{\prime}(\nabla^{k} u) \cdot \nabla^{k}\phi \, \mathrm d x =0 $$ holds for all $\phi \in \mathrm W_{0}^{k}( \Omega , \mathbb R^{N})$, where $\nabla^{k} u$ is the absolutely continuous part of the vector measure $D^{k}u$.
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Concentration effects of BV gradients have gradient structure
We prove that the concentration effects arising from weakly-* convergent sequences of gradients of maps of bounded variation have gradient structure. This is in stark contrast with the corresponding oscillation phenomena.
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Automatic quasiconvexity of homogeneous isotropic rank-one convex integrands
We consider the class of non-negative rank-one convex isotropic integrands on $\mathbb{R}^{n\times n}$ which are also positively $p$-homogeneous. If $p \leq n = 2$ we prove, conditional on the quasiconvexity of the Burkholder integrand, that the integrands in this class are quasiconvex at conformal matrices. If $p \geq n = 2$, we show that the positive part of the Burkholder integrand is polyconvex. In general, for $p \geq n$, we prove that the integrands in the above class are polyconvex at conformal matrices. Several examples imply that our results are all nearly optimal.
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Oscillation and concentration in sequences of PDE constrained measures
We show that for constant rank partial differential operators $\mathscr{A}$, generalized Young measures generated by sequences of $\mathscr{A}$-free measures can be characterized by duality with $\mathscr{A}$-quasiconvex integrands of linear growth.
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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\uı 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\uı 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\uı'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.
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Regularity for higher order quasiconvex problems with linear growth from below
We announce new existence and $\varepsilon$-regularity results for minimisers of the relaxation of strongly quasiconvex integrals that on smooth maps $u\colonΩ\subset\mathbb{R}^{n}\to\mathbb{R}^{N}$ are defined by $$u\mapsto \int_ΩF(\nabla^{k}u)dx.$$ The results cover the case of integrands $F$ with $(1,q)$-growth in the full range of exponents $1<q<\frac{n}{n-1}$ for which a measure representation of the relaxed functional is possible and the minimizers belong to the space $BV^k$ of maps whose $k$-th order derivatives are measures.
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Partial Regularity for BV Minimizers
We establish an $\varepsilon$-regularity result for the derivative of a map of bounded variation that minimizes a strongly quasiconvex variational integral of linear growth, and, as a consequence, the partial regularity of such BV minimizers. This result extends the regularity theory for minimizers of quasiconvex integrals on Sobolev spaces to the context of maps of bounded variation. Previous partial regularity results for BV minimizers in the linear growth set-up were confined to the convex situation.
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Sobolev regularity for Convex Functionals on BD
We study Sobolev regularity results for minimisers of autonomous, convex variational of linear growth which depend on the symmetric gradient rather than the full gradient. This extends the results available in the literature for the BV-setting to the case of functionals whose full gradients are a priori not known to exist as matrix-valued Radon measures.
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Lower semicontinuity for an integral functional in BV
We prove a lower semicontinuity result for a functional of linear growth initially defined by \[ \int_ΩF\left(\frac{dDu}{dμ}\right)\,dμ\] for $u\in BV(Ω;\mathbb{R}^N)$ with $Du\ll μ$. The positive Radon measure $μ$ is only assumed to satisfy $\mathcal L^n\ll μ$.
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On the Morse-Sard property and level sets of $W^{n,1}$ Sobolev functions on ${\mathbb R}^n$
We establish Luzin N and Morse--Sard properties for functions from the Sobolev space $W^{n,1}({\mathbb R}^{n})$. Using these results we prove that almost all level sets are finite disjoint unions of $C^1$--smooth compact manifolds of dimension $n-1$. These results remain valid also within the larger space of functions of bounded variation $BV_{n}({\mathbb R}^{n})$. For the proofs we establish and use some new results on Luzin--type approximation of Sobolev and BV--functions by $C^k$--functions, where the exceptional sets have small Hausdorff content.
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On Rank-One Convex Functions that are homogeneous of Degree One
We show that positively $1$--homogeneous rank one convex functions are convex at $0$ and at matrices of rank one. The result is a special case of an abstract convexity result that we establish for positively $1$--homogeneous directionally convex functions defined on an open convex cone in a finite dimensional vector space. From these results we derive a number of consequences including various generalizations of the Ornstein $\LL^1$ non inequalities. Most of the results were announced in ({\em C.~R.~Acad.~Sci.~Paris, Ser.~I 349 (2011), 407--409}).
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