Geometric function theory, inverse problems and fluid dynamics

arXiv A well known result due to Carlson affirms that a power series with finite and positive radius of convergence R has no Ostrowski gaps if and only if the sequence of zeros of its nth sections is asymptotically equidistributed to {|z|=R}. Here we extend this characterization to those power series with null radius of convergence, modulo some necessary normalizations of the sequence of the sections of f.

arXiv We provide a reconstruction scheme for complex-valued potentials in \(H^s(\mathbb{R}^2)\) for \(s>0\). The procedure extends the method of Bukhgeim relying on quadratic exponential solutions. We also see how the new reconstruction formulas proposed improve the convergence on a set of examples.

arXiv We analyze the structure of the set of limiting Carleman weights in all conformally flat manifolds, 3-manifolds, and 4-manifolds. In particular we give a new proof of the classification of Euclidean limiting Carleman weights, and show that there are only three basic such weights up to the action of the conformal group. In dimension three we show that if the manifold is not conformally flat, there could be one or two limiting Carleman weights. We also characterize the metrics that have more than one limiting Carleman weight. In dimension four we obtain a complete spectrum of examples according to the structure of the Weyl tensor. In particular, we construct unimodular Lie groups whose Weyl or Cotton-York tensors have the symmetries of conformally transversally anisotropic manifolds, but which do not admit limiting Carleman weights.

arXiv We define a class of Sobolev \(W^{1,p} (\Omega, \mathbb{R}^n)\) functions, with \(p>n-1\), such that its trace on \(\partial \Omega\) is also Sobolev, and do not present cavitation in the interior or on the boundary. We show that if a function in this class has positive Jacobian and coincides on the boundary with an injective map, then the function is itself injective. We then prove the existence of minimizers within this class for the type of functionals that appear in nonlinear elasticity.

arXiv We prove Taylor's conjecture which says that in 3D MHD, magnetic helicity is conserved in the ideal limit in bounded, simply connected, perfectly conducting domains. When the domain is multiply connected, magnetic helicity depends on the vector potential of the magnetic field. In that setting we show that magnetic helicity is conserved for a large and natural class of vector potentials but not in general for all vector potentials. As an analogue of Taylor's conjecture in 2D, we show that mean square magnetic potential is conserved in the ideal limit, even in multiply connected domains.

arXiv We present an algorithm to exactly calculate the \(\mathbb{R}^2 \oplus \mathbb{R}\)-separately convex hull of a finite set of points in \(\mathbb{R}^3\), as introduced in [27]. When \(\mathbb{R}^3\) is considered as certain subset of \)3 \times 2\) matrices, this algorithm calculates the rank-one convex hull. If \)\mathbb{R}^3\) is identified instead with a subset of \(2 \times 3\) matrices, the algorithm is actually calculating the quasiconvex hull, due to a recent result by [24]. The algorithm combines outer approximations based in the locality theorem [29, 4.7] with inner approximations to \(2 + 1\) convexity based on “\((2 + 1)\)-complexes”. The departing point is an outer approximation and by iteratively chopping off “D-prisms”, we prove that an inner approximation to the rank-one convex hull is reached.

arXiv We prove the existence of infinitely many mixing solutions for the Muskat problem in the fully unstable regime displaying a linearly degraded macroscopic behaviour inside the mixing zone. In fact, we estimate the volume proportion of each fluid in every rectangle of the mixing zone. The proof is a refined version of the convex integration scheme presented in [DS10, Sze12] applied to the subsolution in [CCF16]. More generally, we obtain a quantitative h-principle for a class of evolution equations which shows that, in terms of weak*-continuous quantities, a generic solution in a suitable metric space essentially behaves like the subsolution. This applies of course to linear quantities, and in the case of IPM to the power balance \(\mathbf{P}\) (14) which is quadratic. As further applications of such quantitative h-principle we discuss the case of vortex sheet for the incompressible Euler equations.

