In this paper we revisit our previous study of the local theory of prescribed Jacobian equations associated with generating functions, which are extensions of cost functions in the theory of optimal transportation. In particular, as foreshadowed in the earlier work, we provide details pertaining to the relaxation of a monotonicity condition A4w in the underlying convexity theory and the consequent classical regularity. Taking advantage of recent work of Kitagawa and Guillen, we also extend our classical regularity theory to the weak form A3w of the critical matrix convexity conditions.
We prove that immersions of planar domains are uniquely specified by their Jacobian determinant, curl function, and boundary values. This settles the two-dimensional version of an outstanding conjecture related to a particular grid generation method in computer graphics.
We develop the fundamentals of a local regularity theory for prescribed Jacobian equations which extend the corresponding results for optimal transportation equations. In this theory the cost function is extended to a generating function through dependence on an additional scalar variable. In particular we recover in this generality the local regularity theory for potentials of Ma,Trudinger and Wang, along with the subsequent development of the underlying convexity theory.
We prove that two Aleksandrov solutions of a generated prescribed Jacobian equation have the same gradients at points where they are both differentiable. For the optimal transportation case where two solutions can be translated to agree at a point without changing the $g$-subdifferential at that point, we recover the uniqueness up to a constant of solutions. For the general case, our result is a new proof with less regularity assumptions of a key theorem recently used to prove the uniqueness of solutions.
We study the symmetry and uniqueness of maps which minimise the $np$-Dirichlet energy, under the constraint that their Jacobian is a given radially symmetric function $f$. We find a condition on $f$ which ensures that the minimisers are symmetric and unique. In the absence of this condition we construct an explicit $f$ for which there are uncountably many distinct energy minimisers, none of which is symmetric. Even if we prescribe the maps to be the identity on the boundary of a ball we show that the minimisers need not be symmetric. This gives a negative answer to a question of Hélein (Ann. I…
We construct planar bi-Sobolev mappings whose local volume distortion is bounded from below by a given function $f\in L^p$ with $p>1$, i.e. bi-Sobolev solutions for the prescribed Jacobian inequality in the plane for right-hand sides $f\in L^p$. More precisely, for any $1<q<(p+1)/2$ we construct a solution which belongs to $W^{1,q}$ and which preserves the boundary pointwise. For bounded right-hand sides $f\in L^{\infty}$, we provide bi-Lipschitz solutions. The basic building block of our construction are Lipschitz maps which stretch a given compact subset of the unit square by a given factor …
This work presents a general principle, in the spirit of convex integration, leading to a method for the characterization of Young measures generated by gradients of maps in $W^{1,p}$ with $p$ less than the space dimension, whose Jacobian determinant is subjected to a range of constraints. Two special cases are particularly important in the theories of elasticity and fluid dynamics: (a) the generating gradients have positive Jacobians that are uniformly bounded away from zero and (b) the underlying deformations are incompressible, corresponding to their Jacobian determinants being constantly o…
In Dacorogna-Moser theorem on the pullback equation $φ^* (g)=f$ between two prescribed volume forms (with the same total volume), control of support of the solutions can be obtained from that of the initial data, while keeping optimal regularity. This result answers a problem implicitly raised on page 14 of Dacorogna-Moser's original article ("On a partial differential equation involving the Jacobian determinant", Ann. Inst. H. Poincaré Anal. Non Linéaire 7 (1990), 1-26), and fully generalizes the solution to the particular case of $g\equiv 1$ (prescribed Jacobian PDE, $\text{det}\,\nablaφ=f$)…
Variational Principle (VP) forms diffeomorphisms with prescribed Jacobian determinant (JD) and curl. Examples demonstrate that, (i) JD alone can not uniquely determine a diffeomorphism without curl; and (ii) the solutions by VP seem to satisfy properties of a Lie group. Hence, it is conjectured that a unique diffeomorphism can be assured by its JD and curl (Uniqueness Conjecture). In this paper, (1) an observation based on VP is derived that a counter example to the Conjecture, if exists, should satisfy a particular property; (2) from the observation, an experimental strategy is formulated to …
Let $Ω\subset \mathbb{R}^n$ be open and let $\mathcal{R}$ be a partial frame on $Ω$, that is a set of $m$ linearly independent vector fields prescribed on $Ω$ ($m\leq n$). We consider the issue of describing the set of all maps $F:Ω\to\mathbb{R}^n$ with the property that each of the given vector fields is an eigenvector of the Jacobian matrix of $F$. By introducing a coordinate independent definition of the Jacobian, we obtain an intrinsic formulation of the problem, which leads to an overdetermined PDE system, whose compatibility conditions can be expressed in an intrinsic, coordinate indepen…
We study existence and regularity of solutions to the Dirichlet problem for the prescribed Jacobian equation, $\det Du = f$, where $f$ is integrable and bounded away from zero. In particular, we take $f\in L^p$, where $p > 1$, or in $L\log L$. We prove that for a Baire-generic $f$ in either space there are no solutions with the expected regularity.
The original proof of Dacorogna-Moser theorem on the prescribed Jacobian PDE, $\text{det}\,\nablaφ=f$, can be modified in order to obtain control of support of the solutions from that of the initial data, while keeping optimal regularity. Briefly, under the usual conditions, a solution diffeomorphism $φ$ satisfying \[ \text{supp}(f-1)\subset\varOmega\Longrightarrow\text{supp}(φ-\text{id})\subset\varOmega \] can be found and $φ$ is still of class $C^{r+1,α}$ if $f$ is $C^{r,α}$, the domain of $f$ being a bounded connected open $C^{r+2,α}$ set $\varOmega\subset\mathbb{R}^{n}$.
In this paper we study the near-field refractor problem with point source at the origin and prescribed target on the given receiver surface $Σ$. This nonvariational problem can be studied in the framework of prescribed Jacobian equations. We construct the corresponding generating function and show that the Aleksandrov and the Brenier type solutions are equivalent. Our main result establishes local smoothness of Aleksandrov's solutions when the data is smooth and when the medium containing the source has smaller refractive index than the medium containing the target. This is done by deriving th…
We study the problem of constructing systems of hyperbolic conservation laws in one space dimension with prescribed eigencurves, i.e. the eigenvector fields of the Jacobian of the flux are given. We formulate this as a typically overdetermined system of equations for the eigenvalues-to-be. Equivalent formulations in terms of differential and algebraic-differential equations are considered. The resulting equations are then analyzed using appropriate integrability theorems (Frobenius, Darboux and Cartan-Kahler). We give a complete analysis of the possible scenarios, including examples, for syste…
I discuss the prescribed Jacobian equation $Ju=\det\nabla u=f$ for an unknown vector-function $u$, and the connection of this problem to the boundedness of commutators of multiplication operators with singular integrals in general, and with the Beurling operator in particular. A conjecture of T. Iwaniec regarding the solvability for general datum $f\in L^p(R^d)$ remains open, but recent partial results in this direction will be presented. These are based on a complete characterisation of the $L^p$-to-$L^q$ boundedness of commutators, where the regime of exponents $p>q$, unexplored until recent…
In this paper, we prove the existence of classical solutions to second boundary value prob- lems for generated prescribed Jacobian equations, as recently developed by the second author, thereby obtaining extensions of classical solvability of optimal transportation problems to problems arising in near field geometric optics. Our results depend in particular on a priori second derivative estimates recently established by the authors under weak co-dimension one convexity hypotheses on the associated matrix functions with respect to the gradient variables, (A3w). We also avoid domain deformations…
