Baisheng Yan

On a new class of elastic deformations not allowing for cavitation

Abstract Let Ω ⊂ ℝ n be open and bounded and assume that u :Ω → ℝ n satisfies u ∈ W 1, p (Ω, ℝ n ), adj D u ∈ L q (Ω; ℝ n × n ) with p ≧ n − 1, q ≧ n n − 1 . We show that for g ∈ C 1 (ℝ n ; ℝ n ) with bounded gradient, one has the identity ∂ ∂ x j { ( g i ∘ u ) ( adj D u ) i j } = ( div g ) ∘ u det D u in the sense of distributions. As an application, we obtain existence results in nonlinear elasticity under weakened coercivity conditions. We also use the above identity to generalize Sverak’s ( cf . [Sv88]) regularity and invertibility results, replacing his hypothesis q ≧ p p − 1 by q ≧ n n − 1 . Finally if q = n n − 1 and if det D u ≧ 0 a.e., we show that det D u ln (2 + det D u ) is locally integrable.

On rank-one convex and polyconvex conformal energy functions with slow growth

We make some remarks about rank-one convex and polyconvex functions on the set of all real n × n matrices that vanish on the subset K n consisting of all conformal matrices and grow like a power function at infinity. We prove that every non-negative rank-one convex function that vanishes on K n and grows below a power of degree n /2 must vanish identically. In odd dimensions n ≧ 3, we prove that every non-negative polyconvex function that vanishes on K n must vanish identically if it grows below a power of degree n ; while in even dimensions, such polyconvex functions can exist that also grow like a power of half-dimension degree.

A DUALITY METHOD FOR MICROMAGNETICS

We present a new method for micromagnetics based on replacing the nonlocal total energy of magnetizations by a new local energy for divergence-free fields and then studying the dual Legendre functional of this new energy restricted on gradient fields. We establish a Fenchel-type duality principle relevant to the minimization for these problems. The dual functional may be written as a convex integral functional of gradients, and its minimization problem will be solved by standard minimization procedures in the calculus of variations. Special emphasis is placed on the analysis of existence/nonexistence, depending on the applied field and the physical domain. In particular, we describe a precise procedure to check the existence of magnetization of minimal energy for ellipsoid domains.

Stability of weakly almost conformal mappings

We prove a stability of weakly almost conformal mappings in W 1,p(Ω;Rn) for p not too far below the dimension n by studying the W 1,pquasiconvex hull of the set Cn of conformal matrices. The study is based on coercivity estimates from the nonlinear Hodge decompositions and reverse Holder inequalities from the Ekeland variational principle.

CONVEX INTEGRATION AND INFINITELY MANY WEAK SOLUTIONS TO THE PERONA-MALIK EQUATION IN ALL DIMENSIONS ∗

We prove that for all smooth nonconstant initial data the initial-Neumann boundary value problem for the Perona-Malik equation in image processing possesses infinitely many Lipschitz weak solutions on smooth bounded convex domains in all dimensions. Such existence results have not been known except for the one-dimensional problems. Our approach is motivated by reformulating the Perona-Malik equation as a nonhomogeneous partial differential inclusion with linear constraint and uncontrollable components of gradient. We establish a general existence result by a suitable Baires category method under a pivotal density hypothesis. We finally fulfill this density hypothesis by convex integration based on certain approximations from an explicit formula of lamination convex hull of some matrix set involved.

Sharp stability results for almost conformal maps in even dimensions

Let Ω, ⊂Rn and n ≥ 4 be even. We show that if a sequence {uj} in W1,n/2(Ω;Rn) is almost conformal in the sense that dist (∇uj,R+SO(n)) converges strongly to 0 in Ln/2 and if uj converges weakly to u in W1,n/2, then u is conformal and ∇uj → ∇u strongly in Llocq for all 1 < -q < n/2. It is known that this conclusion fails if n/2 is replaced by any smaller exponent p. We also prove the existence of a quasiconvex function f(A) that satisfies 0 ≤ f(A) ≤ C (1 + ¦A¦n/2) and vanishes exactly onR+ SO(n). The proof of these results involves the Iwaniec-Martin characterization of conformal maps, the weak continuity and biting convergence of Jacobians, and the weak-L1 estimates for Hodge decompositions.

On two-dimensional ferromagnetism

We present a new method for solving the minimization problem in ferromagnetism. Our method is based on replacing the non-local non-convex total energy of magnetization by a new local non-convex energy of divergence-free fields. Such a general method works in all dimensions. However, for the two-dimensional case, since the divergence-free fields are equivalent to the rotated gradients, this new energy can be written as an integral functional of gradients and hence the minimization problem can be solved by some recent non-convex minimization procedures in the calculus of variations. We focus on the two-dimensional case in this paper and leave the three-dimensional situation to future work. Special emphasis is placed on the analysis of the existence/non-existence depending on the applied field and the physical domain.

Remarks on W1,p-stability of the conformal set in higher dimensions

In this paper, we study the stability of maps in W1,p that are close to the conformal set K1 = R+ · SO(n) in an averaged sense as described in Definition 1.1. We prove that K1 is W1,p-compact for all p ≥ n but is not W1,p-stable for any 1 ≤ p < n/2 when n ≥ 3. We also prove a coercivity estimate for the integral functional ∫RndK1p(∇ϕ(x))dx on W1,p(Rn; Rn) for certain values of p lower than n using some new estimates for weak solutions of p-harmonic equations.

A Baire's category method for the dirichlet problem of quasiregular mappings

We adopt the idea of Baires category method as presented in a series of papers by Dacorogna and Marcellini to study the boundary value problem for quasiregular mappings in space. Our main result is to prove that for any ∈ > 0 and any piece-wise affine map φ ∈ W 1,n (Ω; R n ) with |Dφ(x)| n ≤ Ldet Dφ(x) for almost every x ∈ Ω there exists a map u ∈ W 1,n (Ω; R n ) such that |Du(x)| n = L det Du(x) a.e. x ∈ Ω, u|∂Ω = φ, ∥u-φ∥L n (Ω) < ∈. The theorems of Dacorogna and Marcellini do not directly apply to our result since the involved sets are unbounded. Our proof is elementary and does not require any notion of polyconvexity, quasiconvexity or rank-one convexity in the vectorial calculus of variations, as required in the papers by the quoted authors.

On Lipschitz solutions for some forward-backward parabolic equations. II: The case against Fourier

As a sequel to the paper Kim and Yan (Ann Inst H Poincaré Anal Non Linéaire. doi:10.1016/j.anihpc.2017.03.001, 2017), we study the existence and properties of Lipschitz solutions to the initial-boundary value problem of some forward–backward diffusion equations with diffusion fluxes violating Fourier’s inequality.