arXiv We show that ideal 2D MHD does not possess weak solutions (or even subsolutions) with compact support in time and non-trivial magnetic field. We also show that the \(\Lambda\)-convex hull of ideal MHD has empty interior in both 2D and 3D; this is seen by finding suitable \(\Lambda\)-convex functions. As a consequence we show that mean-square magnetic potential is conserved in 2D by subsolutions and weak limits of solutions in the physically natural energy space \(L^\infty_t L^2_x\), and in 3D we show the conservation of magnetic helicity by \(L^3\)-integrable subsolutions and weak limits of solutions. However, in 3D the \(\Lambda\)-convex hull is shown to be large enough that nontrivial smooth, compactly supported strict subsolutions exist.

arXiv We consider the inverse impedance tomography problem in the plane. Using Bukhgeim’s scattering data for the Dirac problem, we prove that the conductivity is uniquely determined by the Dirichlet-to-Neuman map.

Preprint We study nonlocal variational problems in \(L^p\), like those that appear in peridynamics. The functional object of our study is given by a double integral. We establish characterizations of weak lower semicontinuity of the functional in terms of nonlocal versions of either a convexity notion of the integrand, or a Jensen inequality for Young measures. Existence results, obtained through the direct method of the Calculus of variations, are also established. We cover different boundary conditions, for which the coercivity is obtained from nonlocal Poincaré inequalities. Finally, we analyze the relaxation (that is, the computation of the lower semicontinuous envelope) for this problem when the lower semicontinuity fails. We state a general relaxation result in terms of Young measures and show, by means of two examples, the difficulty of having a relaxation in \(L^p\) in an integral form. At the root of this difficulty lies the fact that, contrary to what happens for local functionals, non-positive integrands may give rise to positive nonlocal functionals.

Preprint We show examples of 2D incompressible isotropic homogeneous hyperelastic materials with a polyconvex stored-energy function that present necking. The construction of the stored-energy function of a material satisfying all those properties requires a fine search. We used the software Algencan to perform numerical experiments and visualize necking for the examples constructed. The algorithm is based on minimization of the elastic energy under the nonconvex constraint of incompressibility.

Preprint Comput. Methods Appl. Mech. Engrg. We present a numerical implementation of a model for void coalescence and fracture in nonlinear elasticity. The model is similar to the Ambrosio-Tortorelli regularization of the standard free-discontinuity variational model for quasistatic brittle fracture. The main change is the introduction of a nonlinear polyconvex energy that allows for cavitation. This change requires new analytic and numerical techniques. We propose a numerical method based on alternating directional minimisation and a stabilized Crouzeix-Raviart finite element discretization. The method is used in several experiments, including void coalescence, void creation under tensile stress, failure in perfect materials and in materials with hard inclusions. The experimental results show the ability of the model and the numerical method to study different failure mechanisms in rubber-like materials.

Preprint Quart. J. Math. In this note we prove that given a continuous Sobolev \( W^{1,p} \) deformation \(f \), with \(1 < p < \infty\), from a planar domain to \(\mathbb{R}^2\) which is injective almost everywhere, we can find a sequence \(f_k\) of diffeomorphisms with \(f_k - f \in W^{1,p}_0 \) such that \(f_k \to f\) uniformly and in the Sobolev norm.

Preprint Calc. Var. Peridynamics is a nonlocal model in Continuum Mechanics, and in particular Elasticity, introduced by Silling (2000). The nonlocality is reflected in the fact that points at a finite distance exert a force upon each other. If, however, those points are more distant than a characteristic length called horizon, it is customary to assume that they do not interact. We work in the variational approach of time-independent deformations, according to which, their energy is expressed as a double integral that does not involve gradients. We prove that the \( \Gamma \)-limit of this model, as the horizon tends to zero, is the classical model of hyperelasticity. We pay special attention to how the passage from the density of the non-local model to its local counterpart takes place.

Preprint J. Funct. Anal. Let \(\mathbf{u}\) be a Sobolev \(W^{1,p}\) map from a bounded open set \(\Omega \subset \mathbb{R}^n \) to \(\mathbb{R}^n \). We assume \(\mathbf{u}\) to satisfy some invertibility properties that are natural in the context of nonlinear elasticity, namely, the topological condition INV and the orientation-preserving constraint \(\det D \mathbf{u} > 0\). These deformations may present cavitation, which is the phenomenom of void formation. We also assume that the surface created by the cavitation process has finite area. If \(p > n-1\), we show that a suitable defined inverse of \(\mathbf{u}\) is a Sobolev map. A partial result is also given for the critical case \(p=n-1\). The proof relies on the techniques used in the study of cavitation.

arXiv (to appear in J. Differential Equations) We find a complete characterization for sets of isotropic conductivities with stable recovery in the \( L^2 \) norm when the data of the Calderón Inverse Conductivity Problem is obtained in the boundary of a disk and the conductivities are constant in a neighborhood of its boundary. To obtain this result, we present minimal a priori assumptions which turn out to be sufficient for sets of conductivities to have stable recovery in a bounded and rough domain. The condition is presented in terms of the integral moduli of continuity of the coefficients involved and their ellipticity bound.

arXiv (to appear in SIMA) We show that complex-valued potentials with jump discontinuities can be recovered from the Dirichlet-to-Neumann map using Bukhgeim's method. We also provide conditional stability estimates for reconstruction given an approximate knowledge of the location of the discontinuities. Combining with known formulas, this enables stable recovery of real-valued potentials from the scattering amplitude at a fixed energy.

Rend. Istit. Mat. Univ. Trieste The paper concerns about the asymptotic behaviour of solutions of initial boundary value problem for a general anisotropic viscoelastic system in the form of integrodifferential sytem of equations with homogeneous mixed boundary condition. We put a usual assumption on the relaxation tensor and assume that the inhomogeneous term of the equation and boundary data are zero. Then, by using the energy method, we show that the solutions decays exponentially with respect to time as it tends to infinity.

arXiv JMAA Let \( \phi \) be a quasiconformal mapping, and let \( T_\phi \) be the composition operator which maps \( f \) to \( f\circ\phi \). Since \( \phi \) may not be bi-Lipschitz, the composition operator need not map Sobolev spaces to themselves. The study begins with the behavior of \( T_\phi\) on \(L^p \) and \(W^{1,p} \) for \(1 < p < \infty \). This cases are well understood but alternative proofs of some known results are provided. Using interpolation techniques it is seen that compactly supported Bessel potential functions in \(H^{s,p} \) are sent to \(H^{s,q} \) whenever \(0 < s < 1 \) for appropriate values of \(q \). The techniques used lead to sharp results and they can be applied to Besov spaces as well.

arXiv Arch. Rational Mech. Anal. We make progress on a problem of R. Coifman, P.-L. Lions, Y. Meyer, and S. Semmes from 1993 by showing that the Jacobian operator \(J\) does not map \(W^{1,n}(\mathbb R^n,\mathbb R^n)\) onto the Hardy space \(\mathcal{H}^1(\mathbb R^n)\) for any \(n \ge 2\). The related question about surjectivity of \(J \colon \dot{W}^{1,n}(\mathbb R^n,\mathbb R^n) \to \mathcal{H}^1(\mathbb R^n)\) is still open.
The second main result and its variants reduce the proof of \(\mathcal{H}^1\) regularity of a large class of compensated compactness quantities to an integration by parts or easy arithmetic, and applications are presented. Furthermore, we exhibit a class of nonlinear partial differential operators in which weak sequential continuity is a strictly stronger condition than \(\mathcal{H}^1\) regularity, shedding light on another problem of Coifman, Lions, Meyer, and Semmes.

arXiv Calc. Var. Let \(\Omega\subset \mathbb{R}^{n}\) be a bounded open set. Given \(1\leq m_1,m_2\leq n-2\), we construct a homeomorphism \(f:\Omega\to \Omega\) that is Hölder continuous, \(f\) is the identity on \(\partial \Omega\), the derivative \(D f\) has rank \(m_1\) a.e. in \(\Omega\), the derivative \(D f^{-1}\) of the inverse has rank \(m_2\) a.e. in \(\Omega\), \(Df\in W^{1,p}\) and \(Df^{-1}\in W^{1,q}\) for \(p<\min\{m_1+1,n-m_2\}\), \(q<\min\{m_2+1,n-m_1\}\). The proof is based on convex integration and laminates. We also show that the integrability of the function and the inverse is sharp.

arXiv (to appear in Commun. Pur. Appl. Anal.) Some new results regarding the Sobolev regularity of the principal solution of the linear Beltrami equation \(\bar{\partial} f = \mu \partial f + \nu \bar{\partial f}\) for discontinuous Beltrami coefficients \(\mu\) and \(\nu\) are obtained, using Kato-Ponce commutators. A conjecture on the cases where the limitations of the method do not work is raised.

arXiv We prove the existence of mixing solutions of the incompressible porous media equation for all Muskat type \(H^5\) initial data in the fully unstable regime.

arXiv Forum of mathematics - Sigma In 1411.4887, we found some necessary conditions for a Riemannian manifold to admit a local limiting Carleman weight (LCW), based upon the Cotton-York tensor in dimension \(3\) and the Weyl tensor in dimension \(4\). In this paper, we find further necessary conditions for the existence of local LCWs that are often sufficient. For a manifold of dimension \(3\) or \(4\), we classify the possible Cotton-York, or Weyl tensors, and provide a mechanism to find out whether the manifold admits local LCW for each type of tensor. In particular, we show that a product of two surfaces admits a LCW if and only if at least one of the two surfaces is of revolution. This provides an example of a manifold satisfying the eigenflag condition of \cite{AFGR} but not admitting \(LCW\).

Preprint Adv. Calc. Var. Let \(\Omega\subset \mathbb{R}^{n}\) be a bounded open set. Given \(2\leq m\leq n\), we construct a convex function \(\phi:\Omega\to \mathbb{R}\) whose gradient \(f= \nabla \phi\) is a H\"older continuous homeomorphism, \(f\) is the identity on \(\partial \Omega\), the derivative \(D f\) has rank \(m-1\) a.e.\ in \(\Omega\) and \(D f\) is in the weak \(L^{m}\) space \(L^{m,w}\). The proof is based on convex integration and staircase laminates.

Preprint Ann. I. H. Poincare (C) We provide Schauder estimates for nonlinear Beltrami equations and lower bounds of the Jacobians for homeomorphic solutions. The results were announced in [1] but here we give detailed proofs.

Preprint Arch. Rational Mech. Anal. We define a class of deformations in \( W^{1,p}(\Omega,\mathbb{R}^n) \), \(p > n-1\), with positive Jacobian that do not exhibit cavitation. We characterize that class in terms of the non-negativity of the topological degree and the equality \( \operatorname{Det} = \det \) (that the distributional determinant coincides with the pointwise determinant of the gradient). Maps in this class are shown to satisfy a property of weak monotonicity, and, as a consequence, they enjoy an extra degree of regularity. We also prove that these deformations are locally invertible; moreover, the neighbourhood of invertibility is stable along a weak convergent sequence in \( W^{1,p} \), and the sequence of local inverses converges to the local inverse. We use those features to show weak lower semicontinuity of functionals defined in the deformed configuration and functionals involving composition of maps. We apply those results to prove existence of minimizers in some models for nematic elastomers and magnetoelasticity.

arXiv DCDS We present a new strategy for proving the Ambrose conjecture, a global version of the Cartan local lemma. We introduce the concepts of linking curves, unequivocal sets and sutured manifolds, and use them to show that any sutured manifold satisfies the Ambrose conjecture. We then prove that the set of sutured Riemmanian manifolds contains a residual set of the metrics on a given smooth manifold of dimension \(3\).

arXiv Inverse Problems and Imaging In the paper AFGR it is shown that the set of Riemannian metrics which do not admit global limiting Carleman weights is open and dense, by studying the conformally invariant Weyl and Cotton tensors. In the paper LS it is shown that the set of Riemannian metrics which do not admit local limiting Carleman weights at any point is residual, showing that it contains the set of metrics for which there are no local conformal diffeomorphisms between any distinct open subsets. This note is a continuation of AFGR, in order to prove that the set of Riemannian metrics which do not admit local limiting Carleman weights \emph{at any point} is open and dense.

arXiv In this paper we show that the homeomorphic solutions to each nonlinear Beltrami equation \(\partial_{\bar{z}}f = \mathcal{H}(z, \partial_z f)\) generate a two-dimensional manifold of quasiconformal mappings \({\mathcal{F}}_{\mathcal{H}}=\{\phi_a(z)\}_{a \in \mathbb{C}} \subset W^{1,2}_{\mathrm{loc}}(\mathbb C)\). The process is reversible. To each family of quasiconformal mappings \(\mathcal{F}\) we can associate a corresponding nonlinear Beltrami equation. Moreover, we show that there is an interplay between the regularity of \(\mathcal H\) with respect to both variables \((z,w)\) and the regularity of \(\mathcal{F}\) with respect to \((z,a)\). Under regularity assumptions the relation between \(\mathcal H\) and \(\mathcal H\) is one-to-one.

arXiv APDE We give a necessary condition for a Riemannian manifold to admit limiting Carleman weights in terms of its Weyl tensor (in dimensions 4 and higher), or its Cotton-York tensor in dimension 3. As an application we provide explicit examples of manifolds without limiting Carleman weights and show that the set of such metrics on a given manifold contains an open and dense set.

Preprint Nonlinearity We consider the \(\overline{\partial}\)-Dirac system that Ablowitz and Fokas used to transform the defocussing Davey--Stewartson system to a linear evolution equation. The nonlinear Plancherel identity for the associated scattering transform was established by Beals and Coifman for Schwartz functions. Sung extended the validity of the identity to functions belonging to \(L^1\cap L^\infty(\mathbb{R}^2)\) and Brown to \(L^2\)-functions with sufficiently small norm. More recently, Perry extended to the weighted Sobolev space \(H^{1,1}(\mathbb{R}^2)\) and here we extend to \(H^{s,s}(\mathbb{R}^2)\) with \(0 < s < 1\).

Preprint Arch. Rational Mech. Anal Our starting point is a variational model in nonlinear elasticity that allows for cavitation and fracture that was introduced by Henao and Mora-Corral (Arch. Rat. Mech. Anal. 197, 617--655, 2010). The total energy to minimize is the sum of the elastic energy plus the energy produced by crack and surface formation. It is a free discontinuity problem, since the crack set and the set of new surface are unknowns of the problem. The expression of the functional involves a volume integral and two surface integrals, and this fact makes the problem numerically intractable. In this paper we propose an approximation (in the sense of \(\Gamma\)-convergence) by functionals involving only volume integrals, which makes a numerical approximation by finite elements feasible. This approximation has some similarities to the Modica--Mortola approximation of the perimeter and the Ambrosio--Tortorelli approximation of the Mumford--Shah functional, but with the added difficulties typical of nonlinear elasticity, in which the deformation is assumed to be one-to-one and orientation-preserving.

arXiv Indiana Univ. Math. J. In this paper we derive a new formulation of the water waves equations with vorticity that generalizes the well-known Zalkarov-Craig-Sulem formulation used in the irrotational case. We prove the local well-posedness of this formulation, and show that it is formally Hamiltonian. This new formulation is cast in Eulerian variable, and in finite depth; we show that it can be used to provide uniform bounds on the lifespan and on the norms of the solutions in the singular shallow water regime. As an application to these results, we derive and provide the first rigorous justification of a shallow water model for water waves in presence of vorticity; we show in particular that a third equation must be added to the standard model to recover the velocity at the surface from the averaged velocity.

arXiv Arch. Rational Mech. Anal. This paper shows finite time singularity formation for the Muskat problem in a stable regime. The framework we exhibit is with a dry region, where the density and the viscosity are set equal to 0 (the gradient of the pressure is equal to \((0,0)\)) in the complement of the fluid domain. The singularity is a splash-type: a smooth fluid boundary collapses due to two different particles evolve to collide at a single point. This is the first example of a splash singularity for a parabolic problem.

arXiv (to appear in DCDS). Guided by numerical simulations, we present the proof of two results concerning the behaviour of SQG sharp fronts and \(\alpha\)-patches. We establish that ellipses are not rotational solutions and we prove that initially convex interfaces may lose this property in finite time.

arXiv (to appear in DCDS) In this paper we show a structural stability result for water waves. The main motivation for this result is that we would like to exhibit a water wave whose interface starts as a graph and ends in a splash. Numerical simulations lead to an approximate solution with the desired behaviour. The stability result will conclude that near the approximate solution to water waves there is an exact solution.

Preprint (to appear in Kokyuroku Bessatsu) We give explicit formulae relating the Dirichlet to Neuman map and the scattering amplitude in two dimensions.

arXiv J. Funct. Anal. We establish optimal conditions under which the G-convergence of linear elliptic operators implies the convergence of the corresponding Dirichlet to Neumann maps. As an application we show that the approximate cloaking isotropic materials from GKLU are independent of the source.

Preprint (to appear in Annales scientifiques de l’École Normale Supérieure) We reconstruct compactly supported potentials with only half a derivative in \(L^2\) from the scattering amplitude at a fixed energy. For this we draw a connection between the recently introduced method of Bukhgeim, which uniquely determined the potential from the Dirichlet-to-Neumann map, and a question of Carleson regarding the convergence to initial data of solutions to time-dependent Schrödinger equations. We also provide examples of compactly supported potentials, with s derivatives in \(L^2\) for any \(s < 1/2\), which cannot be recovered by these means. Thus the recovery method has a different threshold in terms of regularity than the corresponding uniqueness result.

preprint SIAM J. Math. Anal. We prove that the non smooth part of a non compactly supported potential q in the Schrödinger Hamiltonian \(-\Delta +q\), in dimension \(n=2\), is contained in its Born approximation \(q_B\) for backscattering data in a precise sense in terms of continuity: Given q in \(W^{\alpha,2}\) the difference q-q_B is in the H\"older class \(\Lambda^\beta\) for any \(\beta<\alpha\).

J. Geom. Anal. We prove that any countably piecewise affine homeomorphism from an open set of \(\mathbb{R}^2\) can be approximated, together with its inverse, by diffeomorphisms in the \(W^{1,p}\) and the \(L^{\infty}\) norms.

Preprint SIAM J. Math. Anal. We present an existence theory based on minimization of the nonlocal energies appearing in peridynamics, which is a nonlocal continuum model in Solid Mechanics that avoids the use of deformation gradients. We employ the direct method of the calculus of variations in order to find minimizers of the energy of a deformation. Lower semicontinuity is proved under a weaker condition than convexity, whereas coercivity is proved via a nonlocal Poincar\'e inequality. We cover Dirichlet, Neumann and mixed boundary conditions. The existence theory is set in the Lebesgue \(L^p\) spaces and in the fractional Sobolev \(W^{s,p}\) spaces, for \(0 < s < 1\) and \(1 < p < \infty\).

Preprint SIAM J. Math. Anal. We show the well-posedness of a free-discontinuity model in nonlinear elasticity allowing for cavitation. The model, based on global minimization, takes into account the non-interpenetration of matter and the inhomogeneties of the material. Then we prove the existence of a quasistatic evolution corresponding to this variational model that takes into account the irreversibility of the process of the cavity formation.